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 ...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.展开更多
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 mo...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.展开更多
In this paper, we study the number of limit cycles of a near-Hamiltonian system having Za- equivariant quintic perturbations. Using the methods of Hopf and heteroclinic bifurcation theory, we find that the perturbed s...In this paper, we study the number of limit cycles of a near-Hamiltonian system having Za- equivariant quintic perturbations. Using the methods of Hopf and heteroclinic bifurcation theory, we find that the perturbed system can have 28 limit cycles, and its location is also given. The main result can be used to improve the lower bound of the maximal number of limit cycles for some polynomial systems in a previous work, which is the main motivation of the present paper.展开更多
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 ...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.展开更多
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 ...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.展开更多
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 cy...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.展开更多
Consider a three-dimensional autonomous system of the form of =f(u), u∈R^3, (0.1) where f(0)=0, and Df(0)=. By transforming eq. (0.1) into a normal form equation and then unfolding the truncated equation, one can obt...Consider a three-dimensional autonomous system of the form of =f(u), u∈R^3, (0.1) where f(0)=0, and Df(0)=. By transforming eq. (0.1) into a normal form equation and then unfolding the truncated equation, one can obtain a plane system of the form展开更多
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 di...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.展开更多
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 sta...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.展开更多
A criterion of spatial chaos occurring in lattice dynamical systems--heteroclinic cycle--is discussed. It is proved that if the system has asymptotically stable heteroclinic cycle, then it has asymptotically stable ho...A criterion of spatial chaos occurring in lattice dynamical systems--heteroclinic cycle--is discussed. It is proved that if the system has asymptotically stable heteroclinic cycle, then it has asymptotically stable homoclinic point which implies spatial chaos.展开更多
In this paper, we investigate the Poincar bifurcation in cubic Hamiltonian systems with heteroclinic loop, under small general cubic perturbations. We prove that the system has at most two limit cycles and has at leas...In this paper, we investigate the Poincar bifurcation in cubic Hamiltonian systems with heteroclinic loop, under small general cubic perturbations. We prove that the system has at most two limit cycles and has at least two limit cycles, respectively.展开更多
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. Thi...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.展开更多
文摘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.
文摘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.
基金Supported by National Natural Science Foundation of China(Grant Nos.11271261,11461001)
文摘In this paper, we study the number of limit cycles of a near-Hamiltonian system having Za- equivariant quintic perturbations. Using the methods of Hopf and heteroclinic bifurcation theory, we find that the perturbed system can have 28 limit cycles, and its location is also given. The main result can be used to improve the lower bound of the maximal number of limit cycles for some polynomial systems in a previous work, which is the main motivation of the present paper.
文摘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.
文摘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.
文摘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.
基金the National Natural Science Foundation of China.
文摘Consider a three-dimensional autonomous system of the form of =f(u), u∈R^3, (0.1) where f(0)=0, and Df(0)=. By transforming eq. (0.1) into a normal form equation and then unfolding the truncated equation, one can obtain a plane system of the form
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11472212 and 11532011)
文摘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.
文摘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.
文摘A criterion of spatial chaos occurring in lattice dynamical systems--heteroclinic cycle--is discussed. It is proved that if the system has asymptotically stable heteroclinic cycle, then it has asymptotically stable homoclinic point which implies spatial chaos.
文摘In this paper, we investigate the Poincar bifurcation in cubic Hamiltonian systems with heteroclinic loop, under small general cubic perturbations. We prove that the system has at most two limit cycles and has at least two limit cycles, respectively.
文摘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.
