Linearly independent homoclinic bifurcations parameterized by a small function

In this paper we discuss a small nonautonomous perturbation of an autonomous system on Rn which has a homoclinic solution. Regarding the small perturbation as a parameter in an appropriate space of functions we discuss various situations of co-existence of homoclinic orbits. Those conditions of various co-existence actually define bifurcation manifolds in the space of functions for linearly independent homoclinic bifurcations.     

Resonance bifurcation from homoclinic cycles

Differential equations that are equivariant under the action of a finite group can possess robust homoclinic cycles that can moreover be asymptotically stable. For differential equations in R⁴ there exists a classification of different robust homoclinic cycles for which moreover eigenvalue conditions for asymptotic stability are known. We study resonance bifurcations that destroy the asymptotic stability of robust ‘simple homoclinic cycles’ in four-dimensional differential equations. We establish that typically a periodic trajectory near the cycle is created, asymptotically stable in the supercritical case.     

Maslov index for homoclinic orbits of Hamiltonian systems

A useful tool for studying nonlinear differential equations is index theory. For symplectic paths on bounded intervals, the index theory has been completely established, which revealed tremendous applications in the study of periodic orbits of Hamiltonian systems. Nevertheless, analogous questions concerning homoclinic orbits are still left open. In this paper we use a geometric approach to set up Maslov index for homoclinic orbits of Hamiltonian systems. On the other hand, a relative Morse index for homoclinic orbits will be derived through Fredholm index theory. It will be shown that these two indices coincide.     

同宿流形中的奇异轨道

研究较一般的高维退化系统的同宿、异宿轨道分支问题．利用推广的Melnikov函数、横截性理论及奇摄动理论，对具有鞍—中心型奇点的带有角变量的奇摄动系统，在角变量频率产生共振的情况下，讨论其同宿、异缩轨道的扰动下保存和横截的条件．推广和改进了一些文献的结果。     

Inverse Jacobian multipliers and Hopf bifurcation on center manifolds

In this paper we consider a class of higher dimensional differential systems in Rⁿ${\mathbb{R}}^n$ which have a two dimensional center manifold at the origin with a pair of pure imaginary eigenvalues. First we characterize the existence of either analytic or C^∞$C^{\infty }$ inverse Jacobian multipliers of the systems around the origin, which is either a center or a focus on the center manifold. Later we study the cyclicity of the system at the origin through Hopf bifurcation by using the vanishing multiplicity of the inverse Jacobian multiplier.     

Periodic orbits for perturbations of piecewise linear systems

We consider the existence of periodic orbits in a class of three-dimensional piecewise linear systems. Firstly, we describe the dynamical behavior of a non-generic piecewise linear system which has two equilibria and one two-dimensional invariant manifold foliated by periodic orbits. The aim of this work is to study the periodic orbits of the continuum that persist under a piecewise linear perturbation of the system. In order to analyze this situation, we build a real function of real variable whose zeros are related to the limit cycles that remain after the perturbation. By using this function, we state some results of existence and stability of limit cycles in the perturbed system, as well as results of bifurcations of limit cycles. The techniques presented are similar to the Melnikov theory for smooth systems and the method of averaging.     

An existence result for homoclinic solutions to a nonlinear second-order ODE through differential inequalities

In this paper an existence result of homoclinic solutions to a nonlinear second-order ODE is presented. To this end, a method based on differential inequalities is used.     

An example of bifurcation to homoclinic orbits

Consider the equation $\text{x}\text{?}\text{? x + x}{}^2\text{= ?λ}{}_1\text{x + λ}{}_2\text{?(t)}\text{where}\text{?(t + 1) = ?(t)}\text{and}\text{λ = (λ}{}_1\text{, λ}{}_2\text{)}$ is small. For λ = 0, there is a homoclinic orbit Γ through zero. For λ ≠ 0 and small, there can be “strange” attractors near Γ. The purpose of this paper is to determine the curves in λ-space of bifurcation to “strange” attractors and to relate this to hyperbolic subharmonic bifurcations.     

Existence of homoclinic solutions to periodic orbits in a center manifold

Consider a Lagrangian of the form

     

A note on homoclinic orbits for second order Hamiltonian system

Some existence and multiplicity of homoelinic orbits for second order Hamiltonian system x-a(t)x f(t,x)=0 are given by means of variational methods, where the function -1/2a（t）|s|^2∫^t0f（t,s）ds is asymptotically quadratic in s at infinity and subquadratic in s at zero, and the function a (t) mainly satisfies the growth condition limt→∞∫^t 1 t a（t）dt= ∞,VI∈R^1.A resonance case as well as a noncompact case is discussed too.     

Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations

The so-called noose bifurcation is an interesting structure of reversible periodic orbits that was numerically detected by Kent and Elgin in the well-known Michelson system. In this work we perform an analysis of the periodic behavior of a piecewise version of the Michelson system where this bifurcation also exists. This variant is a one-parameter three-dimensional piecewise linear continuous system with two zones separated by a plane and it is also a representative of a wide class of reversible divergence-free systems.     

Global bifurcation of polygonal relative equilibria for masses, vortices and dNLS oscillators

Given a regular polygonal arrangement of identical objects, turning around a central object (masses, vortices or dNLS oscillators), this paper studies the global bifurcation of relative equilibria in function of a natural parameter (central mass, central circulation or amplitude of the oscillation). The symmetries of the problem are used in order to find the irreducible representations, the linearization and, with the help of a degree theory, the symmetries of the bifurcated solutions.     

Infinitely many homoclinic orbits for a class of second‐order damped differential equations

We investigate the existence and multiplicity of homoclinic orbits for the second‐order damped differential equations For Equation 1 where L(t) and W(t,u) are neither autonomous nor periodic in t. Under certain assumptions on g, L, and W, we get infinitely many homoclinic orbits for superquadratic, subquadratic and concave–convex nonlinearities cases by using fountain theorem and dual fountain theorem in critical point theory. These results generalize and improve some existing results in the literature. Copyright © 2015 JohnWiley & Sons, Ltd.     

Periodic orbits on Riemannian manifolds under the action of an at most quadratic potential

In this paper we look for periodic orbits for a Lagrangian system in a complete Riemannian manifold under the action of an eventually unbounded potential. An upper bound on the fixed period is obtained by means of variational tools involving penalization arguments and Morse theory.     

Index estimates for strongly indefinite functionals, periodic orbits and homoclinic solutions of first order Hamiltonian systems

We improve Benci and Rabinowitz's Linking theorem for strongly indefinite functionals, giving estimates for a suitably defined relative Morse index of critical points. Such abstract result is applied to the existence problem of periodic orbits and homoclinic solutions of first order Hamiltonian systems in cases where the Palais-Smale condition does not hold. Received January 27, 1999 / Accepted January 14, 2000 / Published online July 20, 2000     

Stability and uniqueness of periodic orbits produced during homoclinic bifurcation

Under a generic assumption, the existence and the uniqueness of the periodic orbit generating from a homoclinic bifurcation are shown, and the dimensions of its stable and unstable manifolds are given. In the case of a 3-dimensional system, our result revises the stability criterion given in [4,5].Supported by the National Natural Science Foundation of China.     

Codimension-3 bifurcations of a class of homoclinic loop with saddle-point

In this work, bifurcations in the class that the homoclinic orbit connects the strong stable and strong unstable manifolds of a saddle are investigated for four-dimensional vector fields. The existence, numbers, coexistence and incoexistence of 1-homoclinic orbit, 1-periodic orbit, 2ⁿ$2^n$-homoclinic orbit and 2ⁿ$2^n$-periodic orbit are obtained, the approximate expressions of the corresponding bifurcation surfaces and the bifurcation diagrams are also presented.