ALBERT

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.