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.

The construction of homoclinic and heteroclinic orbitals in asymmetric strongly nonlinear systems based on the Pad'e approximant

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.

Existence of heteroclinic orbits in a novel three-order dynamical system

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.

A generalized Padé approximation method of solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators

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.

HETEROCLINIC ORBIT AND SUBHARMONIC BIFURCATIONS AND CHAOS OF NONLINEAR OSCILLATOR

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.

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.

Exponential Dichotomies and Homoclinic Orbits from Heteroclinic Cycles

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.

BIFURCATIONS TO A HETEROCLINIC MANIFOLD WITH NONHYPERBOLIC EQUILIBRIA IN R^n

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.

Bifurcations of Rough Heteroclinic Loops with Three Saddle Points

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.

Bifurcations of heteroclinic loops

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.

Bifurcations of nontwisted heteroclinic loop

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.

Transversal heteroclinic orbits in general degenerate cases

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.

Orbits Heteroclinic to Invariant Manifolds

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].

Heteroclinic orbits and heteroclinic chains for a discrete Hamiltonian system

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.

Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by hyperbolic Lindstedt-Poincaré method

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.

Heteroclinic cycles in lattice dynamical systems

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.

Non-periodic perturbations and transversal heteroclinic orbits

We consider an n-dimensional system x=f(x)+εg(t,x,μ), (1)where f,g∈C_b^2 for x∈ΩR^n,t∈R,0【ε【【1,μ∈R^m,and Ω is some open set.

Homoclinic, heteroclinic and periodic orbits of singularly perturbed systems

The main aims of this paper are to study the persistence of homoclinic and heteroclinic orbits of the reduced systems on normally hyperbolic critical manifolds, and also the limit cycle bifurcations either from the homoclinic loop of the reduced systems or from a family of periodic orbits of the layer systems. For the persistence of homoclinic and heteroclinic orbits, and the limit cycles bifurcating from a homolinic loop of the reduced systems, we provide a new and readily detectable method to characterize them compared with the usual Melnikov method when the reduced system forms a generalized rotated vector field. To determine the limit cycles bifurcating from the families of periodic orbits of the layer systems, we apply the averaging methods.We also provide two four-dimensional singularly perturbed differential systems, which have either heteroclinic or homoclinic orbits located on the slow manifolds and also three limit cycles bifurcating from the periodic orbits of the layer system.

Codimension 2 reversible heteroclinic bifurcations with inclination flips

In this paper, the heteroclinic bifurcation problem with real eigenvalues and two incli- nation-flips is investigated in a four-dimensional reversible system. We perform a detailed study of this case by using the method originally established in the papers "Problems in Homoclinic Bifurcation with Higher Dimensions" and "Bifurcation of Heteroclinic Loops," and obtain fruitful results, such as the existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic loops, R-symmetric homoclinic orbit and R-symmetric periodic orbit. The double R-symmetric homoclinic bifurcation (i.e., two-fold R-symmetric homoclinic bifurcation) for reversible heteroclinic loops is found, and the existence of infinitely many R-symmetric periodic orbits accumulating onto a homoclinic orbit is demonstrated. The relevant bifurcation surfaces and the existence regions are also located.

Exponential dichotomies and heteroclinic bifurcations in degenerate cases

By making use of theory of exponential dichotomies, the theory of heteroclinic bifurcations in higher degenerate cases is investigated. A Melnikov-type vector is given by which the existence of transversal heteroclinic orbits in degenerate cases can be detected. A functional analytical method of proving the transver-sality of heteroclinic orbits in degenerate cases is also provided.