Qualitative behaviors of the continuous-time chaotic dynamical systems describing the interaction of waves in plasma

In this paper, a new Lorenz-like chaotic system describing the interaction of three resonantly coupled waves in plasma is studied. Explicit ultimate boundedness and global attraction domain are derived according to stability theory of dynamical systems. The innovation of the paper is that this paper not only proves this chaotic system is globally bounded for the parameters of this system but also gives a family of mathematical expressions of global exponential attractive sets for this system with respect to the parameters of this system. Furthermore, the exponential rate of the trajectories is also obtained. Finally, numerical localization of attractor is presented.     

Spatial behavior in the electromagnetic theory of microstretch elasticity

This paper is concerned with the electromagnetic theory of microstretch elasticity. First, the initial boundary value problem is formulated in the framework of the linear dynamic theory of microstretch magnetoelectroelastic solids. Then, the spatial behavior of solutions is studied in both bounded and unbounded regions. The obtained result gives an exact idea of the domain of influence, in the sense that for each fixed time in a given interval, the entire activity vanishes at distanced from the support of the given data greater than a time-dependent threshold value. The study of spatial behavior is completed by an exponential decay estimate inside the domain of influence. As a by product a uniqueness result holding for both bounded and unbounded bodies is derived. Finally, the effect of a concentrated microstretch body force is studied.     

有界噪声激励下Josephson系统的混沌运动

利用随机Melnikov方法分析了有界噪声激励下Josephson系统的运动,并运用均方准则得到了系统产生混沌的临界值。结果表明:有界噪声对系统混沌行为的产生起到了加速的作用;且有界噪声的强度越大,混沌吸引子的发散程度就越大。最后利用数值模拟得到系统的庞加莱映射,分析了在不同参数组合下系统庞加莱映射的特征。结果显示:当有界噪声中的一个参数发生改变,系统的庞加莱映射也会发生相应的改变;特别是有界噪声的激励强度增大时,系统庞加莱映射的发散程度也会随之增大。这从侧面验证了理论结果的正确性。     

Active coupling and its circuitry designs of chaotic systems against deteriorated and delayed networks

This paper is concerned with the problem of asymptotic synchronization of a class of chaotic systems in the presence of network deterioration and time-varying delays. Based on adaptive adjustment technique and circuitry principle, a new version of the active coupling as well as its circuit realization is proposed. Then, an approach that is based on application of Lyapunov stability theory for the synchronization error system is introduced to prove the asymptotic synchronization result of the overall chaotic system. Moreover, a condition which denotes that at least one coupling will not be deteriorated for synchronization of the network is provided in the paper. It is shown that, without control inputs, the result can also be established for the deteriorated coupling networks and any time-varying bounded delay under the topological structure satisfying the condition. Finally, the proposed active couplings are physically implemented by circuits and tested by simulation on a Chua??s circuit network.     

A novel bounded 4D chaotic system

This paper presents a novel bounded four-dimensional (4D) chaotic system which can display hyperchaos, chaos, quasiperiodic and periodic behaviors, and may have a unique equilibrium, three equilibria and five equilibria for the different system parameters. Numerical simulation shows that the chaotic attractors of the new system exhibit very strange shapes which are distinctly different from those of the existing chaotic attractors. In addition, we investigate the ultimate bound and positively invariant set for the new system based on the Lyapunov function method, and obtain a hyperelliptic estimate of it for the system with certain parameters.     

Delayed feedback control and bifurcation analysis of Rossler chaotic system

In this paper, from the view of stability and chaos control, we investigate the Rossler chaotic system with delayed feedback. At first, we consider the stability of one of the fixed points, verifying that Hopf bifurcation occurs as delay crosses some critical values. Then, for determining the stability and direction of Hopf bifurcation we derive explicit formulae by using the normal-form theory and center manifold theorem. By designing appropriate feedback strength and delay, one of the unstable equilibria of the Rossler chaotic system can be controlled to be stable, or stable bifurcating periodic solutions occur at the neighborhood of the equilibrium. Finally, some numerical simulations are carried out to support the analytic results.     

Design and analysis of a first order time-delayed chaotic system

The present paper reports the design and analysis of a new time-delayed chaotic system and its electronic circuit implementation. The system is described by a first-order nonlinear retarded type delay differential equation with a closed form mathematical function describing the nonlinearity. We carry out stability and bifurcation analysis to show that with the suitable delay and system parameters the system shows sustained oscillation through supercritical Hopf bifurcation. It is shown through numerical simulations that the system depicts bifurcation and chaos for a certain range of the system parameters. The complexity and predictability of the system are characterized by Lyapunov exponents and Kaplan?CYork dimension. It is shown that, for some suitably chosen system parameters, the system shows hyperchaos even for a small or moderate delay. Finally, we set up an experiment to implement the proposed system in electronic circuit using off-the-shelf circuit elements, and it is shown that the behavior of the time delay chaotic electronic circuit agrees well with our analytical and numerical results.     

A new chaotic attractor and its robust function projective synchronization

We introduce a simple chaotic system that contains one multiplier and one quadratic term. The system is similar to the generalized Lorenz system but is not topologically equivalent. The properties of the proposed chaotic system are examined by theoretical and numerical analysis. An analog chaotic circuit is implemented that realizes the chaotic system for the verification of its attractor. Furthermore, we propose a robust function projective synchronization using time delay estimation. A numerical simulation of synchronization between the proposed system and the Lorenz system demonstrates that the proposed approach provides fast and robust synchronization even in the presence of unknown parameter variations and disturbances.     

Noise-induced chaos in the elastic forced oscillators with real-power damping force

The chaotic behavior of the elastic forced oscillators with real-power exponents of damping and restoring force terms under bounded noise is investigated. By using random Melnikov method, a mean square criterion is used to detect the necessary conditions for chaotic motion of this stochastic system. The results show that the threshold of bounded noise amplitude for the onset of chaos in the system increases as the intensity of the random frequency increases, and decrease as the real-power exponent of damping term increase. The threshold of bounded noise amplitude for the onset of chaos is determined by the numerical calculation via the largest Lyapunov exponents. The effects of bounded noise and real-power exponent of damping term on bifurcation and Poincaré map are also investigated. Our results may provide a valuable guidance for understanding the effect of bounded noise on a class of generalized double well system.     

Adaptive intelligence learning for nonlinear chaotic systems

In this paper, a reinforcement learning algorithm is proposed for a class of nonlinear differential chaotic systems. The nonlinear function of the chaotic systems is assumed to be bounded but the bounds are unknown. The unknown bounds need to be on-line adjusted. An adaptive optimal (or near optimal) control input with the reinforcement signal can be obtained compared with the current adaptive control for chaotic systems. The reinforcement signal is approximated by the neural networks. Based on Lyapunov analysis theory and by using Young’s inequalities, the closed-loop system is guaranteed to be stable. Finally, the simulation results are given to illustrate the effectiveness of the approach.     

Spatial Behavior and Continuous Dependence Results in the Linear Dynamic Theory of Magnetoelectroelasticity

This paper is concerned with some general theorems for the linear dynamic theory of magnetoelectroelasticity. First, the spatial behavior of solutions is studied. In this sense, the so called ??support?? of the given data in a fixed interval of time [0,T] is introduced and an adequate time-weighted surface power function associated with the solution at issue is considered. Using their properties, we get the domain of influence and an exponential decay estimate with time-independent rates inside the domain of influence. As a by-product, a uniqueness result is derived for both bounded and unbounded bodies. Then, the case of non-zero initial conditions is also considered. Under a boundedness restriction on the initial data, an energy estimate is obtained. Finally, a continuous dependence result of solutions upon initial data and body forces is established.     

COMPLEX RELAXATION OSCILLATION TRIGGERED BY BOUNDARY CRISIS IN THE DISCRETE DUFFING MAP

多时间尺度问题具有广泛的工程与科学研究背景，慢变参数则是多时间尺度问题的典型标志之一.然而现有文献所报道的慢变参数问题，其展现出的振荡形式及内部分岔结构，大多较为单一，此外少有文献涉及到混沌激变的现象.本文以含慢变周期激励的达芬映射为例，探讨了一类具有复杂分岔结构的张弛振荡.快子系统的分岔表现为S形不动点曲线，其上、下稳定支可经由倍周期分岔通向混沌.而在一定的参数条件下，存在着导致混沌吸引子突然消失的一对临界参数值.当分岔参数达到此临界值时，混沌吸引子可能与不稳定不动点相接触，也可能与之相距一定距离.对快子系统吸引域分布的模拟，表明存在着导致边界激变（boundary crisis）的临界值，在这些值附近，经由延迟倍周期分岔演化而来的混沌吸引子可与2ⁿ（n=0，1，2，…）周期轨道乃至混沌吸引子共存.当慢变量周期地穿过临界点后，双稳态的消失导致原本处于混沌轨道的轨线对称地向此前共存的吸引子转迁，从而使系统出现了不同吸引子之间的滞后行为，由此产生了由边界激变所诱发的多种对称式张弛振荡.本文的结果丰富了对离散系统的多时间尺度动力学机理的认识.     

Spatio-temporal sine-Wiener bounded noise and its effect on Ginzburg–Landau model

In this work we introduce a spatio-temporal bounded noise derived by the sine-Wiener noise and by the spatially colored unbounded noise proposed by García-Ojalvo, Sancho, and Ramírez-Piscina (GSR noise). We characterize the behavior of the equilibrium distribution of this novel noise by showing its dependence on both the temporal and the spatial autocorrelation lengths. In particular, we show that the distribution experiences a stochastic transition from bimodality to trimodality. Then, we employ the noise here defined to study the emergence of phase transitions in the real Ginzburg–Landau model. Various phenomena are evidenced by means of numerical simulations, among which reentrant transitions, as well as differences in the response of the system to “equivalent” GSR additive noise perturbations.     

有界噪声激励下单摆-谐振子系统的混沌运动

研究了具有同宿轨道和周期轨道的可积单摆-谐振子系统在弱Hamilton摄动(即弱耦合摄动)和弱非Hamilton摄动(即阻尼和有界噪声微扰)下的混沌运动．用Melnikov方程预测Hamilton系统中可能存在混沌运动的参数域，并用Poincare截面验证解析结果．用数值方法计算了有阻尼与有界噪声激励下系统的最大Lyapun0V指数和Poincare截面，结果表明有界噪声在频率上的扩散减小了引发系统产生混沌运动的效应。     

On the spatial behavior in the theory of swelling porous elastic soils

This paper is concerned with the study of the spatial behavior of the processes associated with a mixture consisting of three components: an elastic solid, a viscous fluid and a gas. An appropriate time-weighted surface power function is used in order to describe the spatial behavior of the processes in question. Spatial estimates of Saint–Venant type (for bounded bodies) and Phragmèn–Lindelöf type (for unbounded bodies) with time-dependent and time-independent rates are established. For unbounded bodies the asymptotic spatial behavior of the processes is also studied by means of an appropriate volumetric measure.     

Asymptotic properties of solutions of an equation of order <Emphasis Type="Italic">n</Emphasis> with almost constant coefficients and pulse action

We investigate the asymptotic behavior of solutions of linear differential equations with almost constant coefficients and pulse action at fixed times as t tends to infinity. We establish conditions for the times of pulse action under which there exist values of pulse action for which the solution of the considered Cauchy problem with initial conditions that coincide with the initial conditions for a certain (arbitrary but fixed) solution of the original equation without pulse action is bounded, unbounded, or tending to infinity. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 4, pp. 444–455, October–December, 2005.     

Minimization of semicoercive functions: a generalization of Fichera’s existence theorem for the Signorini problem

The existence theorem of Fichera for the minimum problem of semicoercive quadratic functions in a Hilbert space is extended to a more general class of convex and lower semicontinuous functions. For unbounded domains, the behavior at infinity is controlled by a lemma which states that every unbounded sequence with bounded energy has a subsequence whose directions converge to a direction of recession of the function. Thanks to this result, semicoerciveness plus the assumption that the effective domain is boundedly generated, that is, admits a Motzkin decomposition, become sufficient conditions for existence. In particular, for functions with a smooth quadratic part, a generalization of the existence condition given by Fichera’s theorem is proved.     

一双峰混沌系统非线性动力学行为

通过对一双峰混沌系统的非线性动力学行为的研究,发现随着系统参数的变化,双峰混沌系统由混沌状态开始,经阵发性混沌、不动点、倍周期分岔到受初始值的影响两个混沌吸引子,而后又收敛为另一个不动点,最后再次进入混沌状态。该系统呈现出复杂的非线性动力学行为。     

A new procedure for exploring chaotic attractors in nonlinear dynamical systems under random excitations

Due to uncertain push-pull action across boundaries between different attractive domains by random excitations,attractors of a dynamical system will drift in the phase space,which readily leads to colliding and mixing with each other,so it is very difficult to identify irregular signals evolving from arbitrary initial states.Here,periodic attractors from the simple cell mapping method are further iterated by a specific Poincare’ map in order to observe more elaborate structures and drifts as well as possible dynamical bifurcations.The panorama of a chaotic attractor can also be displayed to a great extent by this newly developed procedure.From the positions and the variations of attractors in the phase space,the action mechanism of bounded noise excitation is studied in detail.Several numerical examples are employed to illustrate the present procedure.It is seen that the dynamical identification and the bifurcation analysis can be effectively performed by this procedure.     

The dynamics of a high-dimensional delayed pest management model with impulsive pesticide input and?harvesting prey at different fixed moments

In this paper, a delayed pest control model with stage-structure for pests by introducing a constant periodic pesticide input and harvesting prey (Crops) at two different fixed moments is proposed and analyzed. We assume only the pests are affected by pesticide. We prove that the conditions for global asymptotically attractive ??predator-extinction?? periodic solution and permanence of the population of the model depend on time delay, pulse pesticide input, and pulse harvesting prey. By numerical analysis, we also show that constant maturation time delay, pulse pesticide input, and pulse harvesting prey can bring obvious effects on the dynamics of system, which also corroborates our theoretical results. We believe that the results will provide reliable tactic basis for the practical pest management. One of the features of present paper is to investigate the high-dimensional delayed system with impulsive effects at different fixed impulsive moments.