ALBERT

Description: The Chirikov diffusion in the sphere–ellipsoid binary asteroids through the spin–orbit resonance model is analytically and numerically studied. The primary and the secondary of the binary system are considered sphere and ellipsoid, respectively. The gravitational potential function is expanded up to the fourth-order. The geography of the first- and second-order resonances is derived and studied for different values of asphericity and dynamical flattening of the secondary asteroid and the semimajor axis and eccentricity of the mutual orbit. The Chirikov diffusion due to overlapping of the first- and second-order resonances is examined. For this end, the previously derived criterion for appearance of large-scale chaos is modified for those binary asteroids in which their secondary has irregular shape.

Description: Multifractal signal processing is now receiving increasing attentions. However, relevant researches mainly focus on differentiability and dimension of the fractal subset, such as fractal dimension and multifractal spectrum, or fractal and multifractal analysis in the fractional domain. In this paper, we propose a fractional domain singularity power spectrum (FDSPS) based on singularity power spectrum (SPS) and fractional order Fourier transform (FrFT). Due to the advantages of FrFT, including fractal fidelity, energy aggregation and signal separation in the fractional domain, the FDSPS can effectively represent the signal power distribution along with singularity exponent in the fractional domain. The discrete approximation algorithm of FDSPS is introduced. Simulation results indicate that fractional domain SPS can reveal effectively FrFT domain singular power distribution of fractal signal and provide application prospect of multifractal signal analysis processing.

Description: This paper complements some of the classic presentations of Euler–Poinsot motion in three ways. First, the state transition matrix for the attitude relative to an arbitrary inertial frame is presented. The state transition matrix elements are shown to be functions of common elliptic, circular, or hyperbolic functions of time depending on the motion state. Second, a complete set of global, time-independent integrals, here referred to as motion constants, are presented. The motion constants are functions involving the system’s momentum and attitude states, and they are valid for any set of attitude coordinates. Third, conditions for the ergodic or periodic nature of the motion are derived. These conditions correspond to one of the motion constants being nonisolating.

Description: Finite-time \(H_{\infty }\) control problem is a fascinating and hot issue in the field of control science. This paper presents a novel framework for finite-time \(H_{\infty }\) stabilization of semi-Markovian switching system. By employing Lyapunov–Krasovskii functional and matrix inequality techniques, together with properties of semi-Markovian process, sufficient conditions are proposed to guarantee finite-time boundedness, \(H_{\infty }\) finite-time boundedness and finite-time \(H_{\infty }\) state feedback stabilization for semi-Markovian switching system. At the same time, a state feedback controller is provided to ensure that the proposed closed-loop system is finite-time \(H_{\infty }\) stabilization. Finally, a numerical example and simulations are given to show the correctness and effectiveness of the proposed results.

Description: This paper presents a simple and quick control strategy for a class of first-order nonholonomic manipulator: planar n -link manipulator with a passive first joint (no gravity). The control target is driving its endpoint from any initial position to any target position. First, we reduce the planar n -link manipulator to a planar three-link one by maintaining the states (angles and angular velocities) of n -3 active links in the initial value all the time. That is, we only adjust the states of the remaining two active links in the whole control process, and these two adjusted active links are chosen to guarantee that the target position is in the reachable region of the planar n -link manipulator by using the enumeration method. Then, we divide the whole control process into two stages for the reduced planar three-link manipulator. In each stage, the manipulator becomes a planar two-link one by maintaining the states of one adjusted active link to be the constant value. The state constraint existing between the passive first link and the adjusted active link is obtained by using the integral characteristics of a planar two-link manipulator. Meantime, the geometric constraint between the position of the endpoint and angles of all joints is obtained based on the homogeneous coordinate transformation method. According to the above two kinds of constraint, the target angles of the two adjusted active links are calculated by employing the particle swarm optimization algorithm. When the two adjusted active links are controlled to their target angles in turn, the control target of the planar n -link manipulator is completed. Finally, simulation results demonstrate that the proposed control strategy is valid and rapid.

Description: A predator–prey system with two delays and stage structure for the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive equilibrium and two boundary equilibria of the system is discussed, respectively. By choosing the time delay as a bifurcation parameter, the existence of Hopf bifurcation with respect to both delays is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the positive equilibrium is feasible. By using the comparison theorem, sufficient conditions are obtained for the global stability of the positive equilibrium and one of the boundary equilibria of the proposed system. Numerical simulations are carried out to illustrate the main results.

Description: The pounding at the deck and abutment interface of a curved bridge, which occurs during earthquakes after closing the gap, directly leads to the deck unseating, in-plane deck rotation and torsional damage to the columns of the bridge. The study in hand assumes the deck of the curved bridge as a completely rigid body to analyse the multi-body oblique frictional impact at the deck–abutment interaction. Extending other authors’ work, a linear complementarity formulation is implemented for computing the unilateral contact to identify the effects of different impact states, such as stick and forward or backward slip, depending on the initial conditions during single or double impact. Stick and slip conditions during single impact are evaluated for different pre-impact rotations and ratios of transverse to normal pre-impact relative velocities. Also, the inward and outward slips during double impact, which represent the possibility of the deck being out-off lane, and the possibility of post-impact deck rotation are estimated for several curved geometries. The results show the curved bridge is very susceptible for in-plane deck rotation and out-off lane misalignment during the deck–abutment pounding.

Description: This paper aims to investigate the strongly nonlinear vibro-impact dynamic system with fractional derivative damping under Gaussian white noise excitation, and this system considers both nonlinear factors and non-smooth factors. With the help of non-smooth transformation, the original system is rewritten as a smooth system, which is an easier handled form. For the altered function, we obtain the approximate stationary solutions analytically by generalized stochastic averaging method. To prove the validity of the approximate analytical methods, an efficient scheme with high accuracy is adopt to simulate the fractional derivative, then the fourth-order Runge–Kutta approach is used to obtain the numerically response statistics. Meanwhile the analytical solutions are verified by the numerical simulation solutions. At last, we research the responses of this system. In this context, we can consider the influences to this system caused by the fractional order, the restitution coefficient and the noise intensity through changing the values of corresponding parameters. The observation and investigation state that fractional derivative term, impact conditions and stochastic excitation can influence the responses of the fractional vibro-impact dynamic system.

Description: This paper discusses iterative identification problems for a class of output nonlinear systems (i.e., Wiener nonlinear systems) with moving average noises from input–output measurement data, based on the Newton iterative method. The basic idea is to decompose a nonlinear system into two subsystems, to replace the unknown variables in the information vectors with their corresponding estimates at the previous iteration, and to present a Newton iterative identification method using the hierarchical identification principle. The numerical simulation results indicate that the proposed algorithms are effective.

Description: The theory of chaos is applied to the construction of substitution boxes used in encryption applications. The synthesis process of the proposed substitution boxes is presented, which is based on chaotic Baker’s map and TDERC chaotic sequences. The objectives of the new substitution box are to provide enhanced resistance against differential and linear cryptanalysis. The constructed substitution boxes uses Galois field elements and relies on discrete chaotic maps while keeping differential and linear approximation probabilities to desired levels.

Description: An efficient fractal method based on discrete wavelet transfer is proposed in this paper. Before encoding, images with one-level wavelet transform are first partitioned into four subbands, which classify the range blocks as four types. An accelerated fractal coding method called variance sorting (VS) scheme with high reconstructed quality is addressed for the low-frequency components. Hereafter, the low-frequency part rebuilt performs one-level wavelet decomposition once again. The first-level high-frequency subbands excluding the diagonal directions are predicted according to the next coarser frequency scale in the same directions. Simulation experimental results compared with other classification algorithms demonstrate that the proposed method can obtain high compression ratio and reduce the encoding time without significant loss in the reconstructed image quality.

Description: This study presents a rule-based cooperative differential evolution (RCDE) learning algorithm to optimize the parameters of a compensatory neural fuzzy network (CNFN) for solving control problems. CNFN uses the compensatory degree for fuzzy reasoning to dynamically adjust fuzzy operators that can make the fuzzy system more adaptive and effective. RCDE decomposes the fuzzy system into multiple rule-based subpopulations where each subpopulation represents a fuzzy rule set, and each individual within each subpopulation evolves by differential evolution (DE) separately. The proposed RCDE uses cooperative behavior among multiple subpopulations for combining their information and building the complete fuzzy system to accelerate the search and increase global search capacity. Finally, the experimental results show that the proposed RCDE method better approximates the global optimal solution and has a faster convergence rate than the other methods.

Description: Traditionally, elastomers are considered to be one of the best material candidates for the design of energy absorbing devices, due to their remarkable visco-elastic and extensibility material properties. This capability can be further enhanced by the design of appropriate Dielectric Elastomer Generators (DEGs). The present paper proceeds to another alternative direction, which consists of the design of appropriate elastomer structures with enhanced energy absorption capabilities, are attributed more to the appropriate non-linear design of the elastomer structure itself, than to the hyperelastic material properties of the elastomer members. For this reason, a simple chi-shaped in-plane elastic structure is considered, comprising one proof mass with four elastomer members attached to it, placed in a chi-shaped configuration, and being properly stretched. A systematic analysis is carried out, with respect to the variation of two basic structure design properties on the dynamic behavior: the orientation angle of the members and their initial pre-stress. The obtained results show that the variation of just these two basic parameters of the structure may lead to a quite rich and interesting non-linear dynamic behavior. More specifically, the traditional always-in-tension concept leads to a linear dynamic system response, even though the elastomer members are considered to have been made of a non-linear hyper-elastic material. Alternatively, a new partially-loose-member operating concept is analyzed. This concept suggests that some individual elastomer members be allowed to become loose for some period of the operating cycle, (but not all of them simultaneously), strongly enhancing the broadband energy absorbing capability of the structure itself.

Description: The problem of sampled-data control is investigated for Takagi–Sugeno (T–S) fuzzy systems with aperiodic sampling intervals based on an enhanced input-delay approach. Delay-dependent stability and stabilizability conditions for the closed-loop continuous nonuniformly sampled-data fuzzy systems are derived by constructing a novel discontinuous Lyapunov–Krasovskii (L–K) functional, which makes good use of not only the upper bound on the variable sampling interval, but also its sawtooth structure information about varying input delay often ignored in previous results. A bounding technique combined by reciprocally convex technics and linear convex combination is presented for acquiring the time derivative of the functional, wherein Jensen’s inequality and Wirtinger’s inequality are integratively employed. And a feasible solution of the obtained criterion formulated as parameterized linear matrix inequalities is ultimately conceived. A numerical example is given to show the effectiveness of the proposed method.

Description: In this paper, we present a new technique, developed using time-delay estimation (TDE) and supervising switching control (SSC), for the control and synchronization of chaos systems. The proposed technique consists of three units: a time-delay estimation unit that cancels system dynamics, a pole placement control unit that shapes error dynamics, and an SSC unit that is activated when the system dynamics are rapidly changing. We prove the stability of the closed-loop system using the Lyapunov analysis method. To verify the control and synchronization performance of the proposed technique (TDE-SSC), we compare it with TDC using numerical simulation. Our results indicate that the proposed scheme is an easily understood, numerically efficient, robust, and accurate solution for the control and synchronization of chaos systems.

Description: Anticipating synchronization is investigated in nonidentical chaotic systems unidirectionally coupled in a master-slave configuration without a time-delay feedback. We show that if the parameters of chaotic master and slave systems are mismatched in such a way that the mean frequency of a free slave system is greater than the mean frequency of a master system, then the phase synchronization regime can be achieved with the advanced phase of the slave system. In chaotic neural systems, this leads to the anticipating spike synchronization: unidirectionally coupled neurons synchronize in such a way that the slave neuron anticipates the chaotic spikes of the master neuron. We demonstrate our findings with coupled Rössler systems as well as with two different models of coupled neurons, namely, the Hindmarsh–Rose neurons and the adaptive exponential integrate-and-fire neurons.

Description: Impedance control provides a unified solution for the position and force control of robot manipulators. The dynamic behavior of a robotic system in response to environment is prescribed by an impedance model formed as Thevenin model. This model is certain and linear while the robot manipulator is highly nonlinear, coupled, and uncertain. Therefore, impedance control must overcome nonlinearity, coupling, and uncertainty to convert the robotic system to the impedance model. To overcome these problems, this paper presents a novel impedance control for electrically driven robots, which is free from the manipulator dynamics. The novelty of this paper is the use of voltage control strategy to develop the impedance control. Compared with the commonly used impedance control, which is based on the torque control strategy, it is computationally simpler, more efficient, and robust. The mathematical verification and simulation results show the effectiveness of the control method.

Description: In this paper, an analytical approximate solution is constructed for a rotor-AMB system that is subjected to primary resonance excitations at the presence of 1:1 internal resonance. We obtain an approximate solution applying the method of multiple scales, and then we conducted the system bifurcation analyses. The stability of the system is investigated applying Lyapunov’s first method. The effects of the different parameters on the system behavior are investigated. The analytical results showed that the rotor-AMB system exhibits a variety of nonlinear phenomena such as bifurcations, coexistence of multiple solutions, jump phenomenon, and sensitivity to initial conditions. Finally, the numerical simulations are performed to demonstrate and validate the accuracy of the approximate solutions. We found that all predictions from analytical solutions are in excellent agreement with the numerical integrations.

Description: In this paper, the analysis problem of adaptive exponential synchronization in p th moment is considered for stochastic complex networks with time varying multi-delayed coupling. By using the Lyapunov–Krasovskii functional, stochastic analysis theory, several sufficient conditions to ensure the mode adaptive exponential synchronization in p th moment for stochastic delayed complex networks are derived. To illustrate the effectiveness of the synchronization conditions derived in this paper, a numerical example is finally provided.

Description: In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.

Description: The complex nonlinear systems appear in many important fields of physics and engineering, which are very useful for cryptography and secure communication. This paper investigates adaptive generalized function projective synchronization (AGFPS) between two different dimensional chaotic complex systems with fully or partially unknown parameters via both reduced order and increased order. Based on the Lyapunov stability theorem and adaptive control technique, a general adaptive controller with corresponding parameter update rule is constructed to achieve AGFPS between two nonidentical chaotic complex systems with distinct orders, and identify the unknown parameters simultaneously. This scheme is then applied to obtain AGFPS between the hyperchaotic complex Lü system and the chaotic complex Lorenz system with fully unknown parameters, and between the uncertain chaotic complex Chen system and the uncertain hyperchaotic complex Lorenz system, respectively. Corresponding simulations results are performed to show the feasibility and effectiveness of the proposed synchronization method.

Description: Nowadays, safety of road vehicles is an important issue due to the increasing road vehicle accidents. Passive safety system of the passenger vehicle is to minimize the damage to the driver and passenger of a road vehicle during an accident. Whereas an active steering system is to improve the response of the vehicle to the driver inputs even in adverse situations and thus avoid accidents. This paper presents a neural network-based robust control system design for the active steering system. Primarily, double-pinion steering system used modeling of the active steering system. Then four control structures are used to control prescribed random trajectories of the active steering system. These control structures are as classical PID Controller, Model-Based Neural Network Controller, Neural Network Predictive Controller and Robust Neural Network Predictive Control System. The results of the simulation showed that the proposed neural network-based robust control system had superior performance in adapting to large random disturbances.

Description: A Volterra series analysis is used to analyse the dispersive behaviour in the frequency domain for the non-linear Schrödinger equation (NLS). It is shown that the solution of the initial value problem for the nonlinear Schrödinger equation admits a local multi-input Volterra series representation. Higher order spatial frequency responses of the nonlinear Schrödinger equation can therefore be defined in a similar manner as for lumped parameter non-linear systems. A systematic procedure is presented to calculate these higher order spatial frequency response functions analytically. The frequency domain behaviour of the equation, subject to Gaussian initial waves, is then investigated to reveal a variety of non-linear phenomena such as self-phase modulation (SPM), cross-phase modulation (CPM), and Raman effects modelled using the NLS.

Description: In this study, we investigate a class of chaotic synchronization and anti-synchronization with stochastic parameters. A controller is composed of a compensation controller and a fuzzy controller which is designed based on fractional stability theory. Three typical examples, including the synchronization between an integer-order Chen system and a fractional-order Lü system, the anti-synchronization of different 4D fractional-order hyperchaotic systems with non-identical orders, and the synchronization between a 3D integer-order chaotic system and a 4D fractional-order hyperchaos system, are presented to illustrate the effectiveness of the controller. The numerical simulation results and theoretical analysis both demonstrate the effectiveness of the proposed approach. Overall, this study presents new insights concerning the concepts of synchronization and anti-synchronization, synchronization and control, the relationship of fractional and integer order nonlinear systems.

Description: In this paper, we investigate the modified Kadomtsev–Petviashvili (mKP) equation for the nonlinear waves in fluid dynamics and plasma physics. By virtue of the rational transformation and auxiliary function, new bilinear form for the mKP equation is constructed, which is different from those in previous literatures. Based on the bilinear form, one- and two-soliton solutions are obtained with the Hirota method and symbolic computation. Propagation and interactions of shock and solitary waves are investigated analytically and graphically. Parametric conditions for the existence of the shock, elevation solitary, and depression solitary waves are given. From the two-soliton solutions, we find that the (i) parallel elastic interactions can exist between the (a) shock and solitary waves, and (b) two elevation/depression solitary waves; (ii) oblique elastic interactions can exist between the (a) shock and solitary waves, and (b) two solitary waves; (iii) oblique inelastic interactions can exist between the (a) two shock waves, (b) two elevation/depression solitary waves, and (c) shock and solitary waves.

Description: In reality, the external computers, in particular, external infected computers are connected to the Internet. Based on this reasonable assumption, a new computer virus propagation model is established. Different from all the previous models, this model regards the external computers as a single compartment to study. Through a qualitative analysis, it is found that (1) this model possesses a unique (viral) equilibrium, and (2) this equilibrium is globally asymptotically stable. Further study shows that, by taking effective measures, the number of infected computers can be made below an acceptable threshold.

Description: Stay cables used in cable-stayed bridge and cable-stayed arch bridge are prone to vibration due to their inherent susceptibility to external deflection. The present work is devoted to the mitigation of a stay cable from the point of view of its nonlinear dynamics. The Galerkin integral, multiple scales perturbation method, and numerical techniques are applied to analyze the primary and subharmonic resonances of the stay cable. The nonlinear dynamic response of the stay cable subjected to parametrical and forced excitations is investigated numerically. The effects of some key parameters of the stay cable, such as initial tension force, damping and inclination angle, and the excitation frequency and amplitude are discussed. The carbon fiber reinforced polymers (CFRP) cable is also studied to understand the effect of the material properties of cable. The results show that these parameters have a considerable effect on the dynamic behavior of the cable. In particular, unreasonable tension force and inclination angle of stay cable may cause excessive vibration. It is suggested that CFRP cable replaces steel cable, which can mitigate the vibration of a stay cable.

Description: In this paper, we propose to use a fractional order model to predict the process output in Smith predictor. The parameters of the model are determined by minimizing the error between its output and one of the processes using a genetic algorithm. After determining the model’s parameters, a fractional PID controller is proposed to improve the controlled system performances. The parameters of the controller are also determined in an optimal way by minimizing the position error taking into account the sensitivity and the complementary sensitivity conditions. Applications on a dead time and multiple lags processes have been performed, where the simulation results show that the proposed Smith predictor enhance the closed loop control system.

Description: Considering the geometrical nonlinearity of an embedded single-walled carbon nanotube, the analytical condition and the numerical results of chaotic vibration of the carbon nanotube are presented in this paper. Firstly, based on the Galerkin approximation method, a Duffing-type model is derived from the equation of motion that describes the oscillation of the embedded single-walled carbon nanotube clamped at both ends under a transverse load. And then, the Melnikov function of the Duffing-type model is derived. From the Melnikov function, the analytical condition of the chaos in the nanotube is obtained. Finally, a structure-preserving difference scheme for the original oscillating model is constructed based on the generalized multi-symplectic framework and the chaotic vibration of the nanotube is reproduced to verify the accuracy and the validity of the analytical condition. The analytical condition obtained in this paper gives some guidance on the property studying and the structure designing of some carbon nanotube devices.

Description: In this note some points for paper [Huabin Chen, Chuanxi Zhu, Peng Hu, Yong Zhang, Delayed-state-feedback exponential stabilization for uncertain Markovian jump systems with mode-dependent time-varying state delays, Nonlinear Dyn. ( 2012 ), doi: 10.1007/s11071-012-0324-3 ] are presented.

Description: Based on the CMC antivirus strategy proposed by Chen and Carley, a mixing propagation model of computer viruses and countermeasures is suggested. This model has two potential virus-free equilibria and two potential endemic equilibria. The existence and global stability of these equilibria are fully investigated. From the obtained results it can be deduced that the CMC strategy is efficacious in deracinating viruses.

Description: Multifractal theory has been widely used in kinds of field. In this paper, methods were proposed to study in the power-law auto-correlation and cross-correlation of power operating data based on multifractal detrended analysis (MF-DFA) and multifractal detrended cross-correlation analysis (MF-DXA). We find that both the price and the load time series in California power market and PJM power market exhibit long-term correlation. And the cross-correlation behaviors of the two series in each power market and between the two markets are also analyzed by the method MF-DXA after testing the existence of the cross-correlation of the above power operating data. However, there are some differences in the cross-correlation behaviors between the two markets. It shows that the cross-correlation of the price and the load is significant in every time periods in California 1999 power market, but in the year of 2000 in the same region market, the cross-correlation is insignificant in most time periods. Meanwhile, we conclude that cross-correlation is weaker in the California market than in the PJM market by studying the two consecutive years of the California 1999–2000 and PJM 2001–2002 power markets. We also discuss how the time intervals affect the cross-correlation exponents of the power operating data based on time-delay MF-DXA. An interesting finding is that the biggest cross-correlation exponent of the two series appeared in about 12 days time delay for the PJM 2001 power market and strongest cross-correlation in the California 1999 power market is found in lots of cyclical time intervals.

Description: In this work, the analysis the response to vibrations of a small building equipped with an electromechanical vibration absorber is investigated. In the first part, the case of harmonic excitation with constant or time dependent frequencies is considered, and in the second part, the case of random earthquakes, respectively. The cross correlation functions and the mean square displacements are calculated for the structure when it is equipped with an electromechanical vibration absorber. The stochastic process which characterizes the earth movements is coupled to the cnoidal method, which delivers the analytical solutions of the nonlinear problem. The interaction between the structure and the energy source is analyzed via the Sommerfeld effect inside the resonance region. The resonance capture and the vibration reduction are displayed by the time history responses of the displacement and angular velocities above the resonance.

Description: We explained the theory about bicomplex numbers, discussed the precondition of that addition and multiplication are closed in bicomplex number mapping of constructing generalized Mandelbrot–Julia sets (abbreviated to M–J sets), and listed out the definition and constructing arithmetic of the generalized Mandelbrot–Julia sets in bicomplex numbers system. And we studied the connectedness of the generalized M–J sets, the feature of the generalized Tetrabrot, and the relationship between the generalized M sets and its corresponding generalized J sets for bicomplex numbers in theory. Using the generalized M–J sets for bicomplex numbers constructed on computer, the author not only studied the relationship between the generalized Tetrabrot sets and its corresponding generalized J sets, but also studied their fractal feature, finding that: (1) the bigger the value of the escape time is, the more similar the 3-D generalized J sets and its corresponding 2-D J sets are; (2) the generalized Tetrabrot set contains a great deal information of constructing its corresponding 3-D generalized J sets; (3) both the generalized Tetrabrot sets and its corresponding cross section make a feature of axis symmetry; and (4) the bigger the value of the escape time is, the more similar the cross section and the generalized Tetrabrot sets are.

Description: The present work is devoted to giving new insights into the Liu chaotic system. The local dynamical entities, such as the number of equilibria, the stability of hyperbolic equilibria, and the stability of the nonhyperbolic equilibrium obtained by using the center manifold theorem, the pitchfork bifurcation, the degenerate pitchfork bifurcation, and Hopf bifurcations, are all analyzed when the parameters are varied in the space of parameters. All the closed orbits of the system are also proven rigorously to be nonplanar but only to be curves in space. Moreover, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated.

Description: This paper focuses the issue of state estimation for a class of switched discrete-time stochastic bidirectional associative memory (BAM) neural networks with time varying delay. The main purpose of this paper is to estimate the neuron states through available output measurements such that the dynamics of the error state system to be robustly exponentially stable. By employing average dwell time approach together with piecewise Lyapunov functional technique, a set of sufficient conditions is derived with respect to all admissible uncertainties, to guarantee the existence of the desired state estimator for the uncertain switched discrete-time BAM delayed neural networks. Specifically, we derive sufficient conditions to achieve robust state estimation with the characterization of complex effects of time delays, parameter uncertainties, and stochastic perturbations. In particular, the parameter uncertainties are assumed to be time varying and unknown, but norm bounded. It should be mentioned that our estimation results are delay dependent, which depend on not only the upper bounds of time delay, but also their lower bounds. More precisely, the desired estimator matrix gain is obtained in terms of the solution of the derived LMIs. Finally, numerical examples with a simulation result are given to illustrate the effectiveness and applicability of the obtained results.

Description: In this paper, an adaptive control strategy for tracking of a direct-current (DC) motor system with a dead-zone is developed. The main contribution of the developed scheme is that we successfully integrate an asymmetric barrier Lyapunov function approach to relax the requirements on the initial conditions. The unknown functions in the DC system are approximated by using the radial basis function neural networks (RBFNN). It is shown that the DC motor can follow a selected trajectory and all the signals are guaranteed to be bounded. Simulation results are provided to confirm the effectiveness of the proposed control.

Description: Spatiotemporal periodic patterns, including phase-locked oscillations, mirror-reflecting waves, standing waves, in-phase or anti-phase oscillations are investigated in a ring of bidirectionally coupled oscillators with neutral delay feedback. It is confirmed that neutral feedback makes Hopf bifurcation occur in a larger domain of parameters. We calculate the normal forms near Hopf bifurcation, D N equivariant Hopf bifurcation and double-Hopf bifurcation in this neutral equation by using the method of multiple scales. Theoretically, the appearance of the in-phase, anti-phase and phase-locked oscillations that we observed in the simulation about a ring of delay coupled Hindmarsh–Rose neurons with neutral feedback is explained.

Description: This paper deals with the adaptive terminal sliding mode control for nonlinear differential inclusion systems subjected to disturbance. The upper bound of the disturbance is unknown. First, the fast terminal sliding mode surface is established and sufficient condition for fast convergence is given. Then the adaptive sliding mode controller is designed to make the state of system arrive at the sliding mode in finite time. A numerical example is provided to show the effectiveness of the proposed method.

Description: Leapfrogging solitary waves are characterized in two capacitively coupled transmission lines that are periodically loaded with Schottky varactors, called coupled nonlinear transmission lines (NLTLs). The coupling implies that a nonlinear solitary wave moving on one of the lines is bounded with the wave moving on the other line, which results in the periodic amplitude/phase oscillation called leapfrogging. In this study, we clarify how the leapfrogging frequency depends on the physical parameters of coupled NLTLs using a numerical model validated through measuring test lines and demonstrate the relaxation of leapfrogging. In addition, coupled Korteweg-de Vries equations are derived by applying the reductive perturbation method to the transmission equations of coupled NLTLs. Using perturbation theory based on the inverse scattering transform, a closed-form expression of leapfrogging frequency is obtained and the parameter values that simulate the properties well are examined. Engineering applications based on leapfrogging are finally discussed.

Description: A recursive terminal sliding mode controller (RTSMC) based on sliding mode disturbance observer (SMDOB) is proposed for the longitudinal dynamics of a generic hypersonic flight vehicles (HFVs) in the presence of parametric uncertainties, measurement noises and external disturbances. First, a sliding mode tracking controller is presented by introducing recursive terminal sliding mode manifolds, in which each manifold will reach zero subsequently in finite time as well as the usual singularity problem will not occur. The RTSMC embraces advantages of both nonsingular terminal sliding mode control and high-order sliding mode control. Next, for the sake of enhancing the robustness of controller for uncertainties, a SMDOB is proposed to estimate and compensate the disturbances. Then, a composite controller that is composed of RTSMC and SMDOB is designed, and its stability is analyzed utilizing Lyapunov function method. Finally, numerical simulation is conducted for cruise flight condition of HFV. Simulation results show the expected control performance.

Description: This paper addresses the problem of stochastic sampled-data stabilization for neural-network-based control systems (NNBCSs) with an optimal guaranteed cost. In order to stabilize the closed-loop system, continuous-time nonlinear plant and three-layer fully connected feed-forward neural networks based on stochastic sampling are connected to the closed loop. By introducing new Lyapunov–Krasovskii functional with triple integral terms and by using second-order reciprocal convex technique, new stability and stabilization criteria for NNBCSs are derived in terms of linear matrix inequalities (LMIs). The desired stochastic sampled-data controllers can be calculated by solving these LMIs. Finally, physical example is given to verify the effectiveness and usefulness of the obtained results.

Description: The perturbation theory approach via the Smoluchowski equation to the nonlinear dielectric relaxation of noninteracting permanent electric dipoles (Coffey and Paranjape, Proc R Ir Acad A 78:17, 1978 ) and the analogous Brownian magnetic relaxation of ferrofluids where Néel relaxation is ignored is revisited for the particular case of a strong dc bias field superimposed on a strong ac field. Unlike weak ac and strong bias dc fields, a frequency-dependent dc term now appears in the response as well as additional nonlinear terms at the fundamental and second harmonic frequencies. These may be experimentally observable particularly in the ferrofluid application. The corresponding results for the dc term for anomalous relaxation based on the fractional Smoluchowski equation are also given.

Description: In this paper, a fractional order dynamical system is constructed that exhibits chaotic and reverse chaotic attractors by changing the sign of the one parameter which involves in the existence of the phase reversal function. A new method of fast projective synchronization of fractional order dynamical systems is introduced. An affine cipher is proposed for secure communication based on the solutions of the synchronized fractional order chaotic systems with the support of the sender’s and receiver’s date of birth. The efficiency and security of an affine cipher are analyzed. Numerical simulations are demonstrated to show the feasibility of the presented theory.

Description: A fractional wave equation replaces the second time derivative by a Caputo derivative of order between one and two. In this paper, we show that the fractional wave equation governs a stochastic model for wave propagation, with deterministic time replaced by the inverse of a stable subordinator whose index is one-half the order of the fractional time derivative.

Description: Recently the discrete fractional calculus (DFC) started to gain much importance due to its applications to the mathematical modeling of real world phenomena with memory effect. In this paper, the delayed logistic equation is discretized by utilizing the DFC approach and the related discrete chaos is reported. The Lyapunov exponent together with the discrete attractors and the bifurcation diagrams are given.

Description: Fractional tumor development is considered in the framework of one-dimensional continuous time random walks (CTRW) in the presence of chemotherapy. The chemotherapy influence on the CTRW is studied by observations of both stationary solutions due to proliferation and fractional evolution in time.

Description: Chaos control and synchronization of second-order nonautonomous fractional complex chaotic systems are discussed in this paper. A novel fractional nonsingular terminal sliding surface which is suitable for second-order fractional systems is proposed. It is proved that once the state trajectories of the system reach to the proposed sliding surface, they will be converged to the origin within a given finite time. After establishing the desired terminal sliding surface, a novel robust single sliding mode control law is introduced to force the system trajectories to reach the terminal sliding surface over a finite time. The stability and robustness of the proposed method are proved using the latest version of the fractional Lyapunov stability theorem. The proposed method is implemented for synchronization of two uncertain different fractional chaotic systems to confirm the theoretical results. Moreover, the fractional-order gyro system is stabilized using the proposed fractional sliding mode control scheme. It is worth noticing that the proposed fractional sliding mode approach is still a general control method and can be applied for control of second- order uncertain nonautonomous/autonomous fractional systems.

Description: Multimedia streaming of three-dimensional (3D) stereoscopic videos over last-generation networks subject to bandwidth limitations is an open problem. The development and spread of communication networks and devices that accept 3D videos is not supported by proper scheduling strategies. Namely the high variability of streams should be considered to reduce effects of network delays, packet losses, shortage of bandwidth resources, and shared use by multiple clients. Then, it is important to improve the characterization of 3D videos for more effective streaming. To this aim, this paper proposes a fractional exponential reduction moments approach based on the statistics of the so-called fractional moments. Each random sequence of frames in 3D videos can be analyzed and reduced to a finite set of parameters, that allow fitting to the sequence by exponential functions and then a characterization and classification of the video by a sort of fingerprint. The method does not depend on the format and the encoding technique of the video. Finally, the approach will allow comparing real streams and numerical data output from fractional dynamical models by means of the reduced parameters. Statistical proximity between time series and a fractional model or between different models simplifies formalization and classification of fractional models.

Description: This paper studies the consensus problem of second-order multi-agent systems under both fixed and switching directed topologies in a more general framework. Specifically, each agent incorporates an intrinsic nonlinear dynamics and the communication topology switches in a state-controlled manner. Novel distributed consensus protocols are proposed based only on relative position information and local velocity information. Through constructing appropriate Lyapunov functions, it is shown that the proposed protocols are feasible for achieving consensus, even under switching topology jointly containing a spanning tree. Moreover, some easily manageable algebraic criteria are presented to unravel the underlying mechanism for reaching consensus. In the special case of multi-agent systems with double-integrator dynamics, we further derive a necessary and sufficient condition for the consensus problem. Finally, two numerical examples are provided to validate the effectiveness of theoretical results.

Description: Both the friction caused by the preload in locked joints and the impact caused by clearance in unlocked joints cause energy dissipation in jointed deployable structures. The energy dissipation of locked joints is studied by analyzing the force on the infinitesimal body of the joint. The jointed beam with an unlocked joint is simplified into an impact mass-spring model with clearance, which considers the coefficient of restitution of impact. The energy dissipations of the joint caused by friction and clearance are transformed into damping ratios by Taylor expansion. Then, the effects of pressure, clearance and the dynamic parameters on the damping of joints are analyzed by utilizing the damping ratio formulation. The damping ratio increases with the preload and the clearance. To validate the damping ratio formulation of joints, experiments on a single jointed beam with preload and double jointed beams with clearance are conducted. Comparison between the experimental results and the model simulation results shows that the friction and impact damping models are accurate for the dynamic calculation of deployable structures. Furthermore, the damping ratio formulations can be directly introduced into the design and dynamic analysis of deployable structures.

Description: In this paper, an adaptive fuzzy backstepping output feedback control approach is developed for a class of SISO nonaffine nonlinear systems with unknown dead-zone input and immeasurable states. Within this scheme, the original nonaffine system is firstly transformed into an affine-like form by using Taylor series expansion, the unknown dead-zone input is treated as a combination of a linear and a bounded disturbance-like term, and a state observer is introduced to estimate the system states. The fuzzy logic systems are utilized to directly approximate the desire virtual control law, and a novel adaptive fuzzy output feedback controller is designed via backstepping. A main advantage of the proposed controller is that only one adaptive parameter needs to be updated online in backstepping design process. Theoretically, it is proved that the presented adaptive fuzzy controller can guarantee that the tracking error converges to a small neighborhood of the origin and all signals in closed-loop are bounded. Finally, the simulation results validate the effectiveness of the proposed scheme.

Description: In driven oscillators, examples are shown that every boundary point of one basin is on the boundary of the two remaining basins and all three boundaries of these basins coincide. When there are more than five basins of attraction, is it possible that every boundary point of one basin is on the boundary of other basins? Is it possible that all basin boundaries coincide? This paper describes some numerical experiments giving evidence of seven Wada basin boundaries and all seven basin boundaries coincide for a driven shallow arch oscillator. The results are verified by the basin cell theory. It suggests that small noises can lead to final state sensitivity and uncertain dynamics in the driven oscillator.

Description: This paper presents a novel fault detection and estimation (FDE) scheme for a class of Lipschitz nonlinear systems subjected to modeling and measurement uncertainties. Many works in this field assume that the states of the system are measurable and the fault function acts as an additive term. The proposed FDE strategy deals with the uncertain nonlinear systems subjected to multiplicative faults, and it does not rely on availability of the full state. Multiplicative fault corresponds to the parameter or structure changes in the system or in the process model. Also, actuator gain fault is another important fault type which can be modeled as a multiplicative fault. The effects of this kind of fault are combined with the inputs and outputs of the system in a multiplicative form which, in turn, make the detection and estimation of the fault complex. The proposed scheme is based on an adaptive diagnostic observer that not only can estimate the states of the system and generate the residual signal simultaneously, but also is able to estimate the characteristic and magnitude of an unknown fault. In the fault detection step, the threshold is derived analytically to ensure the robustness of the proposed detection scheme against uncertainties. Also in the fault estimation step, a robust adaptive law based on the switching \(\sigma \) -modification is developed to estimate the detected fault accurately. Design steps of the proposed estimation scheme are introduced in the form of LMI problem. This formulation provides an effective way to calculate the design parameters. The proposed FDE scheme guarantees that all signals are uniformly ultimately bounded. Simulation results of a single-link, flexible joint, robotic arm show the effectiveness and robustness of the proposed FDE strategy.

Description: The paper tries to give a new scheme for image encryption, which innovatively introduced the idea of Langton’s Ant cellular automaton to scramble the image. We virtualize a chessboard with the size of the image, and let the ant crawls on it by following the rules which Langton gives and steps generated by intertwining logistic map, then to determine the position of the plain image’s pixels in the scrambling image according to the position which the ant stay each time. Lastly, the PWLCM chaos map has been used to diffuse the image. Experimental results and security analysis show that our scheme is secure and can be used in image encryption and transmission.

Description: Friction forces can be advantageously used as a source of passive damping in various mechanical systems. This paper deals with an experimental modelling and numerical simulation of blades interaction by means of a friction element placed in the shroud between the blade heads. The radial force, which represents the centrifugal force acting on the friction element, determines the values of contact forces between the element and blades. The experimental set-up for a couple of non-rotating blades is described in the paper, and the measured dynamic response of two blades is documented. The same situation is modelled by means of a basic and a more complex dynamical model of two blades with a friction element. The effect of friction is studied for the case of harmonic excitation by suitable frequency and subsequent free vibration attenuation. Both mathematical models are based on the finite element method combined with lumped rigid bodies. The interaction of the friction element and blades is described by normal contact and tangential friction forces derived for particular geometrical parameters of the studied mechanical system. The performed comparison of experimental and numerical results shows the satisfactory agreement and the modelling methodology could be used for possible parameter optimization.

Description: In this paper, we are concerned with a singularly perturbed higher-order KdV equation, which is considered as a paradigm in nonlinear science and has many applications in weakly nonlinear and weakly dispersive physical systems. Based on the relation between solitary wave solution and homoclinic orbits of the associated ordinary differential equations, the persistence of the solitary wave solution for the singularly perturbed KdV equation is investigated by using the geometric singular perturbation theory and dynamical systems approach when the perturbation parameter is suitably small.

Description: This work studies the optical solitons with time-modulated nonlinearities and spatiotemporal dispersion. Four types of nonlinearity—Kerr law nonlinearity, parabolic law nonlinearity, power law nonlinearity, and dual-power law nonlinearity—are considered. The non-auto-Bäcklund transformation is obtained. Analytical soliton solutions are constructed. The presented results can be applied in the area of dispersion- and nonlinearity-managed solitons.

Description: In real world, the intrinsic dynamics of velocity for multi-agent systems is usually nonlinear dynamic incorporated time delay. Moreover, each agent can only obtain the information of its neighbors at some disconnected time intervals. To overcome the above essential issue, this paper aims at investigating the formation control for leader–follower second-order multi-agent systems with inherent delayed nonlinear dynamics and intermittent communications. Under the common assumption that the velocity of the active leader cannot be obtained in real time, a distributed observer is designed for each second-order follower agent. Based on the observers, a novel distributed formation control protocol is developed, by using the intermittent neighbors-based information, to guarantee that the follower agents can simultaneously track the leader and form a pre-designed formation under fixed and switching topologies. Furthermore, the similar results are obtained for the multi-agent system in the absence of time delay in corollaries. Simulation examples are provided to show the validity of the theoretical results.

Description: In most practical applications, studying the asymptotic stability of equilibrium points of a system is of utmost importance. Furthermore, in many cases, the response is restricted to only a sector of the state space. For example, positive systems that are common in chemical processes have nonnegative state variables. For such systems, stability analysis of the system using Lyapunov stability is not advised, since this stability is defined for all the points within a neighborhood of the equilibrium point. In this paper, a new notion of stability, called corner stability, is defined as more suitable for studying asymptotic stability of equilibrium points in such systems. In order to derive sufficient conditions of corner stability, a theorem is stated and proven in this paper, and corner stability of three case studies is analyzed and verified.

Description: The coupling effects of fluid-conveying pipe bundle are important issues due to the interactions between the pipes by connecting structures and/or flow environment. However, the researches on the dynamical synchronization of coupled pipes are still insufficient. This paper focuses on the synchronization phenomenon of two equivalent fluid-conveying pipes coupled by a nonlinear spring. The system equations were discretized by applying Galerkin expansion method, and numerical solutions were obtained for two-pipe system with simply supported conditions. The results indicated that the two-pipe system existed several different synchronization patterns including perfect, imperfect, lag, strange synchronization and failed pattern. The results also revealed that the system dynamic behavior is sensitive to flow or structure parameters of the individual pipe. By means of selecting the coupling spring parameters or designing proper structure configurations, the two pipes can be synchronized to similar dynamical states and vibration behaviors, which might be much simpler than that of the individual pipe. Thus, this finding provides theoretically a possible, even effective way to control the two-pipe vibrations and to achieve target dynamics simultaneously.

Description: The problem of designing a sliding-mode controller for a class of fractional-order uncertain linear perturbed systems with Caputo derivative is addressed in this paper. Sufficient conditions for the existence of sliding surfaces guaranteeing the asymptotic stability of the reduced-order sliding-mode dynamics are obtained in terms of linear matrix inequalities, based on which and stability theory of fractional-order nonlinear systems; the corresponding reaching motion controller is proposed, and the reaching time is also obtained. Moreover, the upper bounds of the nonlinear uncertainties are not required to be known in advance, which can be tuned by the designed adaptive law. Meanwhile, some problems for the sliding-mode controller for fractional-order systems in existing literatures are pointed out. A numerical example is presented to demonstrate the validity and feasibility of the obtained results.

Description: In this paper, the complex dynamics of a predator–prey model with group defense and impulsive state feedback control is studied both theoretically and numerically. We obtain the sufficient conditions for the existence and stability of semi-trivial solution and positive periodic solutions by using the Poincaré map and the analogue of the Poincaré criterion. Moreover, the theoretical analysis reveals that the positive periodic solution bifurcates from the semi-trivial solution through a fold bifurcation. Numerical simulations are also illustrated, which agree well with our theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-2, 3, 4, 6, 8 solutions and chaotic solution, cascade of period-doubling bifurcation and inverse period-doubling bifurcation. Moreover, the superiority of impulsive state feedback control strategy is also exhibited over the impulsive fixed-time control.

Description: We investigate the possibility of using a time-delay feedback control to suppress the vortex-induced vibrations of an elastically mounted circular cylinder. An appropriate reduced-order wake-oscillator model is employed to determine the cylinder’s displacement and lift fluctuating coefficient in a coupled manner. A parametric study is performed to investigate the effects of the time-delay feedback control on the cross-flow oscillations of the circular cylinder. To study the effects of this controller on the coupled frequency and damping of the aeroelastic system, a linear analysis is performed. It is demonstrated that the presence of time-delay feedback control can result in an increase or decrease in the coupled damping of the aeroelastic system, varying from negative to positive values periodically. Then, the effects of this time-delay feedback controller on the nonlinear responses of the circular cylinder are determined. The results show that a good choice of time-delayed controller parameters can be efficiently implemented to significantly decrease or suppress the vortex-induced vibrations amplitudes in the lock-in or synchronization region and hence greatly reduce the potential oscillation hazard in engineering applications.

Description: This paper deals with the problems of tracking and \(H_\infty \) control for output-constrained and state-constrained nonlinear switched systems in strict feedback form. Under a mild condition on the initial output tracking error and the simultaneous domination assumption, a novel approach is proposed to design controller such that the output tracking error converges to zero asymptotically and is always within a pre-specified limit range. Smooth or \(p\) -times differentiable unbounded functions are introduced and incorporated in output tracking error transformations to complete the control design. Furthermore, the developed method is extended to the state-constrained \(H_\infty \) control problem for a class of nonlinear switched systems with disturbance input. Finally, simulation examples are provided to demonstrate the applicability of the presented results.

Description: In this paper, we consider a heteroclinic cycle consisting of two hyperbolic equilibria with different indices, one robust heteroclinic connection and a heteroclinic connection within a codimension-2 intersection of the corresponding manifolds of the equilibria, which is called the heterodimensional cycle. By setting up local moving frame systems in some tubular neighborhood of unperturbed heterodimensional cycles, we construct a Poincaré return map under the nongeneric conditions two orbit flips and further obtain the bifurcation equations. By the bifurcation equations, different bifurcation phenomena are discussed under small perturbations. New features produced by the degeneracy that heterodimensional cycles and periodic orbits coexist on the same bifurcation surface are shown. Some known results are extended. An example is given to show the existence of the system which has a heterodimensional with two orbit flips.

Description: Inherent nonlinearities of piezoelectric materials are pronounced in various engineering applications such as sensing, actuation, combined applications for vibration control, and energy harvesting from dynamical systems. The existing literature focusing on the dynamics of electroelastic structures made of piezoelectric materials has explored such nonlinearities separately for the problems of mechanical and electrical excitation. Similar manifestations of softening nonlinearities have been attributed to purely elastic nonlinear terms, coupling nonlinearities, hysteresis alone, or a combination of these effects by various authors. In order to develop a unified nonlinear nonconservative framework with two-way coupling, the present work investigates the nonlinear dynamic behavior of a bimorph piezoelectric cantilever under low to moderately high mechanical and electrical excitation levels in energy harvesting, sensing, and actuation. The highest voltage levels, for near resonance investigation, are well below the coercive field. A distributed parameter electroelastic model is developed by accounting for softening and dissipative nonlinearities to analyze the primary resonance of a soft (e.g., PZT-5A, PZT-5H) piezoelectric cantilever for the fundamental bending mode using the method of harmonic balance. Excellent agreement between the model and experimental investigation is found, providing evidence that quadratic stiffness, damping, and electromechanical coupling effects accurately model predominantly observed nonlinear effects in geometrically linear vibration of piezoelectric cantilever beams. The backbone curves of both energy harvesting and actuation frequency responses for a PZT-5A cantilever are experimentally found to be dominantly of first order and specifically governed by ferroelastic quadratic softening for a broad range of mechanical and electrical excitation levels. Cubic and higher-order nonlinearities become effective only near the physical limits of the brittle and stiff (geometrically linear) bimorph cantilever when excited near resonance.

Description: This paper addresses the problem of stochastic resonance (SR) in a time-delayed bistable system subjected to Gaussian white noise. Differing from the conventional studies, the statistical complexity measure and the normalized Shannon entropy have been defined and employed to quantify SR phenomenon of the time-delayed bistable system. According to the definitions of statistical complexity measure and normalized Shannon entropy, a minimum of entropy illustrates that the motion of system reaches some degree of order and a maximum of statistical complexity measure implies that the system’s intricate pattern tends to complexity. It has been found that for an optimal level of noise intensity, the statistical complexity measure displays a maximum and the normalized Shannon entropy exhibits a minimum, which demonstrates not only the occurrence of SR but also the severity of dynamical complexity. And the effects of different parameters on SR are also studied by means of the statistical complexity measure. To test the validity, the signal-to-noise ratio (SNR) is also calculated and a good agreement has been found between the statistical complexity measure and the SNR, which indicates that the statistical complexity measure is an effective method for quantify SR of the time-delayed bistable system.

Description: A new hyper-chaotic system is presented in this paper by adding a smooth flux-controlled memristor and a cross-product item into a three-dimensional autonomous chaotic system. It is exciting that this new memristive system can show a four-wing hyper-chaotic attractor with a line equilibrium. The dynamical behaviors of the proposed system are analyzed by Lyapunov exponents, bifurcation diagram and Poincaré maps. Then, by using the topological horseshoe theory and computer-assisted proof, the existence of hyperchaos in the system is verified theoretically. Finally, an electronic circuit is designed to implement the hyper-chaotic memristive system.

Description: This paper considers the cooperative dynamic positioning of multiple marine surface vessels in the presence of dynamical uncertainty and time-varying ocean disturbances. The objective is to enable a group of vessels to automatically position themselves in a desired time-varying formation and track a reference position. By employing a dynamic surface control technique, distributed adaptive controllers are derived on the basis of the information of neighboring vessels. Neural network with iterative updating laws is used to accurately identify the dynamical uncertainty and time-varying ocean disturbances. Two types of adaptive laws are proposed and validated: (1) direct iterative updating laws based on the velocity tracking errors; (2) composite iterative updating laws based on the tracking errors and the prediction errors. For both cases, Lyapunov–Krasovskii functionals are employed to analyze the stability of the closed-loop network, and uniform ultimate boundedness of error signals are established. The key features of the proposed controllers are as follows. First, the information exchanges are reduced by employing a distributed control strategy. Second, by using the iterative updating laws, the mixed uncertainty including the internal model uncertainty and external time-varying ocean disturbances can be compensated. Besides, the proposed controllers are easier to implement in digital processors with the derivative-free updating laws. Third, the prediction errors and tracking errors are combined to construct the composite iterative neural control laws, which are able to achieve faster adaptation and improved performance. Comparison studies are given to show the effectiveness of the proposed methods.

Description: This paper presents an adaptive control approach to bridge the gap between unknown time-varying actuator nonlinearities and identical control directions in the area of robotic systems. Technically, a novel Nussbaum gain and its properties are first proposed to pave the way for the formulation of a more advanced tool. Subsequently, a newly developed theorem is summarized and integrated into the stability analysis such that all control units automatically and simultaneously estimate the unknown actuator nonlinearities and directions. Finally, it is rigorously proved that robotic systems asymptotically converge to the desired trajectories.

Description: This paper brings a new mathematical model of the third-order autonomous deterministic dynamical system with associated chaotic motion. Its unique property lies in the existence of circular equilibrium which was not, by referring to the best knowledge of the authors, so far reported. Both mathematical analysis and circuitry implementation of the corresponding differential equations are presented. It is shown that discovered system provides a structurally stable strange attractor which fulfills fractal dimensionality and geometrical density and is bounded into a finite state space volume.

Description: In this paper, the problem of synchronization via sampled-data control is considered for stochastic delayed neural networks with Lévy noise and Markovian switching. The purpose of the problem addressed is to derive a sufficient condition and a sampled-data control law such that the dynamics of the error system is stable in mean square, and thus the synchronization can be achieved for the master system and the slave system. By generalized Itô’s formula and the construction of Lyapunov functional, an LMI-based sufficient condition is established to ensure the synchronization of the two systems. The control law is determined simultaneously, which depends on the switching mode, time delay, and the upper bound of sampling intervals. A numerical example is provided to verify the usefulness of the proposed criterion.

Description: The folded solitary waves are the multivalued localized excitations. Traditionally, to describe these complicated structures, one needs to find the “universal” formula through the multilinear variable separation approach. In this paper, the Boiti–Leon–Pempinelli system is first reduced to the unsteady convection–diffusion equation according to the homogeneous balance principle. And then, the Lie group theory is applied to this equation and the similarity reduction equations, and the similarity solutions are obtained. Thus, the abundant folded solitary waves of the Boiti–Leon–Pempinelli system are investigated. These results illustrate that the multivalued structures can be derived through another skill.

Description: In this paper, complex symmetric and asymmetric period-1 motions in a periodically forced, time-delayed, hardening Duffing oscillator are predicted through an implicit discrete map. The implicit discrete map is obtained by the discretization of a second-order differential equation of the time-delayed Duffing oscillator. Using the theory of nonlinear discrete systems, period-1 motions in the time-delayed Duffing oscillator are determined analytically from the fixed points of the mapping structures of discrete nodes under the computational accuracy of \(10^{-10}\) , and the corresponding stability and bifurcation of period-1 motions are determined by eigenvalue analysis. From the discrete nodes of period-1 motions, the finite discrete Fourier series is employed to determine the frequency–amplitude characteristics of period-1 motions in the time-delayed, Duffing oscillator. The stable and unstable, symmetric and asymmetric period-1 motions are presented, and the corresponding stability and bifurcations are illustrated clearly. Presented are the quantity levels of harmonic amplitudes which indicate the accuracy of the semi-analytical solutions of period-1 motions in such a time-delayed oscillator. From the analytical prediction, numerical simulations of complex period-1 motions in the time-delayed, hardening Duffing oscillator are completed. The amplitude spectrums of period-1 motions in the time-delayed Duffing oscillator are given, and the approximate, analytical expressions of period-1 motions can be obtained. For small excitation frequency, discrete time step size should be reduced to keep the same computational accuracy of discrete nodes. This method can be applied to the time-varying time-delayed nonlinear dynamical systems and other nonlinear dynamical systems.

Description: Based on the theory of Absolute Nodal Coordinate Formulation, this paper proposes a new dynamic computation method to solve the flexible multibody system with uncertain material properties (Young’s modulus and Poisson’s ratio) that may be induced by the material asymmetric distribution. Rather than traditionally considering an uncertain factor as one single variable in the whole system, the material properties vary continuously in the space domain so that they are described by the random field, which is then discretized to countable random variables using the expansion optimization linear estimation method. The uncertain response of the system is approximated by the Polynomial Chaos expansion, numerically implemented by a collocation method. We propose and prove an important theory that the collocation method provides the same results as the Gaussian quadrature formula if the roots of the corresponding orthogonal polynomials are used as the collocation points. As a result, the proposed method is a nonintrusive technique that does not modify the original solver but only adds a preprocess and a postprocess. The uncertain displacement of system is finally illustrated by ellipse and ellipsoid, which visually shows the uncertainty extent and correlation between different coordinates. The numerical examples show that the proposed method has almost an equivalent accuracy of Monte Carlo simulation but much higher efficiency.

Description: The development of suitable mathematical models on the basis of dynamic measurements from dispersed structural systems that may be undergoing significant nonlinear behavior is an important and very challenging problem in the field of Applied Mechanics that has drawn the attention of numerous investigators and motivated the development of many approaches for extracting reduced-order, reduced-complexity models from such systems. However, even though numerous nonlinear system identification techniques that are focused on the class of problems encountered in the structural dynamics field have been developed over the past decades, there are no systematic studies available that rigorously compare the performance and fidelity of such methods under similar operating conditions, and when encountering challenging nonlinear phenomena (such as hysteresis) that are present in physical systems, at different scales. This paper explores a variety of data-driven identification techniques for complex nonlinear systems and provides a much needed critical comparison of the accuracy and performance of each method. The Volterra/Wiener neural network (VWNN), a more recent development in nonlinear identification, is featured and compared against several existing methods, including polynomial-based nonlinear estimators and other artificial neural network systems. A representative three degree-of-freedom structure with nonlinear restoring force elements is used as the primary means of comparison for the different methods, and a variety of nonlinear models were investigated, including bilinear hysteresis, polynomial stiffness, and Bouc–Wen hysteresis. Performance comparisons were based on the ability to estimate the acceleration responses for both training and testing simulations. The results showed that, in general, the VWNN provided better accuracy in its estimates for each model. The VWNN also performed best when evaluated for scenarios in which numerical integration is required to find velocity and displacement information from measured accelerations or sensor noise is present in the measured responses.

Description: This paper proposes a robust adaptive position control scheme for automotive electronic throttle (ET) valve. Compared with the conventional throttle control systems, in this paper, a robust adaptive sliding mode (RASM) control scheme is developed in order to eliminate the effects of the parameter uncertainties and nonlinearities including friction, return-spring limp-home and gear backlash. It is shown that both the lumped uncertainty bound and the control gains are adaptively estimated by the update laws, such that not only the bound information of the lumped uncertainty and the control gains are no longer required, but also a robust tracking performance can be ensured in the presence of the parametric variations and disturbances. The comparative simulation and experimental studies are demonstrated to verify the excellent transient and steady-state tracking performance of the proposed RASM controller for ET systems.

Description: This study proposes a novel nonlinear resilient trajectory control for a quadrotor unmanned aerial vehicle (UAV) using backstepping control and nonlinear disturbance observer. First, a nonlinear dynamic model for the quadrotor UAV that considers external disturbances from wind model uncertainties is developed. A nonlinear disturbance observer is then constructed separately from the controller to estimate the external disturbances and compensate for the negative effects of the disturbances. Based on the estimates from the given observer, a nominal nonlinear backstepping trajectory-tracking position controller is designed to stabilize the subsystems step by step until the ultimate control law is obtained. An extra term is added to the nominal controller to address the problem of actuator effectiveness loss and to ensure system resilience. The stability of the resilient controller is analyzed using Lyapunov stability theory. Simulation results are presented to demonstrate the effectiveness and robustness of the proposed nonlinear resilient controller.

Description: The main purpose of this article is to demonstrate that bursting oscillations can be observed not only in the slow–fast autonomous dynamical systems with multiple scales associated with time domain, but also in the non-autonomous dynamical systems with periodic excitations when an order gap exists between the exciting frequency and the natural frequency, implying multiple scales in frequency domain. Furthermore, we try to investigate the influence of different codimensional bifurcations between the quiescent states (QS) and repetitive spiking states (SP) and the nonlinear structures with different equilibrium branches on the bursting oscillations. By introducing an inductor as well as a periodically changed electrical current source in a traditional Chua’s circuit and taking suitable parameter values, a modified four-dimensional periodically excited oscillator with multiple scales in frequency domain is established. Bursting oscillations for two cases with nonlinear terms up to third and fifth order with codimension-1 and codimension-2 bifurcations have been explored, respectively. It is found that more equilibrium states may exist when higher order nonlinear terms are introduced in the vector field, which may cause multiple quiescent states, and accordingly, multiple forms of repetitive spiking oscillations in one bursting attractor, leading to more complicated bursting phenomena. Furthermore, instead of jumping from one stable equilibrium branch to settle down to another stable equilibrium branch when codimension-1 bifurcations (fold bifurcations) exist between QSs and SPs, codimension-2 bifurcation (fold-Hopf bifurcation) may cause QS approximately located on one stable equilibrium branch to jump to repetitive spiking oscillations surrounding another stable equilibrium branch of the generalized autonomous system.

Description: In this paper, the first-passage problem for nonlinear systems driven by \(\alpha \) -stable Lévy white noises is considered. The path integral solution (PIS) is adopted for determining the reliability function and first-passage time probability density function of nonlinear oscillators. Specifically, based on the properties of \(\alpha \) -stable random variables and processes, PIS is extended to deal with Lévy white noises with any value of the stability index \(\alpha \) . Application to linear and nonlinear systems considering different values of \(\alpha \) is reported. Comparisons with pertinent Monte Carlo simulation data demonstrate the accuracy of the results.

Description: A stochastic differential equation model is considered for nonlinear oscillators under excitations of combined Gaussian and Poisson white noise. Since the solutions of stochastic differential equations can be interpreted in terms of several types of stochastic integrals, it is sometimes confusing about which integral is actually appropriate. In order for the energy conservation law to hold under combined Gaussian and Poisson white noise excitations, an appropriate stochastic integral is introduced in this paper. This stochastic integral reduces to the Di Paola–Falsone integral when the multiplicative noise intensity is infinitely differentiable with respect to the state. The stochastic integral introduced in this paper is applicable in more general situations. Numerical examples are presented to illustrate the theoretical conclusion.

Description: We study theoretically the global chaotic behavior of the generalized Chen–Wang differential system $$\begin{aligned} \dot{x} = y, \quad \dot{y} = z,\quad \dot{z} = -y - b x^2 - x z + 3 y^2 + a, \end{aligned}$$ where \(a,b \in {\mathbb {R}}\) are parameters and \(b\ne 0\) . This polynomial differential system is relevant because is the first polynomial differential system in \({\mathbb {R}}^3\) with two parameters exhibiting chaotic motion without having equilibria. We first show that for \(a〉0\) sufficiently small it can exhibit up to three small amplitude periodic solutions that bifurcate from a zero-Hopf equilibrium point located at the origin of coordinates when \(a=0\) . We also show that the system exhibits two limit cycles emerging from two classical Hopf bifurcations at the equilibrium points \(({\pm }\sqrt{2a}, 0,0)\) , for \(a〉0, b=1/2\) . We also give a complete description of its dynamics on the Poincaré sphere at infinity by using the Poincaré compactification of a polynomial vector field in \({\mathbb {R}}^3\) , and we show that it has no first integrals neither in the class of analytic functions nor in the class of Darboux functions.

Description: This paper aims to study the effect of network eigenvalue on the spread of computer virus. To this end, a node-based propagation model, which incorporates the impact of external computers, is proposed. A dynamical analysis of the model shows that the network eigenvalue plays a key role in controlling viral spread. Specifically, the global stability of virus-free equilibrium and the global attractivity of viral equilibrium depend on the value of the maximum eigenvalue of the propagation network. This result reveals that computer virus would tend to extinction or persist depending on network eigenvalue. Finally, some numerical examples are given to illustrate the main results.

Description: This paper presents a novel layering approach of steganography which implements deception theory by utilizing three types of multimedia, in order to hide the data in video file. In the proposed technique first, audio stream of video file is analyzed and two features are extracted, active zone and silence zone. These features define method numbers where each method number corresponds to one cipher technique which exists in the literature. After selecting the ciphering method, we apply it on the plaintext. In the second layer, first we convert the ciphered text into arbitrary characters and hide the data using chaotic map by generating logo image. In the third layer, we multiply that image with orthogonal matrix which is selected on the basis of silence zones. In the fourth layer, we hide that logo image using DWT with security key. The security key is generated locally by using logistic map, and its initial condition is set on the basis of selected method number. Lastly we hide resultant image into the audio signal using wavelet transform and regenerate the video file. The proposed system is implemented in MATLAB 7.0.  Here, we analyzed the SNR values of the regenerated signal and other statistical results to prove that onion steganography works better than silence interval technique, LSB technique, echo hiding technique, magnitude technique and spectrum technique in many aspects.

Description: In this paper, based on a predator–prey model with Holling’s type III functional response, a pest management system with artificial interference is proposed. We assume that the artificial interference strategy will be taken to control pests when their number reaches a certain threshold. Based on this assumption, the artificial interference strategy of the system with nonlinear state feedback control is analyzed by using the geometric theory of ordinary differential equations. We first study the existence of periodic solutions of the model by successor functions and then the stability of periodic solutions. Finally, numerical simulations are given to illustrate our theoretic conclusions.

Description: The main purpose of this paper is to develop a simple-model moving wheel/rail contact element, so that the sticking, sliding, and separation modes of the wheel/rail contact can be appropriately simulated. In the proposed finite element, the wheel and rail are simulated using the cubic-spline contact element, and a power function normal stiffness and a constant horizontal stiffness are connected to the cubic-spline contact stiffness. The three-dimensional (3D) contact finite element analysis for a realistic wheel and rail was used to accurately model the wheel/rail contact stiffness. The validated examples show that the proposed nonlinear moving wheel element can simulate the complicated sliding, sticking, and separation contact problems with good accuracy. The complicated contact modes, including the multiple contact situation between wheel flange and rail side, can also be simulated accurately. Moreover, the computer memory and CPU time required to achieve this are much less than needed with the 3D finite element contact model.

Description: In this paper, the flocking problem for multiple nonlinear Euler–Lagrange systems is investigated with both parametric uncertainties and time-delays under directed communication graphs. With the help of a new coordinate transformation method, an adaptive controller, which is dependent on position and velocity measurements, is proposed to guarantee flocking. By frequency domain analysis, input–output stability analysis and the final value theorem, the synchronization of positions/velocities of all the agents is guaranteed, and in the meantime all agents ultimately track a designated velocity. It is shown that to ensure flocking the sufficient condition is closely relevant to a constant gain in the proposed controller and the nonzero eigenvalues of the Laplacian matrix whose moduli are less than one. Finally, one numerical simulation is presented to demonstrate the efficiency of the theoretical results.

Description: This paper addresses the problem of optimization of the synchronization of a chaotic modified Rayleigh system. We first introduce a four-dimensional autonomous chaotic system which is obtained by the modification of a two-dimensional Rayleigh system. Some basic dynamical properties and behaviors of this system are investigated. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the proposed system. Correspondences are established between the coefficients of the system model and the components of the electronic circuit. Furthermore, we propose an optimal robust adaptive feedback which accomplishes the synchronization of two modified Rayleigh systems using the controllability functions method. The advantage of the proposed scheme is that it takes into account the energy wasted by feedback coupling and the closed loop performance on synchronization. Also, a finite horizon is explicitly computed such that the chaos synchronization is achieved at an established time. Numerical simulations are presented to verify the effectiveness of the proposed synchronization strategy. Pspice analog circuit implementation of the complete master–slave controller system is also presented to show the feasibility of the proposed scheme.

Description: A class of non-diffusively coupled complex networks consisting of nodes of different dimensions is studied, in which the internal time delays are different from the coupling delays. Proper adaptive controllers are proposed for the stabilization and function matrix projective synchronization of such complex networks, respectively. The symmetric or diffusive conditions for the coupling matrices are not required. Finally, the results are applied to complex networks of chaotic oscillators showing the effectiveness of the proposed controllers.

Description: In the paper “Design of \(\alpha \) -filter based UDE controllers considering finite control bandwidth” by A. Kuperman (Nonlinear Dynamics, vol. 81, no. 1, July. 2015), performances of a new first-order \(\alpha \) -filter for uncertainty and disturbance estimator (UDE)-based control and a classical first-order UDE filter where examined. It was shown that performance superiority of the \(\alpha \) -filter is not straightforward and takes place for limited operating region only in case practical implementation issues of the filters are taken into account. The purpose of this note is complementing (Kuperman in Nonlinear Dyn 81:411–416, 2015 ) by proving that in case practical implementation restrictions are extended to the overall control system and classical UDE-filter-based controller would always outperform the \(\alpha \) -filter-based controller in terms of disturbance and noise rejection, while the \(\alpha \) -filter-based controller would be superior in terms of stability.

Description: The manipulation of motion and ancillary operations are important tasks in kinematics, robotics, and rigid and flexible multibody dynamics. Motion can be described in purely geometric terms, based on Chasles’ theorem. Representations and parameterizations of motion are also available, such as Euler motion parameters and the vectorial parameterization, respectively. Typical operations to be performed on motion involve the selection of local or global parameterizations and the derivation of the associated expressions for the motion tensor, velocity or curvature vector, composition of motions, and tangent tensors. Many of these tasks involve arduous, error-prone algebra. The use of dual entities has been shown to ease the manipulation of motion, yet this concept has received little attention outside of the fields of kinematics and robotics. This paper presents a comprehensive treatment of the topic using a notation that eliminates the bookkeeping parameter typically used in dual number algebra, thereby recasting all operations within the framework of linear algebra and streamlining the process. The manipulation of geometric entities is recast within this formalism, paving the way for the manipulation of motion. All developments are presented within the framework of dual numbers directly; the principle of transference is never invoked: The manipulation of rotation is a particular case of that of motion, as should be. The problem of interpolation of motion, a thorny issue in finite element applications, is also addressed.

Description: In this paper, parameter estimation of unknown fractional-order memristor-based chaotic systems is concerned. Firstly, the parameter estimation is transformed into a multi-dimensional optimization problem, where the fractional orders are treated as independent variables. Then, a hybrid artificial bee colony algorithm combined with differential evolution and other searching mechanisms is put forward to solve the optimization problem. Finally, to demonstrate the effectiveness of the proposed method, numerical simulations based on two typical fractional-order memristor-based chaotic systems are conducted. The simulation results shows that the proposed approach for parameter estimation of unknown chaotic systems is a successful and promising method with higher calculation accuracy, faster convergence speed, and stronger robustness.

Description: Based on generalized bilinear forms, lump solutions, rationally localized in all directions in the space, to dimensionally reduced p -gKP and p -gBKP equations in (2+1)-dimensions are computed through symbolic computation with Maple. The sufficient and necessary conditions to guarantee analyticity and rational localization of the solutions are presented. The resulting lump solutions contain six parameters, two of which are totally free, due to the translation invariance, and the other four of which only need to satisfy the presented sufficient and necessary conditions. Their three-dimensional plots with particular choices of the involved parameters are made to show energy distribution.

Description: By combining the generalized Padé approximation method and the well-known Lindstedt–Poincaré method, a novel technique, referred to as the generalized Padé–Lindstedt–Poincaré method, is proposed for determining homo-/heteroclinic orbits of nonlinear autonomous oscillators. First, the classical Padé approximation method is generalized. According to this generalization, the numerator and denominator of the Padé approximant are extended from polynomial functions to a series composed of any kind of continuous function, which means that the generalized Padé approximant is not limited to some certain forms, but can be constructed variously in solving different matters. Next, the generalized Padé approximation method is introduced into the Lindstedt–Poincaré method’s procedure for solving the perturbation equations. Via the proposed generalized Padé–Lindstedt–Poincaré method, the homo-/heteroclinic bifurcations of the generalized Helmholtz–Duffing–Van der Pol oscillator and \(\Phi ^{6}\) -Van der Pol oscillator are predicted. Meanwhile, the analytical solutions to these oscillators are also calculated. To illustrate the accuracy of the present method, the solutions obtained in this paper are compared with those of the Runge–Kutta method, which shows the method proposed in this paper is both effective and feasible. Furthermore, the proposed method can be also utilized to solve many other oscillators.

Description: The aim of this paper was to study the dynamic response of a single-degree-of-freedom (SDOF) oscillator exited by a base acceleration and constrained by two unilateral constraints (bumpers). The coupled equations of motion are formulated in dimensional and dimensionless form for the case in which both SDOF oscillator and bumpers exhibit a nonlinear behavior. Two possible states characterize the system’s response: flight , when the mass of the oscillator is not in contact with the bumper, and contact , when the mass touches the bumper. When the system is in the flight state, the independent time evolution of the bumpers’ response is also considered in equations of motion. In the numerical investigations, a viscoelastic SDOF oscillator and viscoelastic perfectly plastic bumpers subjected to a harmonic base excitation are considered. First, the transient versus steady-state dynamic response is considered, and afterward the only steady-state dynamic response is studied by means of pseudo-resonance curves of maximum absolute acceleration and relative displacement excursion of the SDOF oscillator. Furthermore, the continuation technique is applied in some cases in order to highlight in the system’s dynamic response hysteresis ranges, jumps between multi-periodic orbits, and super-harmonics. The dynamic analysis with and without bumpers is compared to observe how unilateral constraints modify the dynamic response of the SDOF oscillator with respect to the absence of bumpers and when their presence may be beneficial with respect to response mitigation.

Description: In the practical application of impulsive differential systems, impulse does not always occur at the fixed-time point; it may occur in a little range of time. Namely, impulse occurs in a time window, which is more general and more nearing to reality than those fixed-time impulses. Therefore, it is necessary to investigate the dynamical behaviors of impulsive differential systems with impulse time windows. In this paper, the exponential stability of these systems is researched. By means of Lyapunov functions, Razumikhin technique and other analysis methods, several novel exponential stability criteria for delayed impulsive functional differential equations with impulse time windows are obtained, which are different from the previously published results for fixed-time impulses. What is more, based on the analysis of this paper, it is worth noting that choosing an efficient impulse time window may be easier and more effective than choosing fixed-time impulsive sequences. Finally, three examples and their simulations are provided to illustrate the effectiveness of our results.

Description: This paper investigates the problem of finite-time stabilization by state feedback for a class of uncertain nonholonomic systems with inputs saturation. Comparing with the existing relevant literature, a distinguishing feature of the systems under investigation is that the x -subsystem is in feedforward-like form. Rigorous design procedure for saturated finite-time state feedback control is presented by using the adding a power integrator and the nested saturation methods. The development of saturated finite-time controller is also presented briefly for a class of dynamic nonholonomic systems in feedforward-like form. An application example for a kinematic hopping robot is provided to illustrate the effectiveness of the proposed approach.

Description: In this work, we extend the well-known Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth systems, defined in three zones separated by two switching manifolds \(x=-\alpha \) and \(x=\beta \) . We suppose that the dynamic in each zone is governed by a smooth system. When a trajectory reaches the switching manifolds, then reset maps describing impacting rules on the switching manifolds will be applied instantaneously before the trajectory enters into the other zone. We also assume that the unperturbed system is a piecewise-defined continuous Hamiltonian system and possesses a pair of heteroclinic orbits transversally crossing the switching manifolds. Then, we study the persistence of the heteroclinic orbits under a non-autonomous periodic perturbation and the reset maps. In order to obtain this objective, we derive a Melnikov-type function by using the Hamiltonian function to measure the distance of the perturbed stable and unstable manifolds in this system. Finally, we employ the obtained Melnikov-type function to study the persistence of a heteroclinic cycle and complicated dynamics near the heteroclinic cycle for a concrete planar piecewise-smooth system.