江苏-西部能源需求-供给模型及其动力学分析

根据江苏省能源需求与西部能源供给及江苏省能源进口量之间相互支持、相互制约的复杂关系建立了江苏-西部能源需求-供给系统．分析了系统的稳态、周期、分叉、混沌等若干动力学行为．通过调节参数可以使系统处于稳定发展或周期震荡区域中．基于Silnikov定理,应用待定系数法,求出了能源系统的同宿轨的表达式．证明了在两个平衡点处均有Silnikov型的同宿轨．根据Silnikov定理确保了能源系统有Smale马蹄和马蹄形混沌．     

Melnikov方法和圆型平面限制性三体问题的横截同宿研究*

本文对由两自由度近可积哈密顿系统经非正则变换而得到的,具有高阶不动点的非哈密顿系统给出了判别横截同宿轨和横截异宿轨存在性的两条判据。对原二体质量比很小时近可积圆型平面限制性三体问题,采用本文判据证明存在横截同宿轨,从而存在横截同宿穿插现象;还在一定假设下证明了存在横截异宿轨;并给出了全局定性相图。     

Bifurcation of Homoclinic Orbits with Saddle-Center Equilibrium

In this paper, the authors develop new global perturbation techniques for detecting the persistence of transversal homoclinic orbits in a more general nondegenerated system with action-angle variable. The unperturbed system is assumed to have saddle-center type equilibrium whose stable and unstable manifolds intersect in one dimensional manifold, and does not have to be completely integrable or near-integrable. By constructing local coordinate systems near the unperturbed homoclinic orbit, the conditions of existence of transversal homoclinic orbit are obtained, and the existence of periodic orbits bifurcated from homoclinic orbit is also considered.     

The existences of transverse homoclinic solutions and chaos for parabolic equations

By using Lyapunov-Schmidt reduction and exponential dichotomies, the persistence of homoclinic orbit is considered for parabolic equations with small perturbations. Bifurcation functions are obtained, where d is the dimension of the intersection of the stable and unstable manifolds. The zeros of H correspond to the existence of the homoclinic orbit for the perturbed systems. Some applicable conditions are given to ensure that the functions are solvable. Moreover the homoclinic solution for the perturbed system is transversal under the applicable conditions and hence the perturbed system exhibits chaos. The basic tools are shadowing lemma which was obtained by Blazquez (see [C.M. Blazquez, Transverse homoclinic orbits in periodically perturbed parabolic equations, Nonlinear Anal. 10 (1986) 1277-1291]).     

Transversal homoclinic orbits for higher dimensional difference equations

In this paper, we use the functional analytic method (theory of exponential dichotomies and Liapunov-Schmidt method) to study the homoclinic bifurcations of higher dimensional difference equations in a degenerate case. We obtain a Melnikov vector mapping for difference equations with the help of which the existence of transversal homoclinic orbits can be detected.     

Transversal homoclinic orbits and chaos for partial functional differential equations

The persistence of degenerate homoclinic orbit is considered for parabolic functional differential equations with small periodic perturbations. Bifurcation functions constructed between two finite-dimensional spaces are obtained. The zeros of the function correspond to the existence of the homoclinic orbit for the perturbed systems. Some applicable conditions are given to ensure that the functions are solvable. Moreover, We show that the homoclinic solution for the perturbed system is transversal and hence the perturbed system exhibits chaos.     

Transversal homoclinic orbits and chaos for partial functional differential equations

The persistence of degenerate homoclinic orbit is considered for parabolic functional differential equations with small periodic perturbations. Bifurcation functions constructed between two finite-dimensional spaces are obtained. The zeros of the function correspond to the existence of the homoclinic orbit for the perturbed systems. Some applicable conditions are given to ensure that the functions are solvable. Moreover, We show that the homoclinic solution for the perturbed system is transversal and hence the perturbed system exhibits chaos.     

Transversal connecting orbits from shadowing

A rigorous numerical method for establishing the existence of a transversal connecting orbit from one hyperbolic periodic orbit to another of a differential equation in is presented. As the first component of this method, a general shadowing theorem that guarantees the existence of such a connecting orbit near a suitable pseudo connection orbit given the invertibility of a certain operator is proved. The second component consists of a refinement procedure for numerically computing a pseudo connecting orbit between two pseudo periodic orbits with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component consists of a numerical procedure to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse. Using this method, existence of chaos is demonstrated on examples with transversal homoclinic orbits, and with cycles of transversal heteroclinic orbits.     

Persistent Homoclinic Orbits for Perturbed Nonlinear Schroding Equation With Derivative Term

In recent years there have been existence studies on the extensive of homoclinic orbits for mearly dissipative PDEs,which are chosely related to chaos.In this work,we consider the perturbe     

Bifurcation and chaotic dynamics of homoclinic systems in ℝ3

We consider perturbations which may or may not depend explicitly on time for the three-dimensional homoclinic systems. We obtain the existence and bifurcation theorems for transversal homoclinic points and homoclinic orbits, and illustrate our results with two examples.     

BIFURCATION OF DEGENERATE HOMOCLINIC ORBITS

ＢＩＦＵＲＣＡＴＩＯＮＯＦＤＥＧＥＮＥＲＡＴＥＨＯＭＯＣＬＩＮＩＣＯＲＢＩＴＳ￥ＺｅｎｇＷｅｉｙａｏ（ＨｕｎａｎＬｉｇｈｔｉｎｄｕｓｔｒｙＣｏｌｌｅｇｅ，４１０００７）Ａｂｓｔｒａｃｔ：Ｔｈｅｍａｉｎｐｕｒｐｏｓｅｏｆｔｈｉｓｐａｐｅｒｉｓｔｏｉｎｖ...     

The transversal homoclinic solutions and chaos for stochastic ordinary differential equations

We consider the persistence of a transversal homoclinic solution and chaotic motion for ordinary differential equations with a homoclinic solution to a hyperbolic equilibrium under an unbounded random forcing driven by a Brownian force. By Lyapunov–Schmidt reduction, the persistence of transversal homoclinic solution is reduced to find the zeros of some bifurcation functions defined between two finite spaces. It is shown that, for almost all sample paths of the Brownian motion, the perturbed system exhibits chaos.     

Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers

We consider 4-dimensional, real, analytic Hamiltonian systems with a saddle center equilibrium (related to a pair of real and a pair of imaginary eigenvalues) and a homoclinic orbit to it. We find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle center equilibrium. We also consider one-parameter families of reversible, 4-dimensional Hamiltonian systems. We prove that the set of parameter values where the system has homoclinic orbits to a saddle center equilibrium has no isolated points. We also present similar results for systems with heteroclinic orbits to saddle center equilibria. © 1997 John Wiley & Sons, Inc.     

Homoclinic Bifurcations for Partial Differential Equations in Unbounded Domains

The connection between low-dimensional chaos in ordinary differential equations, and turbulence in fluids and other systems governed by partial differential equations, is one that is in many circumstances not clear. We discuss some examples of turbulent fluid flow, and consider ways in which they may be related to much simpler sets of ordinary differential equations, whose behavior can be reasonably well understood. (We are not advocating drastic Fourier truncation.) The generation of aperiodic solutions through the occurrence of homoclinic orbits is briefly analysed for ordinary differential equations, and the same kind of heuristic analysis is sketched for partial differential equations (in one space dimension). It is suggested that such an analysis can explain certain features of chaos, which have been observed in real fluids.     

在连续的时间系统中不存在Shilnikov型混沌

在n维的、时间连续的光滑系统中,得到了不存在同宿轨道和异宿轨道的条件.基于此结论并用一个基本实例,推断出如下结论:在多项式常微分方程系统中,有着以不存在同宿轨道和异宿轨道为特征的第4类混沌.     

一类Hamilton系统的同宿解存在性

We study the existence of homoclinic orbits for some Hamiltonian system.A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a sequence of systems of differential equations.     

Melnikov method and detection of chaos for non-smooth systems

We extend the Melnikov method to non-smooth dynamical systems to study the global behavior near a non-smooth homoclinic orbit under small time-periodic perturbations. The definition and an explicit expression for the extended Melnikov function are given and applied to determine the appearance of transversal homoclinic orbits and chaos. In addition to the standard integral part, the extended Melnikov function contains an extra term which reflects the change of the vector field at the discontinuity. An example is discussed to illustrate the results.     

Homoclinic orbits for nonlinear difference equations containing both advance and retardation

In this paper we discuss how to use the critical point theory to study the existence of a nontrivial homoclinic orbit for nonlinear difference equations containing both advance and retardation without any periodic assumptions. Moreover, if the nonlinearity is an odd function, the existence of an unbounded sequence of homoclinic orbits is obtained.     

Homoclinic Orbits of Nonlinear Functional Difference Equations

In this paper, by using the critical point theory, we obtain the existence of a nontrivial homoclinic orbit which decays exponentially at infinity for nonlinear difference equations containing both advance and retardation without any periodic assumptions. Moreover, if the nonlinearity is an odd function, the existence of an unbounded sequence of nontrivial homoclinic orbits which decay exponentially at infinity is obtained.