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
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 ...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.展开更多
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 i...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.展开更多
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...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.展开更多
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...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.展开更多
Exact heteroclinic breather-wave solutions for Davey-Stewartson （DSI, DSII） system with periodic boundary condition are constructed using Hirota＇s bilinear form method and generalized ansatz method. The heteroclini...Exact heteroclinic breather-wave solutions for Davey-Stewartson （DSI, DSII） system with periodic boundary condition are constructed using Hirota＇s bilinear form method and generalized ansatz method. The heteroclinic structure of wave is investigated.展开更多
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 ...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.展开更多
In this paper, we discuss a type of chaotic system with delays. We study the equilibrium points and the existence of heteroclinic orbit of the system. Heteroclinic orbit existence theorem is proposed and proved by app...In this paper, we discuss a type of chaotic system with delays. We study the equilibrium points and the existence of heteroclinic orbit of the system. Heteroclinic orbit existence theorem is proposed and proved by applying the undetermined coefficient method, which shows the complex dynamical properties of this system.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
Investigation of new orbit geometries exhibits a very attractive behavior for a spacecraft to monitor space weather coming from the Sun. Several orbit transfer mechanisms are analyzed as potential alternatives to moni...Investigation of new orbit geometries exhibits a very attractive behavior for a spacecraft to monitor space weather coming from the Sun. Several orbit transfer mechanisms are analyzed as potential alternatives to monitor solar activity such as a sub-solar orbit or quasi-satellite orbit and short and long heteroclinic and homoclinic connections between the triangular points L4 and L5 and the collinear point L3 of the CRTBP （circular restricted three-body problem） in the Sun-Earth system. These trajectories could serve as channels through where material can be transported from L5 to L3 by performing small maneuvers at the departure of the Trojan orbit. The size of these maneuvers at L5 is between 299 m/s and 730 m/s depending on the transfer time of the trajectory and does not need any deterministic maneuvers at L3. Our results suggest that material may also be transported from the Trojan orbits to quasi-satellite orbits or even displaced quasi-satellite orbits.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
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 split...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.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
In this paper, we study the bifurcation problems of rough heteroclinic loups cormecting threc saddle points for a higher-dimensional system. Under some transversal conditions and the nontwisted condition. the existenc...In this paper, we study the bifurcation problems of rough heteroclinic loups cormecting threc saddle points for a higher-dimensional system. Under some transversal conditions and the nontwisted condition. the existence. uniqueness. nd incoexistencc of thc l-heteroclinic loop with threc or two saddle pomts. l-homoclinic orbit and l-periodic orbit near T are obtained. Nleanwhile, the bifurcation surfaces and existence regions are also given. Moreover. the above bifurcation results are extended to the case for heteroclinic loop with l saddle points.展开更多
By generalizing the Floquet method from periodic systems to systems with exponential dichotomy, a local coordinate system is established in a neighborhood of the heteroclinic loop \%Γ\% to study the bifurcation probl...By generalizing the Floquet method from periodic systems to systems with exponential dichotomy, a local coordinate system is established in a neighborhood of the heteroclinic loop \%Γ\% to study the bifurcation problems of homoclinic and periodic orbits. Asymptotic expressions of the bifurcation surfaces and their relative positions are given. The results obtained in literature concerned with the 1\|hom bifurcation surfaces are improved and extended to the nontransversal case. Existence regions of the 1\|per orbits bifurcated from Γ are described, and the uniqueness and incoexistence of the 1\|hom and 1\|per orbit and the inexistence of the 2\|hom and 2\|per orbit are also obtained.展开更多
Bifurcations of nontwisted and fine heteroclinic loops are studied for higher dimensional systems. The existence and its associated existing regions are given for the 1-hom orbit and the 1-per orbit, respectively, and...Bifurcations of nontwisted and fine heteroclinic loops are studied for higher dimensional systems. The existence and its associated existing regions are given for the 1-hom orbit and the 1-per orbit, respectively, and bifurcation surfaces of the two-fold periodic orbit are also obtained. At last, these bifurcation results are applied to the fine heteroclinic loop for the planar system, which leads to some new and interesting results.展开更多
A geometrical method using the exponential dichotomy and the invariant manifold thoery is given to set up the criteria for the existence of transversal and tangential heterodinic orbits under the most general degenera...A geometrical method using the exponential dichotomy and the invariant manifold thoery is given to set up the criteria for the existence of transversal and tangential heterodinic orbits under the most general degenerate cases. Conclusions given here extend and contain the relevant known results.展开更多
Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic i...Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic invariant tori. These extend and improve the corresponding results obtained in [3-5].展开更多
In the present work we prove some existence results of heteroclinic orbits and heteroclinic chains for a second order discrete Hamiltonian system of the form Δ2q(t-1)+V(q(t))=0,t∈Z.The methods we use are variational...In the present work we prove some existence results of heteroclinic orbits and heteroclinic chains for a second order discrete Hamiltonian system of the form Δ2q(t-1)+V(q(t))=0,t∈Z.The methods we use are variational in nature.Our results show that under general conditions,for each maximum point β of V,the above system possesses multiple heteroclinic orbits joining β and some other maximum points of V.We also prove that for any pair of distinct maximum points η and ξ of V,there exists at least one heteroclinic chain from η to ξ.展开更多
The hyperbolic Lindstedt-Poincaré method is applied to determine the homoclinic and heteroclinic solutions of cubic strongly nonlinear oscillators of the form x + c1 x + c3 x 3= ε f (μ,x,x).In the method,the hy...The hyperbolic Lindstedt-Poincaré method is applied to determine the homoclinic and heteroclinic solutions of cubic strongly nonlinear oscillators of the form x + c1 x + c3 x 3= ε f (μ,x,x).In the method,the hyperbolic functions are employed instead of the periodic functions in the Lindstedt-Poincaré procedure.Critical value of parameter μ under which there exists homoclinic or heteroclinic orbit can be determined by the perturbation procedure.Typical applications are studied in detail.To illustrate the accuracy of the present method,its predictions are compared with those of Runge-Kutta method.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.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.11072168 and 10872141)
文摘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.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61170037 and 61074192)
文摘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.
文摘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.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11172093 and 11372102)the Hunan Provincial Innovation Foundation for Postgraduate,China(Grant No.CX2012B159)
文摘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.
基金Supported by Chinese Natural Science Foundation under Grant No. 10661002Yunnan Natural Science Foundation under Grant No. 2006A0082M
文摘Exact heteroclinic breather-wave solutions for Davey-Stewartson （DSI, DSII） system with periodic boundary condition are constructed using Hirota＇s bilinear form method and generalized ansatz method. The heteroclinic structure of wave is investigated.
文摘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.
基金Supported by National Natural Science Foundation of China under Grant No. 70271068
文摘In this paper, we discuss a type of chaotic system with delays. We study the equilibrium points and the existence of heteroclinic orbit of the system. Heteroclinic orbit existence theorem is proposed and proved by applying the undetermined coefficient method, which shows the complex dynamical properties of this system.
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Investigation of new orbit geometries exhibits a very attractive behavior for a spacecraft to monitor space weather coming from the Sun. Several orbit transfer mechanisms are analyzed as potential alternatives to monitor solar activity such as a sub-solar orbit or quasi-satellite orbit and short and long heteroclinic and homoclinic connections between the triangular points L4 and L5 and the collinear point L3 of the CRTBP （circular restricted three-body problem） in the Sun-Earth system. These trajectories could serve as channels through where material can be transported from L5 to L3 by performing small maneuvers at the departure of the Trojan orbit. The size of these maneuvers at L5 is between 299 m/s and 730 m/s depending on the transfer time of the trajectory and does not need any deterministic maneuvers at L3. Our results suggest that material may also be transported from the Trojan orbits to quasi-satellite orbits or even displaced quasi-satellite orbits.
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.
文摘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.
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.
基金Project supported byr the National Natural Science Foundation of China (100710122)Shanghai Municipal Foundation of Selected Academic Research.
文摘In this paper, we study the bifurcation problems of rough heteroclinic loups cormecting threc saddle points for a higher-dimensional system. Under some transversal conditions and the nontwisted condition. the existence. uniqueness. nd incoexistencc of thc l-heteroclinic loop with threc or two saddle pomts. l-homoclinic orbit and l-periodic orbit near T are obtained. Nleanwhile, the bifurcation surfaces and existence regions are also given. Moreover. the above bifurcation results are extended to the case for heteroclinic loop with l saddle points.
文摘By generalizing the Floquet method from periodic systems to systems with exponential dichotomy, a local coordinate system is established in a neighborhood of the heteroclinic loop \%Γ\% to study the bifurcation problems of homoclinic and periodic orbits. Asymptotic expressions of the bifurcation surfaces and their relative positions are given. The results obtained in literature concerned with the 1\|hom bifurcation surfaces are improved and extended to the nontransversal case. Existence regions of the 1\|per orbits bifurcated from Γ are described, and the uniqueness and incoexistence of the 1\|hom and 1\|per orbit and the inexistence of the 2\|hom and 2\|per orbit are also obtained.
文摘Bifurcations of nontwisted and fine heteroclinic loops are studied for higher dimensional systems. The existence and its associated existing regions are given for the 1-hom orbit and the 1-per orbit, respectively, and bifurcation surfaces of the two-fold periodic orbit are also obtained. At last, these bifurcation results are applied to the fine heteroclinic loop for the planar system, which leads to some new and interesting results.
基金Project supported by the National Natural Science Foundation of China.
文摘A geometrical method using the exponential dichotomy and the invariant manifold thoery is given to set up the criteria for the existence of transversal and tangential heterodinic orbits under the most general degenerate cases. Conclusions given here extend and contain the relevant known results.
基金Supported by the National Natural Science Foundation of China Shanghai Natural Science Foundation.
文摘Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic invariant tori. These extend and improve the corresponding results obtained in [3-5].
文摘In the present work we prove some existence results of heteroclinic orbits and heteroclinic chains for a second order discrete Hamiltonian system of the form Δ2q(t-1)+V(q(t))=0,t∈Z.The methods we use are variational in nature.Our results show that under general conditions,for each maximum point β of V,the above system possesses multiple heteroclinic orbits joining β and some other maximum points of V.We also prove that for any pair of distinct maximum points η and ξ of V,there exists at least one heteroclinic chain from η to ξ.
基金supported by the National Natural Science Foundation of China (Grant Nos.10672193, 10972240)Fu Lan Scholarship of Sun Yat-sen University,and the University of Hong Kong (CRGC grant)
文摘The hyperbolic Lindstedt-Poincaré method is applied to determine the homoclinic and heteroclinic solutions of cubic strongly nonlinear oscillators of the form x + c1 x + c3 x 3= ε f (μ,x,x).In the method,the hyperbolic functions are employed instead of the periodic functions in the Lindstedt-Poincaré procedure.Critical value of parameter μ under which there exists homoclinic or heteroclinic orbit can be determined by the perturbation procedure.Typical applications are studied in detail.To illustrate the accuracy of the present method,its predictions are compared with those of Runge-Kutta method.
