Chaos in the second—order autonomous Birkhoff system with a heteroclinic circle

Chaotic behaviour in a second-order autonomous Birkhoff system with a heteroclinic circle under weakly periodic perturbation is studied using the Melnikov method.The equations of heteroclinic orbits and the criteria for chaos are given.One example is also presented to illustrate the application of the results.     

Heteroclinic Orbit Existence on a Type of Chaotic System with Delays

In this paper, we discuss a type of chaotic system with delays. We study the equilibrium points and the existence of heteroclinic orbit of the system. Heteroclinic orbit existence theorem is proposed and proved by applying the undetermined coefficient method, which shows the complex dynamical properties of this system.     

谐和激励与有界噪声作用下具有同宿和异宿轨道的Duffing振子的混沌运动

研究了具有同宿轨道、异宿轨道的双势阱Duffing振子在谐和激励与有界噪声摄动下的混沌运动.基于同宿分叉和异宿分叉，由Melnikov理论推导了系统出现混沌运动的必要条件及出现分形边界的充分条件.结果表明：当Wiener过程的强度参数大于某一临界值时，噪声增大了诱发混沌运动的有界噪声的临界幅值，相应地缩小了参数空间的混沌域，且产生混沌运动的临界幅值随着噪声强度的增大而增大.同时数值计算了最大Lyapunov指数，由最大Lyapunov指数为零从另一角度得到了系统出现混沌运动的有界噪声的临界幅值，发现在Wi 关键词： 混沌 同宿和异宿分叉 随机Melnikov方法 最大Lyapunov指数     

非扩散洛伦兹系统的周期轨道

混沌系统的奇怪吸引子是由无数条周期轨道稠密覆盖构成的,周期轨道是非线性动力系统中除不动点之外最简单的不变集,它不仅能够体现出混沌运动的所有特征,而且和系统振荡的产生与变化密切相关,因此分析复杂系统的动力学行为时获取周期轨道具有重要意义.本文系统地研究了非扩散洛伦兹系统一定拓扑长度以内的周期轨道,提出一种基于轨道的拓扑结构来建立一维符号动力学的新方法,通过变分法数值计算轨道显得很稳定.寻找轨道初始化时,两条轨道片段能够被用作基本的组成单元,基于整条轨道的结构进行拓扑分类的方式显得很有效.此外,讨论了周期轨道随着参数变化时的形变情况,为研究轨道的周期演化规律提供了新途径.本研究可为在其他类似的混沌体系中找到并且系统分类周期轨道提供一种可借鉴的方法.     

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.     

Time-delay feedback control in a delayed dynamical chaos system and its applications

The feedback control of a delayed dynamical system, which also includes various chaotic systems with time delays, is investigated. On the basis of stability analysis of a nonautonomous system with delays, some simple yet less conservative criteria are obtained for feedback control in a delayed dynamical system. Finally, the theoretical result is applied to a typical class of chaotic Lorenz system and Chua circuit with delays. Numerical simulations are also given to verify the theoretical results.     

Discovery of closed orbits of dynamical systems with the use of computers

In this paper we derive a general criterion which can be used for the discovery with the use of a computer of closed orbits of systems of ordinary differential equations. We apply this criterion to the Lorenz model and show rigorously the existence of a closed orbit for the case under consideration. In a subsequent paper we shall show how the stable manifold of this orbit determines the boundary of the stochastic attractor.     

在有心力场中行星运行轨道稳定性的证明

从粒子的基本运动方程出发,利用非线性动力学的方法证明了行星在有心力场中运行轨道的稳定性,得到结论:粒子在有心力F=-k2m2r-n的作用下运行时,轨道稳定的条件是n<3.     

Equilibrium points and bifurcation control of a chaotic system

Based on the Routh-Hurwitz criterion, this paper investigates the stability of a new chaotic system. State feedback controllers are designed to control the chaotic system to the unsteady equilibrium points and limit cycle. Theoretical analyses give the range of value of control parameters to stabilize the unsteady equilibrium points of the chaotic system and its critical parameter for generating Hopf bifurcation. Certain nP periodic orbits can be stabilized by parameter adjustment. Numerical simulations indicate that the method can effectively guide the system trajectories to unsteady equilibrium points and periodic orbits.     

Equilibrium points and bifurcation control of a chaotic system

Based on the Routh--Hurwitz criterion, this paper investigates the stability of a new chaotic system. State feedback controllers are designed to control the chaotic system to the unsteady equilibrium points and limit cycle. Theoretical analyses give the range of value of control parameters to stabilize the unsteady equilibrium points of the chaotic system and its critical parameter for generating Hopf bifurcation. Certain nP periodic orbits can be stabilized by parameter adjustment. Numerical simulations indicate that the method can effectively guide the system trajectories to unsteady equilibrium points and periodic orbits.     

一种非线性动力学系统混沌现象的深入研究

本文在实验教学中引入一种非线性混沌摆系统,通过调节混沌摆的驱动力周期演示了该非线性动力学系统出现混沌现象的过程,从而让学生了解混沌现象的参数敏感性、相图特点、频谱特性等基本特性.为了进一步了解该混沌摆的特性,本文建立了该非线性摆系统的简化动力学方程,在数值上对其进行了研究.基于动力学方程的数值模拟,克服了实验上相关参数定量改变困难、摆动稳定性不易控制、实验时间周期长等问题.在数值模拟中,通过改变不同参数得到了相图、频谱图以及分岔图,比较深入详细地对这种混沌摆的相关特性进行了描述,也有利于学生加深对混沌摆的理解.     

Chaos and chaotic control in a relative rotation nonlinear dynamical system under parametric excitation

This paper studies the chaotic behaviours of a relative rotation nonlinear dynamical system under parametric excitation and its control. The dynamical equation of relative rotation nonlinear dynamical system under parametric excitation is deduced by using the dissipation Lagrange equation. The criterion of existence of chaos under parametric excitation is given by using the Melnikov theory. The chaotic behaviours are detected by numerical simulations including bifurcation diagrams, Poincar map and maximal Lyapunov exponent. Furthermore, it implements chaotic control using non-feedback method. It obtains the parameter condition of chaotic control by the Melnikov theory. Numerical simulation results show the consistence with the theoretical analysis. The chaotic motions can be controlled to period-motions by adding an excitation term.     

连续时间系统同宿轨的搜索算法及其应用

同宿轨的求解是非线性系统领域的核心问题之一, 特别是对动力系统分岔与混沌的研究有重要意义. 根据同宿轨的几何特点, 采用轨线逼近的方式, 通过定义逼近轨线与鞍点的距离, 将同宿轨的求解转化为求距离最小值的无约束非线性优化问题. 为了提高优化结果的完整性, 还提出了基于区间细分的搜索算法和实现方法, 并找出了Lorenz系统, Shimizu-Morioka系统和超混沌Lorenz系统等的多个同宿轨道和对应参数, 验证了本文方法的有效性. 关键词： 混沌 同宿轨 非线性系统 数值计算     

一个基于Chen系统的新混沌系统的分析与同步

通过在Chen系统的第一个方程中加入一个可变系数的乘积项, 构造了一个新的三维自治混沌系统.新系统可通过调节其可变系数实现不同系数组合下系统的混沌产生或混沌抑制, 即调节该乘积项的可变系数, 可使不出现混沌的Chen系统产生混沌现象, 同时也可使产生混沌运动的Chen系统不再产生混沌现象.详细分析了新系统的特性, 研究了新系统的混沌同步问题, 并给出了相应的仿真结果.     

Generation of a novel spherical chaotic attractor from a new three-dimensional system

A new three-dimensional （3D） system is constructed and a novel spherical chaotic attractor is generated from the system. Basic dynamical behaviors of the chaotic system are investigated respectively. Novel spherical chaotic attractors can be generated from the system within a wide range of parameter values. The shapes of spherical chaotic attractors can be impacted by the variation of parameters. Finally, a simpler 3D system and a more complex 3D system with the same capability of generating spherical chaotic attractors are put forward respectively.     

The reliability of a stochastically complex dynamical system

The first-passage problem is an important issue in the theory of dynamical systems; it determines whether the system is safe and reliable or destroyed. Therefore, taking a suitable force to control first-passage failure is very significant for such systems. In this paper, we study the reliability of a stochastically complex dynamical system by using the stochastic averaging method. The reliability results show that the reliability function is monotonically decreasing with respect to time, and that the reliability of the system is relatively worse, with external and parametric white-noise excitations. A bang–bang feedback control is used to improve the reliability for the original system. Numerical results display that even a small control force can greatly enhance the reliability of the original system.     

Phase space reconstruction of chaotic dynamical system based on wavelet decomposition

In view of the disadvantages of the traditional phase space reconstruction method,this paper presents the method of phase space reconstruction based on the wavelet decomposition and indicates that the wavelet decomposition of chaotic dynamical system is essentially a projection of chaotic attractor on the axes of space opened by the wavelet filter vectors,which corresponds to the time-delayed embedding method of phase space reconstruction proposed by Packard and Takens.The experimental results show that,the structure of dynamical trajectory of chaotic system on the wavelet space is much similar to the original system,and the nonlinear invariants such as correlation dimension,Lyapunov exponent and Kolmogorov entropy are still reserved.It demonstrates that wavelet decomposition is effective for characterizing chaotic dynamical system.     

一类新混沌系统的线性状态反馈控制

研究了一类新混沌系统的控制问题.利用Routh-Hurwitz准则对受控系统进行了稳定性分析，结合线性状态反馈方法理论上严格证明了达到控制目标反馈系数的选择原则.数值研究证明了该方法能够有效地控制混沌系统到失稳的平衡点或周期解，同时控制效果在弱噪声干扰下具有很强的鲁棒性. 关键词： 新混沌系统 线性状态反馈控制 Routh-Hurwitz准则 噪声     

Implementation of a novel two-attractor grid multi-scroll chaotic system

This paper proposed a method of generating two attractors in a novel grid multi-scroll chaotic system.Based on a newly generated three-dimensional system,a two-attractor grid multi-scroll attractor system can be generated by adding two triangular waves and a sign function.Some basic dynamical properties,such as equilibrium points,bifurcations,and phase diagrams,were studied.Furthermore,the system was experimentally confirmed by an electronic circuit.The circuit simulation results and numerical simulation results verified the feasibility of this method.