Transversal Heteroclinic Bifurcation in Hybrid Systems with Application to Linked Rocking Blocks

In this paper, we study heteroclinic bifurcation and the appearance of chaos in time-perturbed piecewise smooth hybrid systems with discontinuities on finitely many switching manifolds. The unperturbed system has a heteroclinic orbit connecting hyperbolic saddles of the unperturbed system that crosses every switching manifold transversally, possibly multiple times. By applying a functional analytical method, we obtain a set of Melnikov functions whose zeros correspond to the occurrence of chaos of the system. As an application, we present an example of quasiperiodically excited piecewise smooth system with impacts formed by two linked rocking blocks.     

Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems

We discuss bifurcation of periodic orbits in discontinuous planar systems with discontinuities on finitely many straight lines intersecting at the origin and the unperturbed system has either a limit cycle or an annulus of periodic orbits. Assume that the unperturbed periodic orbits cross every switching line transversally exactly once. For the first case we give a condition for the persistence of the limit cycle. For the second case, we obtain the expression of the first order Melnikov function and establish sufficient conditions on the number of limit cycles bifurcate from the periodic annulus. Then we generalize our results to systems with discontinuities on finitely many smooth curves. As an application, we present a piecewise cubic system with 4 switching lines and show that the maximum number of limit cycles bifurcate from the periodic annulus can be affected by the position of the switching lines.     

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.     

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。     

Melnikov向量与退化情形下的横截异宿轨道

利用指数二分性和泛函分析方法,我们研究了当未扰动系统不具有异宿流形的退化异宿分支.我们利用Melnikov型向量给出了系统在退化情形下的横截异宿轨道存在的充分条件.     

Melnikov向量与退化情形下的横截异宿轨道

利用指数二分性和泛函分析方法,我们研究了当未扰动系统不具有异宿流形的退化异宿分支.我们利用Melnikov型向量给出了系统在退化情形下的横截异宿轨道存在的充分条件.     

具有临界两点异宿环的二次系统

本文给出二次系统存在临界两点异宿环的充要条件,并证明二次系统的临界两点异宿环必由双曲线的一支和直线或由椭圆和直线构成,其内部的奇点必是中心。推广所研究的这种系统,本文对[1]中提出的一个公开问题也给出了解答。     

Approximate implicitization and recursive surface intersection algorithms

Most published work on intersection algorithms for Computer Aided Design (CAD) systems addresses transversal intersections [1], situations where the surface normals of the surfaces intersected are well separated along all intersection curves. For transversal intersections the divide and conquer strategy of recursive subdivision, Sinha's theorem [2] and the convex hull property of NonUniform Rational B-Spline surfaces (NURBS) efficiently identify all intersection branches. However, in singular or near singular intersections, situations where the surfaces are parallel or near parallel in an intersection region, along an intersection curve or in an intersection point, even deep levels of subdivision will frequently not sort out the intersection topology. The paper will focus on the novel approach of Approximate Implicitization to address these challenges. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A three-dimensional nonlinear system with a single heteroclinic trajectory

The study for singular trajectories of three-dimensional (3D) nonlinear systems is one of recent main interests. To the best of our knowledge, among the study for most of Lorenz or Lorenz-like systems, a pair of symmetric heteroclinic trajectories is always found due to the symmetry of those systems. Whether or not does there exist a 3D system that possesses a single heteroclinic trajectory? In the present note, based on a known Lorenz-type system, we introduce such a 3D nonlinear system with two cubic terms and one quadratic term to possess a single heteroclinic trajectory. To show its characters, we respectively use the center manifold theory, bifurcation theory, Lyapunov function and so on, to systematically analyse its complex dynamics, mainly for the distribution of its equilibrium points, the local stability, the expression of locally unstable manifold, the Hopf bifurcation, the invariant algebraic surface, and its homoclinic and heteroclinic trajectories, etc. One of the major results of this work is to rigorously prove that the proposed system has a single heteroclinic trajectory under some certain parameters. This kind of interesting phenomenon has not been previously reported in the Lorenz system family (because the huge amount of related research work always presents a pair of heteroclinic trajectories due to the symmetry of studied systems). What"s more key, not like most of Lorenz-type or Lorenz-like systems with singularly degenerate heteroclinic cycles and chaotic attractors, the new proposed system has neither singularly degenerate heteroclinic cycles nor chaotic attractors observed. Thus, this work represents an enriching contribution to the understanding of the dynamics of Lorenz attractor.     

On the approach of Holmes and Sanders to the Melnikov procedure in the method of averaging

In 1963 Melnikov has offered a computational approach to the problem of finding transversal heteroclinic points for certain plane periodic systems. Recently Holmes has proposed a similar scheme for a different type of equations which are related to the method of averaging. In the sequence Sanders has pointed out that the reasoning of Holmes is not conclusive. In this note we review the problem based on a different generalization of the Melnikov scheme.     

Necessary and sufficient conditions for the topological conjugacy of surface diffeomorphisms with a finite number of orbits of heteroclinic tangency

We consider diffeomorphisms of orientable surfaces with the nonwandering set consisting of a finite number of hyperbolic fixed points and the wandering set containing a finite number of heteroclinic orbits of transversal and nontransversal intersection. We distinguish a meaningful class of diffeomorphisms and present a complete topological invariant for this class. The invariant is a scheme consisting of a set of numerical parameters and a set of geometric objects.     

Persistent switching near a heteroclinic model for the geodynamo problem

Modelling chaotic and intermittent behaviour, namely the excursions and reversals of the geomagnetic field, is a big problem far from being solved. Armbruster et al. [5] considered that structurally stable heteroclinic networks associated to invariant saddles may be the mathematical object responsible for the aperiodic reversals in spherical dynamos. In this paper, invoking the notion of heteroclinic switching near a network of rotating nodes, we present analytical evidences that the mathematical model given by Melbourne et al. [19] contributes to the study of the georeversals. We also present numerical plots of solutions of the model, showing the intermittent behaviour of trajectories near the heteroclinic network under consideration.     

Transversal wave maps and transversal exponential wave maps

Transversal wave maps and wave maps are different. There are wave maps which are not transversal wave maps, and vice versa. However, if f is a wave map under certain circumstance, then f is a transversal wave map. We show that if f is a transversal exponential wave map, then the associated energy–momentum is transversally conserved. We finally obtain the relationship among transversal wave maps, transversal exponential wave maps and certain second order symmetric tensors.     

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.     

Effective construction of Poincar\'e--Bendixson regions

This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincar\''e--Bendixson regions by using transversal curves, that enables us to prove the existence of a limit cycle that has been numerically detected. We apply our results to several known systems, like the Brusselator one or some Li\''{e}nard systems, to prove the existence of the limit cycles and to locate them very precisely in the phase space. Our method, combined with some other classical tools can be applied to obtain sharp bounds for the bifurcation values of a saddle-node bifurcation of limit cycles, as we do for the Rychkov system.     

Dynamics of complexity of intersections

The topological complexity of the intersection of a submanifold, moved by a dynamical system, with a given submanifold of the phase space, can increase with time. It is proved that the Morse and Betti numbers of the transversal intersections generically grow at most exponentially, while for some special infinitely smooth systems the topological complexity of the intersections can become larger than any given function of time (for a growing sequence of integer time moments).To S. Smale on the occasion of his 60th birthday     

空间同宿环和异宿环的稳定性

关于平面同（异）宿环的稳定性已有不少文献讨论过，但关于空间同（异）宿环的稳定性尚没有任何结果．本文在可定义回复映射的条件下给出了同（异）宿环在其部分邻域中是渐近稳定的判据．这些结果在某种意义下是平面系统相应结果的推广，包括并推广了［２］，［３］的结果．本文最后讨论了Ｌｏｒｅｎｚ系统同宿环和三种群竞争系统异宿环的稳定性，所得结果和Ｓｐａｒｒｏｗ与Ｍａｙ等的数值结果相吻合．     

The Conley index along heteroclinic trajectories of reaction–diffusion equations

It is well known that hyperbolic equilibria of reaction–diffusion equations have the homotopy Conley index of a pointed sphere, the dimension of which is the Morse index of the equilibrium. A similar result concerning the homotopy Conley index along heteroclinic solutions of ordinary differential equations under the assumption that the respective stable and unstable manifolds intersect transversally, is due to McCord. This result has recently been generalized by Dancer to some reaction–diffusion equations by using finite-dimensional approximations. We extend McCord?s result to reaction–diffusion equations. Additionally, an error in the original proof is corrected.