Heteroclinic Cycles in a Class of 3-Dimensional Piecewise Affine Systems

This paper explores the existence of heteroclinic cycles and corresponding chaotic dynamics in a class of 3-dimensional two-zone piecewise affine systems. Moreover, the heteroclinic cycles connect two saddle foci and intersect the switching manifold at two points and the switching manifold is composed of two perpendicular planes.

The construction of homoclinic and heteroclinic orbitals in asymmetric strongly nonlinear systems based on the Pad'e approximant

In this paper, the extended Pade approximant is used to construct the homoclinic and the heteroclinic trajectories in nonlinear dynamical systems that are asymmetric at origin. Meanwhile, the conservative system, the autonomous system, and the nonautonomous system equations with quadratic and cubic nonlinearities are considered. The disturbance parameter ~ is not limited to being small. The ranges of the values of the linear and the nonlinear term parameters, which are variables, can be determined when the boundary values are satisfied. New conditions for the potentiality and the convergence are posed to make it possible to solve the boundary-value problems formulated for the orbitals and to evaluate the initial amplitude values.

Existence of heteroclinic orbits in a novel three-order dynamical system

In this paper, we design a novel three-order autonomous system. Numerical simulations reveal the complex chaotic behaviors of the system. By applying the undetermined coefficient method, we find a heteroclinic orbit in the system. As a result, the Si＇lnikov criterion along with some other given conditions guarantees that the system has both Smale horseshoes and chaos of horseshoe type.

A generalized Padé approximation method of solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators

An intrinsic extension of Pad′e approximation method, called the generalized Pad′e approximation method, is proposed based on the classic Pad′e approximation theorem. According to the proposed method, the numerator and denominator of Pad′e approximant are extended from polynomial functions to a series composed of any kind of function, which means that the generalized Pad′e approximant is not limited to some forms, but can be constructed in different forms in solving different problems. Thus, many existing modifications of Pad′e approximation method can be considered to be the special cases of the proposed method. For solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators, two novel kinds of generalized Pad′e approximants are constructed. Then, some examples are given to show the validity of the present method. To show the accuracy of the method, all solutions obtained in this paper are compared with those of the Runge–Kutta method.

Homoclinic （Heteroclinic） Orbit of Complex Dynamical System and Spiral Structure

Starting from iterated systems, it is shown that the homoclinic (heteroclinic) orbit is a kind of spiral structure. The emphasis is laid to show that there are homoclinic or heteroclinic orbits in complex discrete and continuous systems, and these homoclinic or heteroclinic orbits are some kind of spiral structure.

Heteroclinic cycles in a new class of four-dimensional discontinuous piecewise affine systems

It is a huge challenge to give an existence theorem for heteroclinic cycles in the high-dimensional discontinuous piecewise systems（DPSs）. This paper first provides a new class of four-dimensional（4 D） two-zone discontinuous piecewise affine systems（DPASs）, and then gives a useful criterion to ensure the existence of heteroclinic cycles in the systems by rigorous mathematical analysis. To illustrate the feasibility and efficiency of the theory, two numerical examples, exhibiting chaotic behaviors in a small neighborhood of heteroclinic cycles, are discussed.

HETEROCLINIC ORBIT AND SUBHARMONIC BIFURCATIONS AND CHAOS OF NONLINEAR OSCILLATOR

Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes yon der Pol damping, is very complex. In this paper, Melnikov's method is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system-far various resonant cases finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.

Exponential Dichotomies and Homoclinic Orbits from Heteroclinic Cycles

In this paper, we investigate the homoclinic bifurcations from a heteroclinic cycle by using exponential dichotomies. We give a Melnikov—type condition assuring the existence of homoclinic orbits form heteroclinic cycle. We improve some important results.

Heteroclinic Orbit Existence on a Type of Chaotic System with Delays

在这份报纸，我们与延期讨论混乱系统的一种类型。我们学习平衡点和系统的 heteroclinic 轨道的存在。Heteroclinic 轨道存在定理被建议并且由使用未经决定的系数方法证明了，它显示出这个系统的复杂动态性质。

Heteroclinic Breather-Wave Solutions for Davey-Stewartson Equation

Exact heteroclinic breather-wave solutions for Davey-Stewartson (DSⅠ, DSⅡ) system with periodic boundarycondition are constructed using Hirota's bilinear form method and generalized ansatz method. The heteroclinic structureof wave is investigated.

BIFURCATIONS TO A HETEROCLINIC MANIFOLD WITH NONHYPERBOLIC EQUILIBRIA IN R^n

The authors study bifurcations from a heteroclinic manifold connecting two non-hyperbolic equilibrium P-0 and P-1 for a n-dimensional dynamical system. They show that under some assumptions, each equilibrium P-i splits into two equilibria <(P)over tilde (i)> and P-i(alpha), i = 0, 1, and find the Melnikov vector conditions assuring the existence of a heteroclinic orbit from P-1(alpha) to P-0(alpha) along directions that are tangent to the strong unstable (resp.strong stable) manifold of P-1(alpha) (resp.P-0(alpha)). The exponential trichotomy and the unified and geometrical method are used to prove their results.

一种三阶分段线性混沌系统及其在图像加密中的应用

基于异宿环Shilnikov定理,该文提出了一个三阶分段线性混沌系统,构建了该系统两个平衡点之间的异宿环,证明了异宿环存在的充分条件,分析了混沌吸引子的形成机理,验证了系统的可实现性.应用该系统产生混沌数据序列,构建置乱矩阵和随机扩散矩阵对图像进行加密.在MATLAB平台进行了密钥敏感性分析、相关性分析、信息熵分析和抗噪声分析等.实验结果表明:该算法有较好的加密效果与抗攻击能力.

Probability Density Analysis of Nonlinear Random Ship Rolling

Ship rolling in random waves is a complicated nonlinear motion that contributes substantially to ship instability and capsizing.The finite element method(FEM)is employed in this paper to solve the Fokker Planck(FP)equations numerically for homoclinic and heteroclinic ship rolling under random waves described as periodic and Gaussian white noise excitations.The transient joint probability density functions(PDFs)and marginal PDFs of the rolling responses are also obtained.The effects of stimulation strength on ship rolling are further investigated from a probabilistic standpoint.The homoclinic ship rolling has two rolling states,the connection between the two peaks of the PDF is observed when the periodic excitation amplitude or the noise intensity is large,and the PDF is remarkably distributed in phase space.These phenomena increase the possibility of a random jump in ship motion states and the uncertainty of ship rolling,and the ship may lose stability due to unforeseeable facts or conditions.Meanwhile,only one rolling state is observed when the ship is in heteroclinic rolling.As the periodic excitation amplitude grows,the PDF concentration increases and drifts away from the beginning location,suggesting that the ship rolling substantially changes in a cycle and its stability is low.The PDF becomes increasingly uniform and covers a large region as the noise intensity increases,reducing the certainty of ship rolling and navigation safety.The current numerical solutions and analyses may be applied to evaluate the stability of a rolling ship in irregular waves and capsize mechanisms.

毒素环境下水生生物持续生存的状态反馈脉冲控制分析

建立了一类具有Holling-Ⅱ型功能反应函数的状态反馈脉冲模型,描述在毒素环境下通过收获食饵和投放捕食者使物种持续生存的策略。使用后继函数等定性方法,得到阶1周期解的存在性、稳定性的充分条件及异宿环的存在性。数值模拟结果验证了理论结果的正确性。

多频激励软弹簧型Duffing系统中的混沌

研究了多频激励下的软弹簧型Duffing系统的混沌动力学,发现混沌产生的根本原因是系统相空间中横截异宿环面的存在· 建立了双频激励情况下二维环面上的Poincar啨映射、稳定流形和不稳定流形,应用Melnikov方法给出了稳定流形和不稳定流形横截相交的条件,并将此方法推广到激励包含有限多个频率的情形· 推广了Melnikov方法在高维系统中的应用,给出了Smale马蹄意义下混沌存在的判据· 同时证明。

随机激励滞后非线性汽车悬架系统的混沌运动

研究了具有滞后非线性的汽车悬架系统在路面随机激励下发生混沌运动的可能性,推导了随机Melnikov过程,得到了均方意义下出现混沌运动的临界激励幅值,并讨论了噪声对系统混沌运动的影响。通过最大L ya-punov指数和Poincare截面进行了数值验证。研究结果表明,汽车在正常行驶时不会出现混沌运动。

一个非线性电力系统的混沌振荡

分析了一个非线性三参数电力系统振荡的异宿分枝,给出Melnikov函数的留数计算法,并获得电力系统发生混沌振荡的锥性参数区域和带形参数区域。

非线性弹性杆的异常动态响应

讨论了拉伸速度呈周期变化的受拉非线性弹性直杆的动力行为· 采用Melnikov方法研究时发现 ,材料的非线性使得动力响应发生异常 ,对确定的直杆而言 ,当拉伸速度超过某个临界值时 ,动力系统将出现次谐分岔和混沌·

具有滞后非线性的汽车悬架中的混沌

研究了具有滞后非线性的单自由度汽车悬架在发生路面单频正弦激励下的受迫振动时的混沌运动。采用位移和速度三次方的数学模型 ,描述汽车悬架中的滞后非线性阻尼力。对运动微分方程进行无量纲化后 ,利用 Mel-nikov方法给出了发生混沌运动的临界条件 ,并进行了数值模拟。分析了非线性阻尼力中的各系数对混沌的影响 ,并通过 Poincare截面、相轨迹、时间历程和功率谱图及 L yapunov指数 。