ALBERT

Description: The paper introduces a technique that decomposes the dynamics of a nonlinear system about an equilibrium into low-order components, which then can be used to reconstruct the full dynamics. This is a nonlinear analogue of linear modal analysis. The dynamics is decomposed using Invariant Spectral Foliation (ISF), which is defined as the smoothest invariant foliation about an equilibrium and hence unique under general conditions. The conjugate dynamics of an ISF can be used as a reduced order model. An ISF can be fitted to vibration data without carrying out a model identification first. The theory is illustrated on a analytic example and on free-vibration data of a clamped-clamped beam.

Description: The stick–slip model of a Girling brake is composed of nonlinear and coupled differential equations that reproduce the friction occurring in this mechanical system. The brake is equivalent to a body sliding on a belt. The problem is very interesting since the possible solutions, which are very sensitive to the parameters of the system, show a chaotic behaviour. In this contribution, the model, which is designed following network method rules, is explained in detail and runs on standard electrical circuit simulation software to provide the displacement and the velocity of the sliding body and the phase planes. In comparison with other models, the considered system does not include dampers to get a more unstable behaviour. Furthermore, a suitable selection of parameters is implemented to reduce the computational time.

Description: The drivers in actual traffic differ in the aggressiveness in driving behaviors, which will finally be reflected in the interaction between vehicles and fluctuations in flows. In this paper, we try to study the effect of driving aggressiveness on the traffic stability by proposing an extended microscopic car-following model, in which the optimal velocity is reconstructed to divide the drivers in traffic system into two groups according to driving aggressiveness of each individual. The stability condition of the proposed model is derived to explore its ability against a small perturbation by use of the linear stability theory. We obtain the neutral stability lines for different percentages of drivers with a higher driving aggressiveness, finding that the traffic flow trends to stable with the increase in the percentage for higher driving aggressiveness drivers when the average headway is less than a critical value or greater than another critical value, but when the average headway falls into the intermediate range between the two critical values, the traffic flow becomes more and more unstable with increase in the percentage of drivers with a higher driving aggressiveness. Finally, numerical simulations are conducted to verify these theoretical results and examine how the percentage of vehicles driven by higher driving aggressiveness drivers affects the traffic flux of the vehicle system.

Description: This paper investigates adaptive synchronization of stochastic time-varying delay dynamical networks with complex-variable systems. By using the complex inequality, stochastic analysis theory and two kinds of random disturbances, several sufficient conditions to ensure adaptive synchronization for stochastic time-varying delay networks with complex-variable systems. To illustrate the effectiveness of the synchronization conditions derived in this paper, numerical examples are provided finally.

Description: This paper presents the unified cooperative strategies for the salvo attack of multiple missiles on the basis of the traditional proportional navigation (PN) algorithm. The cooperative guidance laws are developed in a quite simple formulation consisting of a PN component for target capture and a coordination component for simultaneous arrival. The centralized coordination algorithms have better performance when the global information of time-to-go can be obtained by each missile, whereas the decentralized coordination algorithms come into effect when the group member is only able to collect information from its neighbors. The cooperative strategies can achieve a simultaneous attack against both stationary and maneuvering targets. Numerical simulations are used to demonstrate that the proposed approaches are feasible and flexible for the salvo attack of multiple missiles. The capturability analyses of the cooperative guidance laws are also performed by discussing the selection of gain parameters.

Description: In this work we complete the integrability conditions (i.e., conditions for the existence of a local analytic first integral) for a family of a resonant saddle perturbed with homogeneous quintic nonlinearities studied in a previous work. In order to obtain the necessary conditions we use modular arithmetic computations.

Description: The chaotic behavior of the secondary asteroid in a system of binary asteroids due to the asphericity and orbital eccentricity is investigated analytically and numerically. The binary asteroids are modeled with a sphere–ellipsoid model, in which the secondary asteroid is ellipsoid. The first-order resonance is studied for different values of asphericity and eccentricity of the secondary asteroid. The results of the Chirikov method are verified by Poincare section which show good agreement between analytical and numerical methods. It is also shown that asphericity and eccentricity affect the size of resonance regions such that beyond the threshold value, the resonance overlapping occurs and widespread chaos is visible.

Description: An efficient and secure three-party authenticated key agreement protocol is proposed to enable two users to establish a common secret key for exchanging confidential and authenticated information with the help of a trusted server. The proposed protocol only employs extended chaotic maps and hash operations, i.e., it does not require a server public key, symmetric cryptosystems, time-consuming modular exponential computations, or time-consuming modular exponential computations and scalar multiplications on elliptic curve. A round-efficient version of the proposed protocol is also implemented by rearranging and sending the messages in parallel. The session security of the proposed protocol is based on the Chebyshev chaotic map-based Diffie–Hellman assumption. Compared to related chaotic map-based approaches, the proposed protocol not only requires lower computational cost, but also has fewer transmissions.

Description: In this paper, a novel adaptive neural network control approach is presented for a class of uncertain discrete-time nonlinear strict-feedback systems with input saturation. By combining single neural network approximation and minimal learning parameter technique, the proposed approach is able to eliminate the complexity growing problem and alleviate the explosion of learning parameters. An auxiliary design system is incorporated into the control scheme to overcome the problem of input saturation constraints. Following this approach, the designed controller contains only one actual control law and one adaptive law, the numbers of input variables and weights of neural network updated online are decreased drastically, and the number of parameter updated online for whole system is reduced to only one. Compared with the existing methods, the adaptive mechanism with much simpler controller structure and minimal learning parameterization is achieved; therefore, the computational burden is lighter. It is shown via Lyapunov theory that all signals in the closed-loop system are uniformly ultimately bounded. Finally, simulation results via two examples are employed to illustrate the effectiveness and merits of the proposed scheme.

Description: This paper investigates the problem of global coordinated tracking of a multi-agent system with input additive uncertainties and disturbances via bounded control inputs. Scheduled low-and-high gain feedback-based distributed coordinated tracking protocols are developed. It is shown that, under the assumptions that each agent is asymptotically null controllable with bounded controls and the network is connected, global coordinated tracking of the multi-agent system can be achieved. We finally show some numerical simulations to verify and illustrate the theoretical results.

Description: In this paper, a four-neuron delayed system with inertial terms is considered. By studying the distribution of the eigenvalues of the associated characteristic equation, we derive the critical values where double Hopf bifurcation occurs. Then by employing the perturbation-incremental scheme for the system, bifurcation diagrams are obtained. Furthermore, we carry out bifurcation analysis showing that there exist a stable fixed point, two stable periodic solutions, co-existence of a pair of stable periodic solutions and quasi-periodic motion in the neighborhood of the double Hopf critical point. We also find some interesting phenomena that the dynamical period switching occurs in some delayed regions. Finally, some numerical simulations are performed to support the theoretical analysis.

Description: Analytical spatiotemporal soliton solutions of the coupled nonlinear Schrödinger equation are derived in the (3  \(+\)  1)-dimensional inhomogeneous \(\mathcal {PT}\) -symmetric nonlinear couplers by means of the Hirota bilinear method. Based on these analytical solutions, we construct abundant spatiotemporal soliton structures and discuss the expansion and compression behaviors of spatiotemporal soliton structures in a periodic modulation system.

Description: In this paper we develop a systematic and self-consistent procedure based on a set of compatibility conditions for identifying all maximal (eight parameter) and non-maximal (one and two parameter) symmetry groups associated with the mixed quadratic-linear Liénard-type equation, \(\ddot{x} + f(x){\dot{x}}^{2} + g(x)\dot{x}+h(x)= 0\) , where \(f(x),\,g(x)\) and h ( x ) are arbitrary functions of x . With the help of this procedure we show that a symmetry function b ( t ) is zero for non-maximal cases, whereas it is not so for the maximal case. On the basis of this result the symmetry analysis gets divided into two cases, (i) the maximal symmetry group \((b\ne 0)\) and (ii) non-maximal symmetry groups \((b=0)\) . We then identify the most general form of the mixed quadratic linear Liénard-type equation in each of these cases. In the case of eight-parameter symmetry group, the identified general equation becomes linearizable. In the case of non-maximal symmetry groups the identified equations are all integrable. The integrability of all the equations is proved either by providing the general solution or by constructing time-independent Hamiltonians.

Description: This paper reveals the dynamical behaviors of a coupled network consisting of four neural sub-networks and multiple two-way couplings of neurons between the individual sub-networks. Different types of time delays are introduced into the internal connections within the individual sub-networks and the couplings between the sub-networks. Stability switches of the network equilibrium and Hopf bifurcation are analyzed by discussing the associated characteristic equation. The conditions for the existence of different patterns of oscillations are discussed. By using the Lyapunov’s second method, the global stability of the network equilibrium is studied. Numerical simulations are performed to validate the theoretical results, and rich dynamical behaviors are observed, such as synchronous oscillations, multiple stability switches between the rest state and oscillations, phase-locked oscillations, asynchronous oscillations, and the coexistence of different patterns of bifurcated oscillations.

Description: A significant impediment to the deployment of vibration-based energy harvesting devices has been the limitation of most low-frequency transducers, usually in the form of unimorph or bimorph cantilever beam, to extract energy from a very narrow bandwidth around the transducer’s fundamental frequency. In such devices, a slight deviation from the fundamental frequency causes a significant reduction in the level of harvested power by several orders of magnitudes. Additionally, most of the current research efforts on the design of piezoelectric energy harvesters have had limited success in achieving low resonance frequency. To overcome these challenges, we introduce an enhanced broadband low-frequency piezomagnetoelastic energy harvester. This harvester consists of a partially covered piezoelectric cantilever beam with a fixed magnet mass at the top of the magnet tip mass. A nonlinear distributed-parameter model based on Euler–Bernoulli beam theory and Galerkin discretization is developed. This electromechanical model is validated with previous experimental measurements for a specific value of the spacing distance between the two magnets. A parametric study is performed to determine the effects of the spacing distance between the two magnets on the static position of the harvester, natural frequency, and level of the harvested power. It is demonstrated that a decrease between the two attractive magnets results in a decrease in the natural frequency of the harvester with a strong softening behavior which gives the opportunity to harvest energy at broadband low-frequency range. The results also show that the presence and importance of the softening behavior depends on the electrical load resistance. In conclusion, the results show that depending on the available low excitation frequency, an enhanced piezoelectric energy harvester can be tuned and optimized by changing the spacing distance between the two tip magnets.

Description: In this paper, we are interested to justified two typical hypotheses that appear in the convergence analysis, \(|\lambda |\le 2\) and \(z_0\) sufficient close to \(z^*\) . In order to proof these ideas, the dynamics of a damped two-step Newton-type method for solving nonlinear equations and systems is presented. We present the parameter space for values of the damping factor in the complex plane, focusing our attention in such values for which the fixed points related to the roots are attracting. Moreover, we study the stability of the strange fixed points, showing that there exists attracting cycles and chaotical behavior for some choices of the damping factor.

Description: We consider memelements (memristors and memductors) with special periodic responses (mixed-mode oscillations) and 2D one-period loops yielding constant parameters describing the memelements as single units or components of oscillating circuits. One of the parameters is the action parameter having the dimensions of energy   \(\times \)   time and the SI unit Joule   \(\times \)   second . The remaining loops and parameters correspond to energy of magnetic and electric fields, power and rms current and voltage values. Special mixed one-period loops are also analyzed with pairs of signals associated with two different components of the circuits. The areas enclosed by various loops result in special equations which can be derived from the underlying ODE model of the circuits. The action of a memelement is equivalent to the time integral of the Lagrangian \(L(w,w')\) , where w is the internal state variable of a memelement. The analysis of memristive circuits and their parameters is considered in the framework of mixed-mode oscillations. Also, the unit of action for memelements is proposed to be called Chua ( \(=\) Joule \(\,\times \)   second ) to honor L.O. Chua for his work on memristors and memristive circuits.

Description: In this paper, the binary Bell polynomials are employed to find the bilinear form, bilinear Bäcklund transformation and Lax pair for the (3+1)-dimensional BKP equation. Based on Hirota’s bilinear form and three-wave method, multi-soliton solutions are presented. Furthermore, a new bilinear Bäcklund transformation is constructed via applying a gauge transformation to the Bäcklund transformation in bilinear form.

Description: In this paper, a \((2 + 1)\) -dimensional generalized shallow water wave equation is investigated through bilinear Hirota method. Interestingly, the breather-type and lump-type soliton solutions are obtained. Furthermore, dynamic properties of the soliton waves are revealed by means of the asymptotic analysis. Based on Hirota bilinear method and Riemann theta function, we succeed in constructing quasi-periodic wave solutions with a generalized form. We also display the asymptotic properties of these quasi-periodic wave solutions and point out the relation between the quasi-periodic wave solutions and the soliton solutions.

Description: In this paper, the classical problem and the inverse problem of generalized synchronization for different dimensional chaotic dynamical systems in discrete time are investigated. The generalized synchronization results have been derived using active control method and Lyapunov stability theory. Numerical simulations are performed to verify the effectiveness of the proposed schemes.

Description: Under investigation in this paper is a generalized \((2+1)\) -dimensional Korteweg–de Vries equation, which could describe many nonlinear phenomena in plasma physics. By virtue of the Bell’s polynomials, a straightforward way is presented to succinctly construct its bilinear form, bilinear Bäcklund transformation and Lax pairs. Once the Lax pairs obtained, the important infinite conservation laws of the equation are directly found. Moreover, based on the bilinear formalism, we construct the Riemann theta function periodic wave solutions and soliton solutions. Finally, the relationships between the periodic wave solutions and soliton solutions are strictly established, and the asymptotic behavior of the periodic waves is also presented with detailed proof.

Description: In the present paper, we study regular and chaotic dynamics from planar oscillations of a dumbbell satellite under the influence of the gravity field generated by an oblate body, considering the effect of the zonal harmonic parameter \(J_{2}\) . We theoretically show the existence of chaotic oscillations provided that the eccentricity becomes arbitrarily small, and the parameter \(J_{2}\) is of the same order of magnitude as the eccentricity. This is carried out by applying the so-called Melnikov method. Finally, for arbitrarily chosen values for the parameters involved in such a problem, we study the transition from regular to chaotic oscillations for a dumbbell satellite via the analysis of chaotic maps and Poincaré surfaces of section, respectively.

Description: We obtain some results on transitivity for cyclically permuted direct product maps, that is, maps of the form \(F\left( x_{1},x_{2},\ldots ,x_{n}\right) =\left( f_{\sigma (1)}\left( x_{\sigma (1)}\right) ,f_{\sigma (2)}\left( x_{\sigma (2)}\right) ,\ldots ,f_{\sigma (n)}\left( x_{\sigma (n)}\right) \right) \) , defined from the Cartesian product \(I^n\) onto itself, where \(I=[0,1]\) , \(\sigma \) is a cyclic permutation of \(\{1,2,\ldots ,n\}\) \((n\ge 2)\) and each map \(f_{\sigma (j)}:I\rightarrow I\) is continuous, \(j\in \{1,\ldots ,n\}\) . In particular, we prove that for \(n\ge 3\) the transitivity of F is equivalent to the total transitivity, and if \(n=2\) , we give a splitting result for transitive maps. Moreover, we extend well-known properties of transitivity from interval maps to cyclically permuted direct product maps. To do it, we use the strong link between F and the compositions \(\varphi _j=f_{\sigma (j)}\circ \ldots \circ f_{\sigma ^n(j)},\, j\in \{1,\ldots ,n\}.\)

Description: In this paper, targeted energy transfer from a nonlinear continuous system to a nonlinear energy sink (NES) is investigated. For this purpose, the equation of a nonlinear beam with simply supported ends, on which the NES is attached, is derived using Rayleigh–Ritz method and the Lagrange equation. Then, parameters of the NES are optimized by both sensitivity analysis and particle swarm optimization (PSO) method. Analysis of the energy transfer between the nonlinear beam and the NES is presented, using the complexification-averaging method, too. Attaching an NES to a nonlinear continuous system and using the PSO method to obtain the optimized parameters of the NES is a new development, presented in this work. Also, here, more than one mode of the beam has been considered for analysis of energy transfer between the NES and different modes of the primary system.

Description: In this work we present a pseudo-random Bit Generator via unidimensional multi-modal discrete dynamical systems called k -modal maps. These multi-modal maps are based on the logistic map and are useful to yield pseudo-random sequences with longer period, i.e., in order to attend the problem of periodicity. In addition the pseudo-random sequences generated via multi-modal maps are evaluated with the statistical suite of test from NIST and satisfactory results are obtained when they are used as key stream. Furthermore, we show the impact of using these sequences in a stream cipher resulting in a better encryption quality correlated with the number of modals of the chaotic map. Finally, a statistical security analysis applied to cipher images is given. The proposed algorithm to encrypt is able to resist the chosen-plaintext attack and differential attack because the same set of encryption keys generates a different cipher image every time it is used.

Description: In this paper, we numerically study the effect of ion channel noise on the synchronization of delayed Newman–Watts network of stochastic Hodgkin–Huxley neurons. It is found that time delay can induce synchronization transitions only when channel noise intensity is intermediate, and the synchronization transitions become most obvious when channel noise intensity is optimal. As channel noise intensity is varied, the neurons can also exhibit synchronization transitions, and the phenomenon is enhanced when time delay is tuned at around time delays that can induce synchronization transitions. Moreover, this phenomenon becomes most obvious when the value of coupling strength or the value of network randomness is optimal. These results show that channel noise has a subtle regulation effect on the synchronization of the neuronal network. These findings may find potential implications for the normal and pathological functions in the brain and the information processing and transmission in neural systems.

Description: Using the tool of Turing instability for partial differential equations, we investigate the spatiotemporal distributions for solutions of a predator–prey-type reaction–diffusion model with spatiotemporal delay. The linear stability conditions of Turing instability, which induce bifurcation patterns in this model, are obtained. Moreover, according to these conditions, we numerically calculate the bifurcation diagrams by using time delay and the predator rate as parameters. The effects of two parameters in the different bifurcation diagrams are also demonstrated through numerical computations and lead to some spatiotemporal patterns of this model, which enrich the pattern formation of predator–prey models.

Description: A new two-lane traffic lattice hydrodynamic model is proposed with the consideration of the global average-and-optimal flux difference effect based on the local relative flux two-lane lattice model. First, the influence of the global average-and-optimal flux difference on the stability of traffic flow is investigated through linear stability theory. The results reveal that the unstable region will be shrunk by taking the global average-and-optimal flux difference effect into account. Additionally, by using the reductive perturbation method, the mKdV equation near the critical point is derived and traffic jam transition can be described by its kink–antikink soliton solution. The good agreement between the numerical simulations and the analytical results shows that traffic congestion can be suppressed efficiently by considering the global average-and-optimal flux difference and the local relative flux effects in two-lane traffic system and the local relative flux is more important than the global average-and-optimal flux difference in stabilizing traffic flow.

Description: Under investigation in this paper is a \((2+1)\) -dimensional Korteweg–de Vries-type equation, which can describe the propagation of nonlinear waves such as the shallow-water waves, surface and internal waves. By virtue of the Bell polynomials, symbolic computation and auxiliary independent variable, the bilinear forms, Bäcklund transformations and soliton solutions are obtained. Solitonic propagation and elastic collisions between/among the two- and three-solitons are discussed analytically and graphically. It can be seen that, after each collision, solitonic shapes and amplitudes keep invariant except for some phase shifts, and the smaller-amplitude soliton moves faster and overtakes the larger.

Description: Based on the optimal velocity model, a new continuum model considering traffic jerk effect is presented in this paper. Then, the critical condition for the steady traffic flow is deduced. Near the neutral stability line, nonlinear analysis is taken to derive the KdV-Burgers equation for describing the density wave, and one of the solutions is given. Numerical simulation is carried out to study the influence about the traffic jerk effect.

Description: The nonlinear dynamical behavior of the hysteretic rheological device proposed in Carboni et al. (J Eng Mech 2014 ) is investigated. The device can provide nonlinear hysteretic forces to a one-degree-of-freedom (one-dof) mass through suitable assemblies of NiTiNOL and steel wire ropes subject to tension–flexure cycles. The simultaneous occurrence of interwire friction, phase transformations and geometric nonlinearities is the key feature of the obtained material behavior. Frequency-response curves (FRCs) of the system subject to base excitation are obtained numerically via a continuation procedure together with stability analysis and experimentally by carrying out shaking table tests, respectively. The phenomenological identification of the material behaviors through force–displacement cycles, reported in Carboni et al. (J Eng Mech 2014 ), is employed for the computation of the FRCs and the equivalent damping ratios as function of the displacement amplitude. The different restoring forces give rise to whole new families of nonlinear hysteretic oscillators governed by softening, hardening and softening–hardening behaviors depending on the oscillation amplitude.

Description: The modeling of the piezoelectric actuator is very important for the fast and accurate nano-positioning control. However, the complex creep, vibration and hysteresis dynamics make the modeling very difficult. Because there are often model uncertainties in the first-principle model, the model identification from the experimental data is very necessary. Most identification approaches only consider partial dynamics of the creep, vibration and hysteresis. Though they can be combined together after they are individually identified from different experiments, it is difficult to design the completely decoupled experiment. In this study, an incremental Hammerstein-like modeling approach is proposed to identify the creep, vibration and hysteresis dynamics from one experiment. The model is called nonlinear–linear–linear-Hammerstein-like model, where one dynamic nonlinear system is used to model the hysteresis and two dynamic linear systems are used to model the creep and vibration. The creep and vibration are assumed to be decoupled, and there is a known upper bound on the order of the creep model. A two-stage incremental modeling approach is proposed to reduce the modeling complexity, where the slow dynamics of the hysteresis and creep are estimated first and then the residual of this model is used to estimate the fast dynamics of the vibration. In each stage, the model structure and order are determined by a locally regularized orthogonal least-squares-based model term selection algorithm and the parameters are estimated using a regularized least-squares method. The effectiveness of the proposed modeling approach is verified by the simulations and experiments on typical piezoelectric actuators.

Description: This paper investigates the problem of harvesting energy from transverse base vibration by a laminated piezoelectric beam. The beam is fixed at one end and subjected to a compressive load at the other end. The electromechanical model is set up, and the corresponding equations are derived by extended Hamilton principle. The simulation is carried out, and the results are obtained. The response under harmonic excitation exhibits that the system experiences period-doubling bifurcation and chaos as frequency varies, and the buckled state can generate more power compared to the unbuckled one. Coherence resonance is proved and can be observed when the system is subjected to Gauss white noise excitation, which can be used to optimize the efficiency of energy harvesting.

Description: In this paper, we present a new scheme for the secured transmission of discrete information based on hyperchaotic discrete dynamics. The system is a modified-Henon hyperchaotic discrete-time oscillator considered as transmitter and a delayed step-by-step observer used as receiver. The transmitter parameters play the role of secret keys of the transmission scheme. To increase the robustness of the secure data transmission against known plain-text attacks, the message to be sent is encrypted by additional secret keys and inserted by inclusion method in the chaotic discrete-time system dynamics. By this way, the parameters used as secret keys cannot be identified with usual techniques. Simulation results are presented to highlight the performances of the proposed method. One of the main contributions of this paper is to demonstrate the feasibility of discrete realization of a chaotic observer-based secured transmission scheme. Indeed, experimental implementation results using Arduino Uno board validate the proposed approach, since it exhibits good performances of throughput and cost in terms of resources used.

Description: This paper is concerned with the problem of complex function projective synchronization for uncertain networked chaotic complex systems. Based on Lyapunov stability, an adaptive control method is proposed for complex modified projective synchronization, which guarantees that the general drive-response networked chaotic complex systems are synchronized up to a complex scaling function matrix. Moreover, a complex fuzzy logic-based observer is designed to compensate for the model uncertainties and the external disturbances that exist in response networks, without prior information about uncertain factors. Numerical simulations are presented to demonstrate the effectiveness of the proposed method.

Description: We investigate the spatiotemporal behaviour of a network where the local dynamics at the nodes (sites) is governed by piecewise linear maps. The local maps we consider exhibit the interesting and potentially useful property of robust chaos. We study the coupled system of such maps with varying fraction of random non-local connections, where the random links may be static, or may change over time. While this system is always unsynchronized under regular connections, synchronized chaos emerges when some of the links are rewired randomly. Further, increasing the frequency of link changes and fraction of random links significantly enhances the range of synchronization. Additionally, dynamic random links are also found to suppress unbounded dynamics in parameter regimes where blow-ups occurred under regular coupling.

Description: Chaotic delay systems are abundant in nature and play a significant role in engineering applications and in describing global behaviors of physical systems. This work presents novel first-order chaotic delay systems with the simplest nonlinearities. The exponential, absolute value, and hyperbolic and signum functions, which arise in many systems like electronic circuits, are utilized to generate chaotic delay systems. The practical realization of chaotic delay systems is carried out with all-pass filters and diode-based electronic circuits. Bifurcation diagrams using numerical simulations and experimental results are provided to verify the existence and feasibility of the novel chaotic delay systems. It is expected that the novel chaotic delay systems and the novel electronic implementation circuits will contribute to some practical applications and modeling of physical systems or events in different fields.

Description: Based on the state-dependent Riccati equation (SDRE) technique, in this paper, a suboptimal pinning control scheme is proposed to synchronize linearly coupled complex networks. The Lyapunov direct method is used to analyze the stability of the closed-loop control system, where it leads to a LMI criterion for pinning synchronization. It is shown that the time interval for synchronization of the proposed SDRE controllers is faster comparing with the results in the latest literatures. It is also shown that the minimum required coupling weights for the network synchronization in a finite desired time is decreased when some specified nodes in the network are pinned with the SDRE controllers. Based on the proposed criterion for pinned nodes selection, the network performances for different topological structures are investigated and the results are compared. The results indicate that the coupling weights for network synchronization in finite desired time in random Erdos Reiny networks are minimum when the pinned nodes are selected based on the minimum matching theorem. In small-world and scale-free networks, the minimum required coupling weight for network synchronization in finite desired time decreases when the highest degree nodes are pinned.

Description: In recent years, digital image processing has become commonplace with growing powerful and available image editing software. People without any professional technique can also manipulate and forge digital images easily. One of the most popular manners of digital image forgeries is the copy–move image forgery. Extensive researches in detecting copy–move forgery have made a deal of achievements, but most presented methods based on these researches have been only focus on some simple composite forgeries and not able to detect different types of post-processed forgeries. In this paper, we aim to deal with the post-processed forgery operations and scenarios, mainly geometric distortion. We introduce analytical Fourier–Mellin transform (AFMT) and focus on its discretization. We propose discrete analytical Fourier–Mellin transform (DAFMT). We also pay attention to high performance of DAFMT in detecting the copy–move image forgeries. Due to the AFMT described in polar coordinate, so we need to convert coordinate system from polar to Cartesian coordinates. To be computed conveniently, we define an auxiliary disk template to accomplish this conversion. We devote to the use of our proposed DAFMT in detection of image forgeries. A great deal of researches and experiments show that the proposed DAFMT can effectively resist translation, rotation, scaling, and added Gaussian noise operations. Compared with other relevant up-to-date methods, experiments also prove that DAFMT has made a progress in detecting and identifying the forgery images which are suffered from geometric distortion operations.

Description: The problem of controlling synchrony in bistable networks, which possess coherent and incoherent attractors in a certain range of parameters, is considered. Along with the known Kuramoto-type models, we introduce the bistable networks consisting of all-to-all coupled noisy FitzHugh-Nagumo neurons as well as chaotic Rulkov neurons. We suggest two different algorithms to switch the bistable networks from the stable coherent state to the stable incoherent state. One of them is an act-and-wait control method, which utilizes the mean field measurements and homogeneous time delayed feedback perturbations with the periodically switched on and off feedback gain. We show that this algorithm is efficient for the globally coupled populations. Another algorithm is based on the multisite coordinated reset stimulation. The algorithm is nonfeedback, but it uses inhomogeneous perturbations and is efficient even for the networks with a complex scale-free topology. In addition to the numerical analysis of finite-size networks, the analytical results for the Kuramoto-type models in the thermodynamic limit are presented.

Description: Considering topography conditions, economic factors and driving safety, in real traffic, a road may be built as curved road. Traffic flow on curved road is different from the one on straight road. And it is worth to investigate the influencing mechanism of traffic flow on curved road. In order to investigate traffic flow on curved road analytically, in this paper, an extended one-dimensional lattice hydrodynamic model for traffic flow on curved road is proposed. The stability condition is obtained by the use of linear stability analysis. It is shown that the stability of traffic flow varies with the radian, friction coefficient and curvature radius of curved road. The Burgers, Korteweg–de Vries and modified Korteweg–de Vries equations are derived to describe the nonlinear density waves in the stable, metastable and unstable regions, respectively. The simulations are given to verify the analytical results. The results, which obtained from the theoretical analysis and numerical simulations, show that traffic flow may be affected by the angle going into curved road, the increment of angle, friction coefficient and curvature radius of curved road. And the maximal theoretical flux and velocity of traffic flow are influenced by the above factors as well.

Description: This paper focuses on the adaptive modified hybrid function projective synchronization with complex function transformation matrix (CMHFPS) for different dimensional chaotic (hyperchaotic) systems with complex variables and unknown complex parameters. The chaotic systems are considerably different from those in the existing closely related literature. Moreover, the transformation matrix in this type of chaos synchronization is not a square matrix, and its elements are complex functions. In particular, by constructing appropriate Lyapunov functions dependent on complex variables, the adaptive controllers are designed to synchronize different dimensional complex chaos (hyperchaos) with complex parameters in the sense of CMHFPS, and the complex update laws for estimating unknown complex parameters of complex chaotic systems are also given. Finally, two examples are presented to illustrate the effectiveness and feasibility of the theoretical results.

Description: In this paper, an experimental verification of the vibro-impact capsule model proposed by Liu et al. in (Int. J. Mech. Sci, 66:2–11; 2013a , Int. J. Mech. Sci, 72:39–54; 2013b , Int. J. Non-Linear Mech., 70, 30–46; 2015 ) is presented. The capsule dynamics is investigated experimentally by varying the stiffness of the support spring, and the frequency and the amplitude of excitation. The novel design of the experimental set-up is discussed, and comparisons between the experiments and numerical simulations are presented showing a good agreement. The conducted bifurcation analysis indicates that the behaviour of the system is mainly periodic and that a fine tuning of the control parameters can significantly improve the performance of the system. The main findings provide a better insight into the vibro-impact systems subject to nonlinear friction, and the experimental rig can be used to predict the dynamic behaviour of these systems.

Description: Based on the stability theory of fractional-order systems, using the fractional-order Lyapunov direct method, in this paper, the Mittag–Leffler stability for fractional-order Lorenz and fractional-order Lorenz-family system is investigated, respectively. We propose the definition of the Mittag–Leffler stability; some sufficient conditions for the Mittag–Leffler stability are obtained by selecting properly the parameters and the initial values of system. Finally, numerical simulations are performed to verify the theoretical analysis.

Description: In this paper, cluster synchronization on multiple sub-networks of complex networks with nonidentical nodes, stochastic disturbances and time-varying delays is investigated. Based on the leader–follower model, an improved network structure model for realizing the cluster synchronization on multiple sub-networks of complex networks is presented. In this improved network model, the complex networks are divided into multiple pairs of matching sub-networks, each of which consists of a leaders’ sub-network and a followers’ sub-network, such that the dynamics of the nodes belonging to the same pair of matching sub-networks are identical, while the ones belonging to different pairs of unmatched sub-networks are nonidentical. Furthermore, the nodes in a sub-network may be inevitably influenced by some stochastic disturbances and time-varying delays. In this new setting, the aim is to design some suitable pinning controllers on the chosen nodes of each followers’ sub-network, such that the proposed method for realizing the cluster synchronization has good immunity to the influence of these stochastic factors. By using the Lyapunov stability theory and stochastic differential equation theory, some cluster synchronization criteria, and a pinning scheme that the nodes with very large or low degrees are good candidates for applying pinning controllers, are established such that all the nodes in each sub-network are exponentially synchronized to the average state of their matching leaders. Then, the attack and robustness of the pinning scheme are discussed. Finally, some simulation examples are presented to verify the theoretical analysis.

Description: This paper introduces a new segmentation technique to segment incomplete nutrient-deficient crop images by imputing missing pixels. Usually, each image contains pixels holding information about intensity, but sometimes image can miss some pixels (that is, the pixel without an appropriate intensity value). An image with missing pixels is called an incomplete image. Intuitionistic fuzzy clustering algorithm is a useful tool for clustering images, but it is not directly applicable for incomplete images. For example, segmentation of nutrient deficiency portions in the presence of missing pixels leads to error in segmentation. Crop images with nutrient deficiency might have missing pixels due to inherent defects in imaging equipment or due to environmental conditions. In this paper, nutrient deficiency in crop images is segmented after imputation of missing pixels based on intuitionistic fuzzy C-means color clustering algorithm. Experiments are performed on various incomplete crop images. Through the derived results and evaluated comparisons with other methods, namely K-means, fuzzy K-means, principal component analysis, regularized expectation maximization and fuzzy C-means algorithms, it has been proven that the proposed method performs well.

Description: Theoretical analysis of a suspended nano-mechanical membrane subject to an optical driving field in an optomechanical cavity is presented, which is confirmed through numerical simulations. In the presence of an optical field between its mirrors, the high-finesse optomechanical resonator acts as an oscillator driven by a radiation pressure force. The periodic nature of the radiation pressure force makes the nano-mechanical membrane in the optomechanical system as a kicked harmonic oscillator. Mathematically the physical system displays a stochastic web map that helps to understand several properties of the kicked membrane in classical phase space. We find that our web map is area preserving and displays quasiperiodic symmetrical structures in phase space which we express as q -fold symmetry. It is shown that under appropriate control of certain parameters, namely the frequency ratio and the kicking strength, the dynamics of kicked membrane exhibits chaotic dynamics. We provide the stability analysis by means of Lyapunov exponent and survival probability.

Description: We study the coupled nonlinear Schrödinger equation in the (2+1)-dimensional inhomogeneous \(\mathcal {PT}\) -symmetric nonlinear couplers and obtain \(\mathcal {PT}\) -symmetric and \(\mathcal {PT}\) -antisymmetric vortex soliton solutions. The dynamical behaviors of the completely localized structures (vortex solitons) are analytically and numerically investigated in a diffraction decreasing system with exponential profile. Numerical results indicate that one vortex soliton with different topological charges can stably propagate a long distance. The space between two humps and the modulation depth of vortex solitons add when the topological charge increases. However, the change tendency of the amplitude and width of vortex solitons is same with the increase in topological charge.

Description: The detection of shaft cracks within operating rotors is an old subject of scientific research. Several signal- and model-based approaches are developed. Here two approaches used for crack detection in rotating machinery, model-based and signal-based approaches, are compared. Strength and weak points are discussed and compared for the two approaches using two representative applicable methods, in order to achieve a comparative overview of these two available techniques. The PI-observer-based method is considered, as modern model-based technique, to give indication about possibilities and limitations of such kind of methods. A novel signal-based approach is introduced, based on SVM and wavelets as an example for a modern machine learning technique. The concepts of severity estimation and service life prediction are investigated in the proposed approach. Furthermore, a brief comparative discussion is presented in the contribution, including ideas for combination of the introduced approaches, in order to achieve more comprehensive and more robust monitoring system applied to the detection of shaft cracks in rotating machines.

Description: This paper proposes a novel approach to study the problem of master–slave synchronization for chaotic delayed Lur’e systems with sampled-data feedback control. Specifically, first, it is assumed that the sampling intervals are randomly variable but bounded. By getting the utmost out of the usable information on the actual sampling pattern and the nonlinear part condition, a newly augmented Lyapunov–Krasovskii functional is constructed via a more general delay-partition approach. Second, in order to obtain less conservative synchronization criteria, a novel integral inequality is developed by the mean of the new adjustable parameters. Third, a longer sampling period is achieved by using a double integral form of Wirtinger-based integral inequality. Finally, three numerical examples with simulations of Chua’s circuit are given to demonstrate the effectiveness and merits of the proposed methods.

Description: In this article, new and general exact traveling wave solutions including soliton-like solutions, triangular-type solutions, single and combined nondegenerate Jacobi elliptic wave function-like solutions, doubly periodic-like solutions are obtained for integrable (2+1)-dimensional Maccari system. This system is frequently introduced to define the motion of the isolated waves, localized in a very small part of space, in many fields such as quantum field theory, hydrodynamics, in plasma physics to describe the behavior of the sonic Langmuir solitons, and also in nonlinear optics. Based on the generalized elliptic equation, an algebraic method is used to construct a series of exact solutions. Being concise and straightforward, the calculations demonstrate the effectiveness and convenience of the method for solving other nonlinear partial differential equations.

Description: A new kind of vibration isolation and mitigation device with high damping is designed and fabricated for reducing dynamic responses of a platform under the excitation with wide frequency range. Tests are carried out to consider the influence of the frequency and amplitude of excitation on the properties of the device. The dynamic model of the platform incorporating with the vibration isolation and mitigation devices is established, and dynamic responses of the mitigated platform are calculated under different excitations. It can be shown from comparison between responses of the mitigated and unmitigated structures in time domain and frequency domain that the devices can significantly reduce dynamic responses of the platform.

Description: The nonlinear longitudinal wave equation, describing the propagation of optical solitons in magneto-electro-elastic circular rod, is investigated analytically. Two integration tools that are traveling wave hypothesis and G \(^{\prime }\) /G expansion scheme are recruited to extract explicit soliton solutions. The existence conditions are derived.

Description: We derive variable separation solutions of the ( \(1+1\) )-dimensional KdV-type model by means of the modified tan h -function method with three different ansätz. Superficially speaking, the positive and negative power-symmetric ansatz seems to be better than the positive-power ansatz and radical sign combined ansatz because the positive and negative power-symmetric ansatz can construct the most forms of variable separation solutions. Actually, we find that various “different” solutions obtained by the modified tanh-function method are not independent, and many of the so-called new solutions are equivalent to one another. Moreover, in the two- or multi-component system, if we construct localized coherent structures for a special component based on variable separation solutions, we must note the structure constructed by the other component for the same equation in order to avoid the appearance of some divergent and un-physical structures. We hope that these results above contribute to the analysis on exact solutions and the construction of localized structures for other nonlinear models.

Description: This work presents an analytical study of dynamic behaviors of solitons in the electrical transmission line. The nonlinear dynamical model that is the modified Zakharov–Kuznetsov equation with an external force is investigated. Via the Riccati equation mapping scheme, explicit twenty seven traveling wave solutions, which include periodic solutions, rational wave solution, soliton solutions as well as soliton-like solutions, are constructed for the first time.

Description: Many modern applications of the flexible multibody systems require formulations that can effectively solve problems that include large displacements and deformations having the ability to model nonlinear materials. One method that allows dealing with such systems is continuum-based absolute nodal coordinate formulation (ANCF). The objective of this study is to formulate an efficient method of modeling nonlinear nearly incompressible materials with polynomial Mooney–Rivlin models and volumetric energy penalty function in the framework of the ANCF. The main part of this paper is dedicated to the examination of several ANCF fully parameterized beam elements under incompressible regime. Moreover, two volumetric suppression methods, originating in the finite element analysis, are proposed: a well-known selective reduced integration and F -bar projection. It is also presented that the use of these methods is crucial for performing reliable analysis of models with bending-dominated loads when lower-order elements are employed. The results of the simulations carried on with considered elements and proposed methods are compared with the results obtained from commercial finite element package and existing ANCF implementation. The results show important improvement as compared with previous applications and good agreement with reference results.

Description: In this paper, we focus on discussing the fractional Noether symmetries and fractional conserved quantities for non-conservative Hamilton system with time delay. Firstly, the fractional Hamilton canonical equations of non-conservative system with time delay are established; secondly, based upon the invariance of the fractional Hamilton action with time delay under the infinitesimal transformations of group, we obtain the definitions and criterion of fractional Noether symmetric transformations, fractional Noether quasi-symmetric transformations and fractional generalized Noether quasi-symmetric transformations in phase space; finally, the relationship between the fractional Noether symmetries and fractional conserved quantities with time delay in phase space is established. At the end of the paper, some examples are given to illustrate the application of the results.

Description: Descriptor systems, which are also called differential-algebraic systems, singular systems, degenerate systems, constrained systems and so on, have been one of the major research topics in engineering field. However, problems related to the local stability and the Hopf bifurcation of nonlinear descriptor systems with time delay have not been thoroughly investigated. In this paper, we consider the dynamical behavior of two-dimensional nonlinear descriptor systems with time delay. First, local stability of the system is analyzed using the location of the roots of the characteristic equation of the corresponding linearized system. It is well known that the equilibrium of the linearized system is locally asymptotically stable if all roots of the corresponding characteristic equation locate in the left half of the complex plane, and they are uniformly bounded away from the imaginary axis. Otherwise, the equilibrium is unstable. The conditions for the existence of Hopf bifurcation are investigated in detail by using the time delay as a bifurcation parameter. Furthermore, based on the descriptor and the “neutral-type” model transformation, some special neutral differential equations can be equivalently transformed into the nonlinear descriptor systems. Finally, the correctness and the effectiveness of the theoretical analysis are justified by numerical examples.

Description: A 3D jerk system with only one stable equilibria was presented and discussed. Some periodic orbits and chaotic behaviors of this system are obtained. Meanwhile, a delayed feedback control scheme for this system was proposed. By using the method of projection for center manifold computation, Hopf bifurcation for the delayed feedback control system was analyzed and obtained. The simulation results demonstrate the correctness of the Hopf bifurcation analysis and the effectiveness of the proposed delayed feedback control strategy.

Description: In this paper, two simple 3D chaotic systems are constructed by introducing nonlinear functions into Sprott A system, which all have no equilibrium, and multi-scroll hidden attractors can be obtained. In the case of simple sine function without restricting its nonlinear dynamical range, the number of multi-scroll attractors has nothing to do with the system equilibria, which is only determined by the transient simulation time. In the other case, the introduced nonlinear function is composed of nonlinear part and linear part; thus, the nonlinearity of improved Sprott A system is limited in certain range, as a result, the number of multi-scroll hidden attractors can be selected arbitrarily within finite transient simulation time. Furthermore, the dynamical properties of two systems are studied through phase plane, time series, Poincaré map and frequency spectra. Finally, an electronic circuit of improved Sprott A system is implemented in Pspice, and the results of electronic circuit are consistent with that of the numerical simulation.

Description: Theoretical analysis and experimental verification of nonlinear monostable energy harvesters are presented to enhance energy harvesting performance from low-level ambient vibrations. Harmonic balance analysis of these harvesters demonstrates that there are two stable orbits (high-branch orbit and low-branch orbit) in a specific multi-solution range when the excitation level is larger than a threshold. Under the same excitation, these nonlinear harvesters may finally vibrate at different orbits depending on different initial conditions or disturbances. The energy harvesting efficiency will be greatly improved if high-branch oscillations could be induced in the multi-solution range. Numerical investigations into the influence of a disturbance method and initial conditions on high-branch oscillations are performed for energy harvesting enhancement of nonlinear monostable energy harvesters. Experimental results verify the effectiveness of adding disturbances by using an impact method for improving energy harvesting performance and the capability of nonlinear monostable energy harvesters in maintaining high-branch oscillations at low-level vibrations.

Description: This paper investigates the event-driven observer-based output control of networked control systems. By introducing a tuning parameter and a weighting matrix, an extended event-driven threshold is proposed. To obtain the controller and observer gains in a convex manner, a constructive strategy is employed to extract the controller matrix coupled with Lyapunov variables and system matrices. Based on these approaches, a sufficient condition for the closed-loop system to be the global uniform ultimate boundedness is ensured in terms of linear matrix inequalities. The validity of the proposed method is illustrated via a numerical example.

Description: In an attempt to investigate the nonlinear dynamic mechanism of financial market microstructure, a stochastic interacting epidemic system is applied to establish an agent-based financial price dynamics, in which the spread of viruses and the physical condition of humans in the interacting epidemic system are, respectively, utilized to imitate the dispersal of the information and the investment attitude of the investors in a stock market. Combined with the ensemble empirical mode decomposition, the composite multiscale entropy analysis is applied to analyze the fluctuation complexity of financial time series, including the proposed model data and seven real stock indices. Further, the Zipf fluctuation behaviors of these time series are also investigated. The comparatively empirical results of the real indices and the simulation data show the similar fluctuation behaviors, indicating that this agent-based financial model can imitate some important properties of stock markets.

Description: This paper presents an inductor-free memristive circuit, which is implemented by linearly coupling an active band-pass filter (BPF) with a parallel memristor and capacitor filter. Mathematical model is established, and numerical simulations are performed. The results verified by hardware experiments show that the active BPF-based memristive circuit exhibits the dynamical behaviors of point, period, chaos, and period-doubling bifurcation route. Most important of all, the newly proposed memristive circuit has a line equilibrium and its stability closely relies on memristor initial condition, which results in the emergence of extreme multistability. Stability distribution related to memristor initial condition is numerically estimated and the coexistence of infinitely many attractors is intuitively captured by numerical simulations and PSIM circuit simulations.

Description: The relationship of equations of motion of a Lagrangian \(\phi (L)\) to those of L is studied in the non-autonomous case, and the question of the existence of a function \(\phi \) such that \(\phi (L)\) is dynamically equivalent to L is answered.

Description: The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, unforced cubic–quintic Duffing oscillator is solved exactly, providing exact expressions for the period and the solution. The period is given in terms of the complete elliptic integral of the first kind and the solution involves Jacobi elliptic functions. Some particular cases obtained varying the parameters that characterize this oscillator are presented and discussed. The behaviour of the period as a function of the initial amplitude is analysed, the exact solutions and velocities for several values of the initial amplitude are plotted, and the Fourier series expansions for the exact solutions are also obtained. All this allows us to conclude that the quintic term appearing in the cubic–quintic Duffing equation makes this nonlinear oscillator not only more complex but also more interesting to study.

Description: We study the dynamics of a particle on an elastic surface carrying a harmonic traveling wave. The particle–surface interaction is assumed to be inelastic both normally and tangentially, and the surface is assumed to provide a kinematic boundary condition to the particle. In this setting, we search for periodic hopping solutions of the particle. The results point to a rich variety of possible particle trajectories which include periodic and chaotic motions. Interestingly, coexisting distinct multi-period attractors, (one with period-3 and its multiples, and including coexisting periodic–chaotic attractors) are observed over a frequency band. In the case of a mutually non-interacting multi-particle system, we observe ballistic and diffusive transport in inter-leaved frequency bands. These results indicate tunability of transport without/with mixing.

Description: When synchronizing chaotic systems, an interesting phenomenon that may occur is the coexistence of several synchronization types. Referring to discrete-time chaotic (hyperchaotic) systems, this paper presents a new approach to rigorously study the coexistence of some synchronization types between maps with different dimensions. By exploiting a Lyapunov-based approach, the paper first analyzes the coexistence of full state hybrid projective synchronization and generalized synchronization when the drive system is a three-dimensional map and the response system is a two-dimensional map. Successively, the coexistence of three different synchronization types is illustrated, i.e., inverse projective synchronization, inverse generalized synchronization and Q–S synchronization are proved to coexist between two-dimensional drive systems and three-dimensional response systems. Finally, by exploiting a pole placement technique, the coexistence of antiphase synchronization and dislocated full state hybrid projective synchronization is proved to be achieved between three-dimensional drive system maps and two-dimensional response system maps. Since synchronization is achieved in finite time, this last case represents an example of dead beat coexistence of some synchronization types in maps with different dimensions. Several numerical examples of coexistence of synchronization types are illustrated, with the aim to show the effectiveness of the novel approach developed herein.

Description: Generalizations of fractional derivatives of noninteger orders for N -dimensional Euclidean space are proposed. These fractional derivatives of the Riesz type can be considered as partial derivatives of noninteger orders. In contrast to the usual Riesz derivatives, the suggested derivatives give the usual partial derivatives for integer values of orders. For integer values of orders, the partial fractional derivatives of the Riesz type are equal to the standard partial derivatives of integer orders with respect to coordinate. Fractional generalizations of the nonlinear equations such as sine-Gordon, Boussinesq, Burgers, Korteweg–de Vries and Monge–Ampere equations for nonlocal continuum are considered.

Description: In this paper, to investigate the lateral vibrations of a footbridge under crowd excitation Nakamura’s model and its modified version are studied by using the energy method. The periodic solutions of Nakamura’s model and its modified version are obtained analytically, the efficiency of which is demonstrated by comparing with numerical results. Our analytical solutions are very convenient to apply to the structural calculation and design of footbridges. Our analysis shows that the introduction of the time delay of interaction between pedestrians and the bridge in Nakamura’s model makes the model accord better with the measurement data. From the modified Nakamura’s model, the increase in the time delay decreases the lateral amplitude of a footbridge under crowd excitation, which suggests us that increasing the time delay of the interaction between pedestrians and the bridge maybe a new approach to reduce the lateral vibrations of a footbridge. In addition, our calculations indicate that delay differential equations can be solved more simply by using the energy method than the Galerkin method.

Description: Very recently, the KdV6 was extended to a \((2+1)\) - dimensional equation by a direct way. Here, by using the “ \(\hat{kp}\) -operator” the extension of that equation is done in a rigorous mathematical formulation. Polynomial solutions to the closed form equation are obtained via the unified method. It is found that via nonlinear interactions of traveling waves the solutions of the constructed equation may not be unique. Also, these solutions may show solitary or explosive shock waves.

Description: We consider the wave motion in a partially nonlocal and inhomogeneous nonlinear medium, and a (3+1)-dimensional nonlocal nonlinear Schrödinger equation with variable coefficients is used to govern this dynamics. Based on this model, spatiotemporal Hermite–Gaussian vortex soliton solutions are derived. The evolution behaviors of spatiotemporal Hermite–Gaussian vortex solitons in a diffraction decreasing system are investigated. Results indicate that the topological charge m changes the spiral structures of phase, and its value determines the number of the branch of the spiral phase structures. If the value of parameter n adds, spatiotemporal vortex solitons change their structures. Obviously, the layer of ring solitons along the vertical ( z -axis) direction is decided by \(n+1\) .

Description: This study proposes a new car-following (CF) model considering the effect of visual angle under the non-lane-discipline environment. In particular, a car-following model is proposed to capture the impacts from the visual angle of the driver between the following vehicle and the preceding vehicle as well as its change rate on a road without lane discipline. Stability analysis of the proposed CF model is performed using the perturbation method to obtain the stability condition. Numerical experiments analyze three scenarios: start process, stop process, and evolution process for lane-discipline-based FVD model, non-lane-discipline-based CF model with lateral gap, and the proposed model, respectively. Results from numerical experiments illustrate that the proposed CF model that considers the effects of both lateral gap and visual angle has larger stable region compared with FVD model and the NLBCF model. Also, the responsiveness and smoothness of the proposed CF model is improved with respect to the velocity, and acceleration or deceleration profiles.

Description: The electric activities of neurons are dependent on the complex electrophysiological condition in neuronal system, and it indicates that the complex distribution of electromagnetic field could be detected in the neuronal system. According to the Maxwell electromagnetic induction theorem, the dynamical behavior in electric activity in each neuron can be changed due to the effect of internal bioelectricity of nervous system (e.g., fluctuation of ion concentration inside and outside of cell). As a result, internal fluctuation of electromagnetic field is established and the effect of magnetic flux across the membrane should be considered during the emergence of collective electrical activities and signals propagation among a large set of neurons. In this paper, the variable for magnetic flow is proposed to improve the previous Hindmarsh–Rose neuron model; thus, a four-variable neuron model is designed to describe the effect of electromagnetic induction on neuronal activities. Within the new neuron model, the effect of magnetic flow on membrane potential is described by imposing additive memristive current on the membrane variable, and the memristive current is dependent on the variation of magnetic flow. The dynamics of this modified model is discussed, and multiple modes of electric activities can be observed by changing the initial state, which indicates memory effect of neuronal system. Furthermore, a practical circuit is designed for this improved neuron model, and this model could be suitable for further investigation of electromagnetic radiation on biological neuronal system.

Description: In this paper, the fractional-order generalized Lotka–Volterra (GLV) model and its discretization are investigated qualitatively. A sufficient condition for existence and uniqueness of the solution of the proposed system is shown. Analytical conditions of the stability of the system’s three non-negative steady states are proved. The conditions of the existence of Hopf bifurcation in the fractional-order GLV system are discussed. The necessary conditions for this system to remain chaotic are obtained. Based on the stability theory of fractional-order differential systems, a new control scheme is introduced to stabilize the fractional-order GLV system to its steady states. Furthermore, the analytical conditions of stability of the discretized system are also studied. It is shown that the system’s fractional parameter has effect on the stability of the discretized system which shows rich variety of dynamical analysis such as bifurcations, an attractor crisis and chaotic attractors. Numerical simulations are used to support the analytical results.

Description: This paper proposes an image-based visual servo controller for the quadrotor vertical takeoff and landing unmanned aerial vehicle (UAV). The controller utilizes an estimate of flow of image features as the linear velocity cue and assumes angular velocity and attitude information available for feedback. The image features are selected from perspective image moments and projected on a suitably defined image plane, providing decoupled kinematics for the translational motion. A nonlinear observer is designed to estimate the flow of image features using outputs of visual information. The controller for the translational dynamics is bounded which helps to keep the target points in the field of view of the camera. A smooth asymptotic controller, using the robust integral of the sign of the error method, is designed for the rotational dynamics in order to compensate for the unmodeled dynamics and external disturbances. Furthermore, the proposed approach is robust with respect to unknown image depth through an adaptive scheme and also the yaw information of the UAV is not required. The complete Lyapunov-based stability analysis is presented to show that all states of the system are bounded and the error signals converge to zero. Simulation examples are provided in both nominal and perturbed conditions which show the effectiveness of the proposed theoretical results.

Description: In this paper, we study gait optimization of ballistic walking in order to understand the natural dynamics of an underactuated biped with knees. We also propose applications for our understandings. Our optimization problem is solved by fixing energy levels, and then, we attempt to explain how optimal gaits are formed by examining the role of each joint in speeding up. In addition, we explain some natural characteristics of walking. Based on the results, we propose a new cost function to generate various walking gaits, including the optimum. Finally, we evaluate and discuss the energy efficiency of our ballistic walker and other bipedal walkers including humans.

Description: The homoclinic phenomena in double-layered nanoplates (DLNP) are investigated. Based on the nonlocal continuum theory, the nonlinear dynamical equations for DLNP subjected to in-plane excitation are derived by double-mode Galerkin truncation. The extended Melnikov method is utilized to discuss the homoclinic phenomena and chaotic motion for the buckled DLNP system. The criterions for the existence of transverse homoclinic orbits are established under different four buckling cases (i.e., the first- and second-type synchronous buckling as well as asynchronous buckling). And the results derived by the above-mentioned analysis are verified by molecular dynamics and Lyapunov exponent spectrum. Small-scale effect on the homoclinic motion is mainly inspected. From the result, it is rather novel that the transversality condition is independent of the nonlinear terms in the equations. This fact means that it is not necessary to distinguish whether the boundaries are movable or immovable for the in-plane parametric excitation DLNP. The parametric regime where homoclinic phenomena appear shrinks with the augment of nonlocal parameter for the first-type synchronous buckling case. However, this trend is just opposite for the other three buckling cases. Finally, it can be seen that homoclinic phenomena more likely occur on Mode I (i.e., lower mode) for the second-type synchronous and asynchronous buckling cases. However, the homoclinic motion for the first-type asynchronous buckling most likely takes place on Mode III.

Description: This paper focuses on the attitude tracking control problem of Mars entry vehicle (MEV) with time-varying input delay. The original attitude dynamics of MEV is divided into slow subsystem and fast subsystem. For slow subsystem, the dynamic inversion method is used to generate the angular velocity command. For fast subsystem, a T-S fuzzy model is used to approximate it, and delays-dependent \(H_{\infty }\) attitude tracking control is applied to reduce the effects of delay on attitude dynamics. Specially, a decomposition coefficient of delay integral inequality is introduced in our proposed results, which may further reduce the design algorithm conservatism. Finally, numerical simulations are used to verify the effectiveness of the proposed method.

Description: Discontinuous dynamic characteristics in gear backlash often cause convergence problems when a nonlinear torsional system is simulated. This clearance-type nonlinearity can be mathematically modeled by using smoothening functions which change the dynamic responses of the system under discontinuous ranges into continuous ones. However, the effect of the smoothening functions is not well known and difficult to anticipate under various nonlinear conditions. Thus, a new smoothening function is proposed. The effect and feasibility of the model were investigated with a practical vehicle driveline system. To examine the key factors of the smoothening function, the harmonic balance method was used with an ‘ n ’th order polynomial function and compared with hyperbolic-type smoothening functions. The harmonic balance method and numerical analysis were compared for nonlinear system responses that include much high order of the super-harmonic components with respect to impulsive contact motions to understand the limits of the method. The smoothening function is applicable for simulating gear impact phenomena with limit conditions.

Description: The nonlinear free vibration of a conductive rotating thin circular plate subjected to static loads in magnetic field was investigated. The nonlinear vibration equations of the motion on a spinning circular thin plate were derived. According to the set of a displacement function, the approximate solution of the static problem and the magneto elastic axisymmetric vibration differential equation of the round plate were obtained through the application of Galerkin integral method. The characteristic formula between nonlinear frequency and amplitude using the static load as a parameter was obtained by the method of multiple scales. According to the numerical calculation, the characteristic curves of the relationship between frequency and the parameter such as the static load, the magnetic induction intensity and the speed of rotation were obtained. The nonlinear vibration characteristics of a spinning plate with clamped boundary condition were described according to the curves drawn by Matlab. We also showed that different parameters (including magnetic induction intensity, static load, time and spinning velocity.) have effect on the frequency of the plate.

Description: In the paper, an improved car-following model based on the full velocity difference model considering the influence of optimal velocity for leading vehicle on a single-lane road is proposed. The linear stability condition of the model is obtained by applying the linear stability theory. Through nonlinear analysis, the time-dependent Ginzburg–Landau (TDGL) equation and the modified Korteweg–de Vries (mKdV) equation are derived to describe the traffic flow near the critical point. In addition, the connection between the TDGL and the mKdV equations is also given. Good agreement between the simulation and the theoretical results shows that the improved model can be enhanced the stability of traffic flow.

Description: In this paper, the impact angle constrained guidance problem, focusing on the nonlinear engagement dynamics, is considered. In order to fulfill the terminal constraints, a novel function-based finite-time sliding mode control methodology is introduced to design the guidance law. The main feature of this guidance law is that it achieves the desired impact angle exactly at the time of interception. In addition, it is capable of ensuring a continuous and smooth control action. To improve the tolerance of initial heading errors and broaden the application, a feasible guidance logic is also developed. Numerical simulations in various scenarios have shown that the proposed guidance scheme can realize all-aspect interception against stationary, constant-velocity and maneuvering targets. Furthermore, comparison studies with respect to nonsingular terminal sliding mode control-based methods demonstrate the superiority of the proposed method.

Description: In this paper, we study the number of limit cycles of a class of quartic Liénard systems with polynomial perturbations of degree n , \(20\le n\le 24\) . Let H ( n , 4) denote the maximal number of limit cycles of Liénard system \({\dot{x}}=y,\, {\dot{y}}=-g(x)-\varepsilon f(x)y\) , where \(\varepsilon \ge 0\) is a small parameter, f ( x ), g ( x ) are polynomials in x and \(\deg f=n\) , \(\deg g=m\) . We obtain five better lower bounds of H ( n , 4) for \(20\le n\le 24\) , which greatly improve the existing results.

Description: In this paper, the adaptive robust tracking and model following controller is designed for the uncertain nonlinear systems with disturbances, time-delays and input nonlinearities. A novel control law based on the linear matrix inequalities is proposed for the tracking control of the nonlinear systems with Lipschitz nonlinearities. The selection procedure of the nonlinear integral sliding surface and the existence of the proposed sliding mode are two significant subjects, which have been addressed. This controller guarantees that the tracking error decreases asymptotically to the origin in the presence of time-varying perturbations. By elimination of the reaching mode of the sliding surface, the robustness performance against perturbations is guaranteed right from the initial instant. Furthermore, the knowledge of the upper bounds of the uncertainties, disturbances and time-delays is not required to be determined. Simulation results are provided to illustrate the efficiency of the presented technique compared to the previous methods.

Description: This study proposes a new car-following model considering the effect of electronic throttle opening angle to capture the characteristics of connected autonomous vehicular traffic flow. The proposed model incorporates the opening angle of electronic throttle into the full velocity difference (FVD) model by assuming that the information of electronic throttle dynamics is shared by vehicles through vehicle-to-vehicle communications. The stability condition of the proposed car-following model is obtained using the perturbation method. Numerical experiments are constructed on three scenarios, start, stop, and evolution processes, to analyze the vehicular traffic flow characteristics of the proposed model. The results of numerical experiments illustrate that the proposed car-following model has a larger stable region compared with FVD model. Also, it demonstrates that the proposed car-following model can better present the characteristics of connected and autonomous vehicular flow in terms of the smoothness and stability with respect to the space headway, position, velocity, and acceleration/deceleration profiles.

Description: In this study, Lie’s invariant analysis method is applied to nonlinear time fractional Qiao equation with the Riemann–Liouville derivative. Fractional derivative value \(\alpha \) is considered for three cases and each cases symmetry generators are given. Using these generators group-invariant solutions in explicit form are obtained.