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.

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.

Stability of a homoclinic cycle and a heteroclinic cycle in space

About the stability of a homoclinic cycle or a heteroclinic cycle on a plane there are already many results.However, the stability of a homoclinic cycle or a heteroclinic cycle in space is still unclear up to now. This letter for the first time gives the criterion for determining the stability of a homoclinic cycle or a heteroclinic cycle and the criterion for 3-dimension weak attractor.

THE EXISTENCE AND STABILITY OF THE HETEROCLINIC CYCLE IN A KIND OF BIOLOGICAL SYSTEM

In this paper, we study the n-species biological systemwe get sufficient conditions for the existence of the invariant plane to system (1) whenm=1 and m = 2, we also get sufficient conditions for the eristence and stability ofthe heteroclinic cycle to system (1) when m = 1 and m = 2. In the case m = 1 andn = 3, we get conditions for the existence and stability of the heteroclinic cycle on theinvariant plane of system (1). In this case, we also prove that there is a center insidethe heteroclinic cycle and bounded by this heteroclinic cycle.

THE HETEROCLINIC CYCLE IN THEMODEL OF COMPETITION BETWEENn SPECIES AND ITS STABILITY

This paper gives a sufficient condition for the existence of heteroclinic cycle in the model of competition between n species and a criterion for determining the stability of the heteroclinic cycle. The results given in this paper extend the results obtained by May and Leonard in [1]and by Hofbaner and Sigmund in [2]. A conjecture on the permanence of the model and a open problem on the stability of the heteroclinic cycle for the critical case are given at the end of this paper.

BIFURCATIONS OF LIMIT CYCLES FROM A HETEROCLINIC CYCLE OF HAMILTONIAN SYSTEMS

This paper is concerned with the bifurcations of limit cycles from a heteroclinic cycle of planar Hamiltonian systems under perturbations. The author obtains a simple condition which guarantees the existence of at most two limit cycles near the heteroclinic cycle.

LIMIT CYCLE PROBLEM OF QUADRATIC SYSTEM OF TYPE （ Ⅲ ）m=0, （ Ⅲ ）

To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation,the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through saddle S1 on 1 ＋ax-y = 0,which has the same stability with the limit cycle created by Hopf bifurcation,then the uniqueness of limit cycles in such cases can be proved. If it is a homoclinic cycle passing through saddle N on x= 0,which has the different stability from the limit cycle created by Hopf bifurcation,then it will be a case of two limit cycles. For the case when the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity,the discussion of the paper shows that the number of limit cycles will change from one to two depending on the different values of parameters of system.

COMPLETED AND INCOMPLETED STOCHASTIC WEBS

In this paper, the completed stochastic web and incompleted stochastic web produced by the perturbed saddle separatrix net are given. The structural properties of two kinds of web are discussed by means of the dynamical system theory.

On the Number of Limit Cycles in Small Perturbations of a Piecewise Linear Hamiltonian System with a Heteroclinic Loop

In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n(n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [(n+1)/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.

HETEROCLINIC BIFURCATIONS OF A PREY-PREDATOR FISHERY MODEL WITH IMPULSIVE HARVESTING

In this paper, we consider a prey-predator fishery model with Allee effect and state- dependent impulsive harvesting. First, we investigate the existence of order-1 heteroclinic cycle. Second, choosing p as a control parameter, we obtain the sufficient conditions for the existence and uniqueness of order-1 periodic solution of system （2.3） by using the geometry theory of semi-continuous dynamic systems. Finally, on the basis of the theory of rotated vector fields, heteroclinic bifurcation to perturbed system of system （2.3） is also studied. The methods used in this paper are novel to prove the existence of order-1 heteroclinic cycle and heteroclinic bifurcations.

CHAOTIC BEHAVIOR OF A SYSTEM OF UNIFORM WEB

In this paper, we study the chaotic behavior of a system of uniform web. We give the set of parameters such that the mapping q (for q=3 and 4 ) is chaotic in the sence of Smale horseshoes.