一个分段Sprott系统及其混沌机理分析

提出了一个分段Sprott系统,对其混沌机理进行了分析.根据Shilnikov定理,在满足异宿轨道基本特性、Shilnikov不等式和特征方程条件下,通过寻找该系统中由不稳定流形、异宿点和稳定流形三个几何不变集上所形成的一条异宿轨道,在分段Sprott系统中导出了存在异宿轨道时该系统中各个参数应符合的条件, 并找到了一组对应的实参数,由此证明了异宿轨道的存在性.最后,根据这组对应的实参数,进行了电路设计与实验验证. 关键词： 分段Sprott系统 Shilnikov定理 异宿轨道 电路实验     

A novel methodology for constructing a multi-wing chaotic and hyperchaotic system with a unified step function switching control

This paper aims at developing a novel method of constructing a class of multi-wing chaotic and hyperchaotic system by introducing a unified step function. In order to overcome the essential difficulties in iteratively adjusting multiple parameters of conventional multi-parameter control, this paper introduces a unified step function controlled by a single parameter for constructing various multi-wing chaotic and hyperchaotic systems. In particular, to the best of the authors' knowledge, this is also the first time to find a non-equilibrium multi-wing hyperchaotic system by means of the unified step function control. According to the heteroclinic loop Shilnikov theorem, some properties for multi-wing attractors and its chaos mechanism are further discussed and analyzed. A circuit for multi-wing systems is designed and implemented for demonstration, which verifies the effectiveness of the proposed approach.     

一类3D混沌系统的异宿轨道和backstepping控制

基于异宿轨道Shilnikov准则,分析了一类三维自治微分系统异宿环的存在性,并证明了该系统具有Smale马蹄意义的混沌.然后对系统的分岔,Lyapunov指数,Poincare映射进行了数值分析,同时利用自适应反步控制方法,对含有三个未知参数的系统给出了一种控制算法.最后通过数值示例进行仿真,对文中论述进行了验证. 关键词： 异宿环 自适应反步 Shilnikov准则 Poincare映射     

OBSERVATIONS OF HOMOCLINIC CHAOS THROUGH ALTERNATING PERIODIC-CHAOTIC SEQUENCES IN A NONLINEAR CIRCUIT

Homoclinic chaos in the alternating periodic-chaotic sequences is observed in a nonlinear circuit with sinusoidal driving force. In particular, a complete Alternating Periodic-Chaotic sequence is recorded with a high-resolution up to P⁽⁸⁾ state. The experimental results, analyzed by constructing the time of flight and the next maximal amplitude return maps, are in good agreement with the scenario described by Shilnikov. The underlying dynamics of homoclinic chaos is determined from the next amplitude return map, to be that of a unimodal map and thus a strong dissipation case.     

Chaos via Shilnikov's saddle-node bifurcation in a theory of the electroencephalogram

We study the bifurcation diagram of a mesoscopic model of the human cortex. This model is known to exhibit robust chaotic behavior in the space of parameters that model exterior forcing. We show that the bifurcation diagram has an unusual degree of organization. In particular, we show that the chaos is spawned by a codimension-one homoclinic bifurcation that was analyzed by Shilnikov in 1969 but has never before been found in a physical application.     

On the Possibility of Chaos in a Generalized Model of Three Interacting Sectors

In this study we extend a model, proposed by Dendrinos, which describes dynamics of change of influence in a social system containing a public sector and a private sector. The novelty is that we reconfigure the system and consider a system consisting of a public sector, a private sector, and a non-governmental organizations (NGO) sector. The additional sector changes the model’s system of equations with an additional equation, and additional interactions must be taken into account. We show that for selected values of the parameters of the model’s system of equations, chaos of Shilnikov kind can exist. We illustrate the arising of the corresponding chaotic attractor and discuss the obtained results from the point of view of interaction between the three sectors.     

Higher-dimensional chaotic dynamics of a composite laminated piezoelectric rectangular plate

The analysis on the chaotic dynamics of a six-dimensional nonlinear system which represents the averaged equation of a composite laminated piezoelectric rectangular plate is given for the first time. The theory of normal form and the energy-phase method are combined to investigate the higher-dimensional chaotic dynamics of the composite laminated piezoelectric rectangular plate. Firstly, the theory of normal form is used to reduce the six-dimensional averaged equation to the simpler normal form. Then, the energy-phase method is extended to analyze the global bifurcations and chaotic dynamics of a six-dimensional nonlinear system. The analysis results indicate that there exist the homoclinic bifurcation and Shilnikov type multi-pulse chaos for the composite laminated piezoelectric rectangular plate. Finally, numerical simulations are also used to investigate the nonlinear dynamic characteristics of the composite laminated piezoelectric rectangular plate. The results of numerical simulations also demonstrate that there exist the chaotic motions and the multi-pulse jumping orbits of the composite laminated piezoelectric rectangular plate.     

Multi-pulse homoclinic orbits and chaotic dynamics of a parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities

In this paper,the complicated dynamics and multi-pulse homoclinic orbits of a two-degree-of-freedom parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities are studied.The damping,parametrical excitation and the nonlinearities are regarded as weak.The averaged equation depicting the fast and slow dynamics is derived through the method of multiple scales.The dynamics near the resonance band is revealed by doing a singular perturbation analysis and combining the extended Melnikov method.We are able to determine the criterion for the existence of the multi-pulse homoclinic orbits which can form the Shilnikov orbits and give rise to chaos.At last,numerical results are also given to illustrate the nonlinear behaviors and chaotic motions in the nonlinear nano-oscillator.     

基于参数切换算法的混沌系统吸引子近似及其电路设计

基于参数切换算法和离散混沌系统, 设计一种新的混沌系统参数切换算法, 给出了两算法的原理. 采用混沌吸引子相图观测法, 研究了不同算法下统一混沌系统和Rössler混沌系统参数切换结果, 最后引入方波发生器, 设计了Rössler混沌系统参数切换电路. 结果表明, 采用参数切换算法可以近似出指定参数下的系统, 其吸引子与该参数下吸引子一致; 基于离散系统的参数切换结果更为复杂, 当离散序列分布均匀时, 只可近似得到指定参数下的系统; 相比传统切换混沌电路, 参数切换电路不用修改原有系统电路结构, 设计更为简单, 输出结果受方波频率影响, 通过加入合适频率的方波发生器, 数值仿真与电路仿真结果一致.     

Incomplete approach to homoclinicity in a model with bent-slow manifold geometry

The dynamics of a model, originally proposed for a type of instability in plastic flow, has been investigated in detail. The bifurcation portrait of the system in two physically relevant parameters exhibits a rich variety of dynamical behavior, including period bubbling and period adding or Farey sequences. The complex bifurcation sequences, characterized by mixed mode oscillations, exhibit partial features of Shilnikov and Gavrilov–Shilnikov scenario. Utilizing the fact that the model has disparate time scales of dynamics, we explain the origin of the relaxation oscillations using the geometrical structure of the bent-slow manifold. Based on a local analysis, we calculate the maximum number of small amplitude oscillations, s, in the periodic orbit of L^s type, for a given value of the control parameter. This further leads to a scaling relation for the small amplitude oscillations. The incomplete approach to homoclinicity is shown to be a result of the finite rate of ‘softening’ of the eigenvalues of the saddle focus fixed point. The latter is a consequence of the physically relevant constraint of the system which translates into the occurrence of back-to-back Hopf bifurcation.     

Chaotic patterns in a coupled oscillator-excitator biochemical cell system

In this paper we examine dynamical modes resulting from diffusion-like interaction of two model biochemical cells. Kinetics in each of the cells is given by the ICC model of calcium ions in the cytosol. Constraints for one of the cells are set so that it is excitable. One of the constraints in the other cell - a fraction of activated cell surface receptors-is varied so that the dynamics in the cell is either excitable or oscillatory or a stable focus. The cells are interacting via mass transfer and dynamics of the coupled system are studied as two parameters are varied-the fraction of activated receptors and the coupling strength. We find that (i) the excitator-excitator interaction does not lead to oscillatory patterns, (ii) the oscillator-excitator interaction leads to alternating phase-locked periodic and quasiperiodic regimes, well known from oscillator-oscillator interactions; torus breaking bifurcation generates chaos when the coupling strength is in an intermediate range, (iii) the focus-excitator interaction generates compound oscillations arranged as period adding sequences alternating with chaotic windows; the transition to chaos is accompanied by period doublings and folding of branches of periodic orbits and is associated with a Shilnikov homoclinic orbit. The nature of spontaneous self-organized oscillations in the focus-excitator range is discussed. (c) 1999 American Institute of Physics.     

CO2激光器对相位共轭波时空混沌系统控制和同步的研究

以一维耦合映象格子为对象,研究了相位共轭波时空混沌系统特性. 基于Lyapunov稳定性定理,通过选取耦合参数,实现了CO₂激光器对相位共轭波时空混沌系统的控制,以及驱动多个相位共轭波时空系统达到并行同步. 数值模拟结果显示,耦合参数对相位共轭波时空混沌系统的控制和同步速度有影响,即耦合参数越大同步时间越短. 关键词： 2激光器')" href="#">CO₂激光器 相位共轭波 时空混沌 控制和同步     

一种基于混沌系统部分序列参数辨识的混沌保密通信方法

本文提出一种混沌保密通信方法,即混沌系统的部分序列用于混沌系统参数辨识其他序列用于通信保密.利用混沌蚁群优化算法对部分序列混沌系统进行参数辨识,以达到了解混沌系统全部信息的目的.在参数辨识过程中引入参数空间和蚁群空间,通过空间变换函数使参数空间与蚁群空间之间相互变换.文中使用Lorenz系统进行数值试验,其结果验证混沌系统部分序列参数辨识及混沌保密通信的可行性.     

Analysis of stochastic bifurcation and chaos in stochastic Duffing--van der Pol system via Chebyshev polynomial approximation

The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing--van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing--van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.     

Phase Locking and Chaos in a Josephson Junction Array Shunted by a Common Resistance

The dynamics of a Josephson junction array shunted by a common resistance are investigated by using numerical methods. Coexistence of phase locking and chaos is observed in the system when the resistively and capacitively shunted junction model is adopted. The corresponding parameter ranges for phase locking and chaos are presented. When there are three resistively shunted junctions in the array, chaos is found for the first time and the parameter range for chaos is also presented. According to the theory of Chernikov and Schmidt, when there are four or more junctions in the array, the system exhibits chaotic behavior. Our results indicate that the theory of Chernikov and Schmidt is not exactly appropriate.     

随机Bonhoeffer-Van der Pol系统的随机混沌控制

讨论了具有有界随机参数的随机Bonhoeffer-Van der Pol系统的随机混沌现象，并利用噪声对其进行控制.首先运用Chebyshev多项式逼近的方法，将随机Bonhoeffer-Van der Pol系统转化为等价的确定性系统，使原系统的随机混沌控制问题转换为等价的确定性系统的确定性混沌控制问题，继而可用Lyapunov指数指标来研究等价确定性系统的确定性混沌现象和控制问题.数值结果表明，随机Bonhoeffer-Van der Pol系统的随机混沌现象与相应的确定性Bonhoeffer-Van der Pol系统极为相似.利用噪声控制法可将混沌控制到周期轨道，但是在随机参数及其强度的影响下也呈现出一些特点.     

Basic structures of the Shilnikov homoclinic bifurcation scenario

We find numerically small scale basic structures of homoclinic bifurcation curves in the parameter space of the Chua circuit. The distribution of these basic structures in the parameter space and their geometrical properties constitute a complete homoclinic bifurcation scenario of this system. Furthermore, these structures and the scenario are theoretically demonstrated to be generic to a large class of dynamical systems that presents, as the Chua circuit, Shilnikov homoclinic orbits. We classify the complexity of primary and subsidiary homoclinic orbits by their order given by the number of their returning loops. Our results confirm previous predictions of structures of homoclinic bifurcation curves and extend this study to high order primary orbits. Furthermore, we identify accumulations of bifurcation curves of subsidiary homoclinic orbits into bifurcation curves of both primary and subsidiary orbits.     

Comparing mechanical analysis with generalized-competitive-mode analysis for the plasma chaotic system

The plasma chaotic system is a dissipative dynamical system modeled by a parametric plasma instability arising from the interaction of the whistler and ion acoustic waves with the plasma oscillation near the lower hybrid resonance. The amplitudes of these three oscillations obey a three-dimensional system of ordinary differential equations that exhibits chaos for certain parameter values. Besides the maximal Lyapunov exponent technique, a generalized-competitive-mode (GCM) technique has been proposed to evaluate parameter values associated with chaos. A mechanical analysis has also been proposed to reveal the mechanisms underlying the different dynamical modes including chaos. In a series of comparisons between the GCM analysis and mechanical analysis, chaos for the plasma chaotic system is determined. The mechanism and causes by which the plasma chaotic system produces different dynamical behaviors are interpreted. Furthermore, using the whistler-parameter variation of the Casimir function and Casimir power for the plasma system, the generating mechanisms of the different orbital modes and the different levels of chaos are uncovered.     

Synchronization properties and effects of parameter mismatches in unidirectionally coupled chaotic vertical-cavity surface-emitting lasers

We numerically investigate the effects of parameter mismatches on chaos synchronization in vertical-cavity surfaceemitting lasers (VCSELs). We assume injection-locked chaos synchronization in a unidirectionally coupled and openloop optical feedback system. The accuracy of chaos synchronization is greatly affected by the mismatches of the device parameters and operation conditions between the two lasers. In particular, the oscillation frequency of the laser is one of the important parameters in a system of injection-locked chaos synchronization. However, the variations of the device characteristics of VCSELs are very large compared with those of other types of semiconductor lasers. We study the effects of parameter mismatches related to the oscillation frequency of VCSELs on chaos synchronization. We proved that mismatches in terms of the birefringence and the injection current play crucial roles for the quality of chaos synchronization.     

Bidirectional chaos synchronization and communication based on mutually coupled semiconductor lasers with asymmetrical bias currents

In this paper, a bidirectional chaos secret communication system, based on mutually coupled semiconductor lasers (MCSLs) with asymmetrical bias currents, is proposed, and the synchronization characteristics and the communication performances of such a system are numerically investigated. The results show that the stable leader-laggard chaos synchronization can be achieved under relatively large asymmetrical bias current levels. Meantime, the influence of the intrinsic parameter variations of the laser on the synchronization quality is also considered, and the simulation reveals that this system still possesses good robustness to the parameter variations. Moreover, the influences of delay time and mutually coupling strength between the two lasers on chaos communication performance have also been discussed. Finally, unidirectional and bidirectional secret communication performances of such a system are examined under the chaos masking (CMS) encryption scheme, and the security of this system is also discussed.