Rich dynamics caused by diffusion

It is an awfully difficult task to design an efficient numerical method for bifurcation diagrams, the graphs of Lyapunov exponents, or the topological entropy about discrete dynamical systems by linear/nonlinear diffusion with the Direchlet/Neumann- boundary conditions. Until now there are less works concerned with such a problem. In this paper, we propose a scheme about bifurcating analysis in a series of discrete-time dynamical systems with linear/nonlinear diffusion terms under the periodic boundary conditions. The complexity of dynamical behaviors caused by the diffusion term are to be determined. Bifurcation diagrams are shown by numerical simulation and chaotic behavior (chaotic Turing patterns) is demonstrated by computing the largest Lyapunov exponent. Our theoretical model can give an interesting case study about the phenomenon: the individuals exhibit a very simple dynamics but the groups with linear/nonlinear coupling can own a complex dynamics including fluctuation, periodicity and even chaotic behavior. We find that diffusion can trigger chaotic behavior in the present system and there exist multiple Turing patterns. It is interesting as regular or chaotic patterns can be reported in this study. Chaotic orbits emerge when exploring further in the diffusion coefficient space, and such a behavior is entirely absent in the corresponding continuous time-space system. The method proposed in the present paper is innovative and the conclusion is novel.

     

Single amplifier biquad based autonomous electronic oscillators for chaos generation

Two simple autonomous chaotic electronic circuits have been proposed in this paper. The core of each of the circuits consists of a single amplifier biquad (SAB). We have proposed two configurations of converting this SAB into chaotic oscillators using suitable passive nonlinear element and a storage element in the form of an inductor. The mathematical models of the proposed chaotic circuits have been constructed, which are fourth order autonomous nonlinear differential equations. The behavior of the proposed circuits has been investigated through numerical simulations, Spice-based circuit simulations and electronic hardware experiments and they agree well with each other. It has been found that both the circuits show complex behaviors like bifurcations and chaos for a certain range of circuit parameters.     

周期激励下广义蔡氏电路混沌运动中的概周期行为

由于广义蔡氏电路存在2个对称的稳定平衡点,周期激励可能导致系统出现相应于不同初值的2种共存的分岔模式. 概周期解由环面破裂进入混沌,混沌吸引子从相位不同步逐渐演化为同步,并进一步随着参数的变化,产生分裂现象. 分裂后的2个相互对称的混沌吸引子仍存在相位同步效应,这2个混沌吸引子再次相互作用后形成扩大了的混沌吸引子,并交替围绕2个子混沌结构来回振荡. 同时,在混沌过程中,其轨迹在相当长的一段时间内严格按照概周期行为振荡,即混沌结构中存在局部概周期行为,这种局部概周期行为随参数的变化会逐步减弱,直至消失.      

Chaotification for switched linear systems with controllers

This paper shows that two or more switched linear systems can generate chaotic dynamical behaviors by an appropriate switching rule as they at least consist of a controllable system and an unstable system with the expanding property. According to the results in the reference （Xie, L. L., Zhou, Y., and Zhao, Y. Criterion of chaos for switched linear systems with antrollers. International Journal of Bifurcation and Chaos, 20（12）, 4105-4109 （2010））, a nonlinear feedback gain is needed to generate chaotic dy- namics. A linear feedback control is usually used to approximate the nonlinear one for simulation. In order to obtain the exact control, as a main result of this paper, the con- troller is constructed by Russell＇s result, and a block diagram is included to interpret the realization of the controller. Numerical simulations are given to illustrate the generated chaotic dynamical behavior of the switched linear systems with some parameters and show the effects of the constructed controller.     

Generating tri-chaos attractors with three positive Lyapunov exponents in new four order system via linear coupling

This paper presents a new hyperchaotic system with three positive Lyapunov exponents (called Tri-Chaos). Via linear coupling, Mathieu, and van der Pol systems are coupled with each other and then become a new four order system??Mathieu?Cvan der Pol autonomous system. As we know, two positive Lyapunov exponents confirm hyperchaotic nature of its dynamics and it means that the system can present more complicated behavior than ordinary chaos. We further generate three positive Lyapunov exponents in a new coupled nonlinear system and anticipate the advanced application in secure communication. Not only a new chaotic system with three Lyapunov exponents is proposed, but also its implementation of an electronic circuit is put into practice in this article. The phase portrait, electronic circuit, power spectrum, Lyapunov exponents, and 2-D and 3-D parameter diagram of tri-chaos with three positive Lyapunov exponents of the new system will be shown in this paper.     

周期激励下分段线性电路的动力学行为

基于四阶自治分段线性电路的分岔特性,探讨了两种幅值周期激励下该电路系统的复杂动力学行为. 给出了弱周期激励下系统共存的两种分岔模式及其产生的原因,讨论了不同分岔模式下动力学行为的演化过程及混沌吸引子相互作用机理. 而随着激励幅值的增大,即强激励作用下,围绕两个原自治系统平衡点的周期轨道不再分裂,从而导致共存的分岔模式消失.指出无论在弱激励还是在强激励下,由于系统的固有频率与外激励频率存在量级上的差距,其相应的各种运动模式,诸如周期运动、概周期运动甚至混沌运动均表现出明显的快慢效应,进而从分岔的角度分析了不同快慢效应的产生机制.      

Distinguishing Periodic and Chaotic Time Series Obtained from an Experimental Nonlinear Pendulum

The experimental analysis of nonlinear dynamical systems furnishes ascalar sequence of measurements, which may be analyzed using state spacereconstruction and other techniques related to nonlinear analysis. Thenoise contamination is unavoidable in cases of data acquisition and,therefore, it is important to recognize techniques that can be employedfor a correct identification of chaos. The present contributiondiscusses the experimental analysis of a nonlinear pendulum, consideringstate space reconstruction, frequency domain analysis and thedetermination of dynamical invariants, Lyapunov exponents and attractordimension. A procedure to construct Poincaré map of the signal ispresented. The analyses of periodic and chaotic motions are carried outin order to establish a difference between them. Results show that it ispossible to distinguish periodic and chaotic time series obtained froman experimental set up employing proper procedures even though noisesuppression is not contemplated.     

Implementation and study of the nonlinear dynamics of a memristor-based Duffing oscillator

An electronic model of Duffing oscillator with a characteristic memristive nonlinear element is proposed instead of the classical cubic nonlinearity. The memristive Duffing oscillator circuit system is mathematically modeled, and the stability analysis presents the evolution of the proposed system. The dynamical behavior of this circuit is investigated through numerical simulations, statistical analysis, and real-time hardware experiments, which have been carried out under the external periodic force. The chaotic dynamics of the circuit is studied by means of phase diagram. It is found that the proposed circuit system shows complex behaviors, like bifurcations and chaos, three tori, transient chaos, and intermittency for a certain range of circuit parameters. The observed phenomena and scenario are illustrated in detail through experimental and numerical studies of memristive Duffing oscillator circuit. The existence of regular and chaotic behaviors is also verified by using 0–1 test measurements. In addition, the robustness of the signal strength is confirmed through signal-to-noise ratio. The numerically observed results are confirmed from the laboratory experiment.     

Analog circuit design and optimal synchronization of a modified Rayleigh system

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.     

Chaotic behavior of a parametrically excited nonlinear mechanical system

The chaotic dynamics of a single-degree-of-freedom nonlinear mechanical system under periodic parametric excitation is investigated. Besides the well known type-I and type-III intermittent transitions to chaos we give numerical evidence that the system can follow an alternative route to chaos via intermittency from an equilibrium state to a chaotic one, which was not found in the previous simulations of the dynamics of the system.     

Determination of the Chaos Onset in Mechanical Systems with Several Equilibrium Positions

Determination of the chaos onset in some mechanical systems with several equilibrium positions are analyzed. Namely, the snap-through truss and the oscillator with a nonlinear dissipation force, under the external periodical excitation, are considered. Two approaches are used for the chaos onset determination. First, Padé and quasi-Padé approximants are used to construct closed homoclinic trajectories for a case of small dissipation. Convergence condition used earlier in the theory of nonlinear normal vibration modes as well conditions at infinity make possible to evaluate initial amplitude values for the trajectories with admissible precision. Mutual instability of phase trajectories is used as criterion of chaotic behavior in nonlinear systems for a case of not small dissipation. The numerical realization of the Lyapunov stability definition gives us a possibility to observe a process of appearance and fast enlargement of the chaotic behavior regions if some selected parameters of the dynamical systems under consideration are changing.     

Nonlinear characteristic of a circular composite plate energy harvester: experiments and simulations

Conservative chaotic systems are rare, especially autonomous smooth dynamical systems. This paper reports two four-dimensional (4D) autonomous conservative systems. The conservation of these two systems has been verified using the trace of Jacobian matrix, perpetual point theory and Hamiltonian energy theory. Numerical analyses, including phase portrait, Poincaré section, Lyapunov exponent spectrum and bifurcation diagram, verify the existence of the chaotic and quasiperiodic flows. Moreover, a electronic circuit in Multisim is built to demonstrate their chaotic dynamics, whose circuit experimental results agree well with the numerical results.     

Chaotic behavior in fractional-order memristor-based simplest chaotic circuit using fourth degree polynomial

In this paper, a memristor with a fourth degree polynomial memristance function is used in the simplest chaotic circuit which has only three circuit elements: a linear passive inductor, a linear passive capacitor, and a nonlinear active memristor. We use second order exponent internal state memristor function and fourth degree polynomial memristance function to increase complexity of the chaos. So, the system can generate double-scroll attractor and four-scroll attractor. Systematic studies of chaotic behavior in the integer-order and fractional-order systems are performed using phase portraits, bifurcation diagrams, Lyapunov exponents, and stability analysis. Simulation results show that both integer-order and fractional-order systems exhibit chaotic behavior over a range of control parameters.     

Conductance and Noncommutative Dynamical Systems

In this work, we introduce the notion of conductance in the context of Cuntz–Krieger C^∗-algebras. These algebras can be seen as a noncommutative version of topological Markov chains. Conductance is a useful notion in the theory of Markov chains to study the approach of a system to the equilibrium state. Our goal is twofold. On one hand, conductance can be used to measure the complexity of dynamical systems, complementing topological entropy. On the other hand, using C^∗-algebras, we can give a natural framework to analyze the path space of a finite graph associated to a Markov shift.     

DC-DC开关功率变换器的非线性动力学行为研究

DC-DC开关功率变换器是一种典型的分段光滑动力学系统, 在一定的工作和参数条件下, 系统会出现各种分岔如倍周期分岔、Hopf分岔、边界碰撞分岔和混沌运动. 系统评述了DC-DC开关功率变换器的非线性动力学行为的研究进展;介绍了离散非线性映射、分段线性模型、平均值模型等3种建模方法;分析了这种电路系统中的分岔特点及通向混沌的途径与机制;结合我们的研究工作, 讨论了对这种电路系统进行混沌控制的必要性及相关策略;最后, 从应用的角度提出了未来的若干研究方向.      

A novel non-equilibrium fractional-order chaotic system and its complete synchronization by circuit implementation

In this paper, we construct a novel four dimensional fractional-order chaotic system. Compared with all the proposed chaotic systems until now, the biggest difference and most attractive place is that there exists no equilibrium point in this system. Those rigorous approaches, i.e., Melnikov??s and Shilnikov??s methods, fail to mathematically prove the existence of chaos in this kind of system under some parameters. To reconcile this awkward situation, we resort to circuit simulation experiment to accomplish this task. Before this, we use improved version of the Adams?CBashforth?CMoulton numerical algorithm to calculate this fractional-order chaotic system and show that the proposed fractional-order system with the order as low as 3.28 exhibits a chaotic attractor. Then an electronic circuit is designed for order q=0.9, from which we can observe that chaotic attractor does exist in this fractional-order system. Furthermore, based on the final value theorem of the Laplace transformation, synchronization of two novel fractional-order chaotic systems with the help of one-way coupling method is realized for order q=0.9. An electronic circuit is designed for hardware implementation to synchronize two novel fractional-order chaotic systems for the same order. The results for numerical simulations and circuit experiments are in very good agreement with each other, thus proving that chaos exists indeed in the proposed fractional-order system and the one-way coupling synchronization method is very effective to this system.     

Bifurcations and chaos of an inclined cable

The nonlinear behavior of an inclined cable subjected to a harmonic excitation is investigated in this paper. The Galerkin’s method is applied to the partial differential governing equations to obtain a two-degree-of-freedom nonlinear system subjected to harmonic excitation. The nonlinear systems in the presence of both external and 1:1 internal resonances are transformed to the averaged equations by using the method of averaging. The averaged equations are numerically examined to obtain the steady-state responses and chaotic solutions. Five cascades of period-doubling bifurcations leading to chaotic solutions, 3-periodic solutions leading to chaotic solution, boundary crisis phenomena, as well as the Shilnikov mechanism for chaos, are observed. In order to study the global dynamics of an inclined cable, after determining the averaged equations of motion in a suitable form, a new global perturbation technique developed by Kova?i? and Wiggins is used. This technique provides analytical results for the critical parameter values at which the dynamical system, through the Shilnikov type homoclinic orbits, possesses a Smale horseshoe type of chaos.     

The chaotic dynamics of the social behavior selection networks in crowd simulation

This paper researches the nonlinear dynamics of the behavior selection networks (BSN) model by virtue of which we can understand the origin of flocking behaviors in social networks. To commentate the notion of BSN, this article introduces a social behavior selection model for evolutionary dynamics of behaviors in social networks that exhibits a rich set of emergent behaviors of evolution. For behavioral networks with different complex networks topology, we analyze the nonlinear dynamics including the chaotic dynamics by the numerical simulation tools. With changing the topological structure, the behavioral networks behave affluent dynamical phenomena. Lastly, we draw the conclusion and paste the prospection about the networks model.