Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero

In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first-order nonperiodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we give some new criteria to guarantee that Hamiltonian systems with asymptotically quadratic terms and spectrum point zero have at least one and a finite number of pairs of homoclinic orbits under some adequate conditions, respectively.     

Analytic and algebraic conditions for bifurcations of homoclinic orbits I: Saddle equilibria

We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but two parameters are needed in general systems. We apply a version of Melnikov?s method due to Gruendler to obtain saddle-node and pitchfork types of bifurcation results for homoclinic orbits. Furthermore we prove that if these bifurcations occur, then the variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under the assumption that the homoclinic orbits lie on analytic invariant manifolds. We illustrate our theories with an example which arises as stationary states of coupled real Ginzburg–Landau partial differential equations, and demonstrate the theoretical results by numerical ones.     

Existence of homoclinic solutions for a class of second-order Hamiltonian systems

A new result for existence of homoclinic orbits is obtained for the second-order Hamiltonian systems under a class of new superquadratic conditions. A homoclinic orbit is obtained as a limit of solutions of a certain sequence of boundary-value problems which are obtained by the minimax methods.     

Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers

We consider 4-dimensional, real, analytic Hamiltonian systems with a saddle center equilibrium (related to a pair of real and a pair of imaginary eigenvalues) and a homoclinic orbit to it. We find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle center equilibrium. We also consider one-parameter families of reversible, 4-dimensional Hamiltonian systems. We prove that the set of parameter values where the system has homoclinic orbits to a saddle center equilibrium has no isolated points. We also present similar results for systems with heteroclinic orbits to saddle center equilibria. © 1997 John Wiley & Sons, Inc.     

Infinitely many homoclinic orbits for the second-order Hamiltonian systems

In this paper, the existence of homoclinic orbits for the second-order Hamiltonian systems without periodicity is studied and infinitely many homoclinic orbits for both superlinear and asymptotically linear cases are obtained.     

DISSIPATIVE SCHEMES WITH PROPERTY OF INHERITING HOMOCLINIC ORBITS FOR 2N-DIMENSIONAL HAMILTONIAN SYSTEMS

1　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｉｓｐａｐｅｒｉｓｄｅｖｏｔｅｄｔｏｓｔｕｄｙｗｈａｔｋｉｎｄｏｆｄｉｓｃｒｅｔｅｓｃｈｅｍｅｓｏｆｔｈｅｆｏｌｌｏｗｉｎｇ 2ｎ ｄｉｍｅｎ ｓｉｏｎａｌＨａｍｉｌｔｏｎｉａｎｓｙｓｔｅｍｓｗｉｔｈｐａｒａｍｅｔｅｒｉｎｎｏｒｍａｌｆｏｒｍ ｕ=Ｊ2ｎ Ｈ ｕＴ,　　Ｈ =Ｈ(ｕ ,λ) ,(1 )ｗｈｅｒｅｕ∈Ｒ2ｎ,λ∈Ｒ ,Ｈ∈Ｃｋ+1(Ｒ2ｎ×Ｒ ,Ｒ) ,ｋ≥ 6,ａｎｄＪ2ｎ =0Ｉｎ-Ｉｎ 0 ,Ｉｎ:ｕｎｉｔｍａｔｒｉｘｏｆｏｒｄｅｒｎｈａｓｔｈｅｐｒｏｐｅｒｔｙｏｆｉｎｈｅｒｉｔｉｎｇｈｏｍ…     

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.     

Multi-bump orbits homoclinic to resonance bands

We establish the existence of several classes of multi-bump orbits homoclinic to resonance bands for completely-integrable Hamiltonian systems subject to small-amplitude Hamiltonian or dissipative perturbations. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The homoclinic orbits, which include multi-bump \v{S}ilnikov orbits, connect equilibria and periodic orbits in the resonance band. The main tools we use in the existence proofs are the exchange lemma with exponentially small error and the existence theory of orbits homoclinic to resonance bands which make only one fast excursion away from the resonance bands.

     

Existence of even homoclinic orbits for second-order Hamiltonian systems

Some existence theorems for even homoclinic orbits are obtained for a class of second-order nonautonomous Hamiltonian systems with symmetric potentials under a class of new superquadratic conditions. A homoclinic orbit is obtained as a limit of solutions of a certain sequence of nil-boundary-value problems which are obtained by the minimax methods.     

Index theory for heteroclinic orbits of Hamiltonian systems

Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic case is well-established, very few results are known in the case of homoclinic orbits of Hamiltonian systems. Moreover, to the authors’ knowledge, no results have been yet proved in the case of heteroclinic and halfclinic (i.e. parametrized by a half-line) orbits. Motivated by the importance played by these motions in understanding several challenging problems in Classical Mechanics, we develop a new index theory and we prove at once a general spectral flow formula for heteroclinic, homoclinic and halfclinic trajectories. Finally we show how this index theory can be used to recover all the (classical) existing results on orbits parametrized by bounded intervals.     

一类Hamilton系统的同宿解存在性

We study the existence of homoclinic orbits for some Hamiltonian system.A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a sequence of systems of differential equations.     

Homoclinic orbits for Hamiltonian systems induced by impulses

In this paper, we consider a class of impulsive Hamiltonian systems with a p‐Laplacian operator. Under certain conditions, we establish the existence of homoclinic orbits by means of the mountain pass theorem and an approximation technique. In some special cases, the homoclinic orbits are induced by the impulses in the sense that the associated non‐impulsive systems admit no non‐trivial homoclinic orbits. Copyright © 2015 John Wiley & Sons, Ltd.     

Melnikov方法和圆型平面限制性三体问题的横截同宿研究*

本文对由两自由度近可积哈密顿系统经非正则变换而得到的,具有高阶不动点的非哈密顿系统给出了判别横截同宿轨和横截异宿轨存在性的两条判据。对原二体质量比很小时近可积圆型平面限制性三体问题,采用本文判据证明存在横截同宿轨,从而存在横截同宿穿插现象;还在一定假设下证明了存在横截异宿轨;并给出了全局定性相图。     

Homoclinic orbits for second-order Hamiltonian systems with subquadratic potentials

In this paper we consider a class of subquadratic second-order Hamiltonian systems and new results about the existence and multiplicity of homoclinic orbits are obtained by using the Minimizing Theorem and the Clark’s Theorem respectively and a new compact imbedding theorem is also proved.     

一类含有参数和有理奇性哈密顿系统的同宿与异宿轨道

研究一类含有两个参数和有理奇性平面哈密顿系统的同宿与异宿轨道,该问题来源于一个关于聚合物流体剪切流动特性的研究．借助常微定性理论和不变流形分析的方法,文中给出了系统存在同宿与异宿轨道的条件,并通过数值计算检验了所得理论结果。     

Hunting for homoclinic orbits in reversible systems: A shooting technique

A dynamical system is said to be reversible if there is an involution of phase space that reverses the direction of the flow. Examples are Hamiltonian systems with quadratic potential energy. In such systems, homoclinic orbits that are invariant under the reversible transformation are typically not destroyed as a parameter is varied. A strategy is proposed for the direct numerical approximation to paths of such homoclinic orbits, exploiting the special properties of reversible systems. This strategy incorporates continuation using a simplification of known methods and a shooting approach, based on Newton's method, to compute starting solutions for continuation. For Hamiltonian systems, the shooting uses symplectic numerical integration. Strategies are discussed for obtaining initial guesses for the unknown parameters in Newton's method. An example system, for which there is an infinity of symmetric homoclinic orbits, is used to test the numerical techniques. It is illustrated how the orbits can be systematically located and followed. Excellent agreement is found between theory and numerics.This paper is presented as an outcome of the LMS Durham Symposium convened by Professor C.T.H. Baker on 4–14 July 1992 with support from the SERC under grant reference number GR/H03964.     

On homoclinic bifurcation emanating from Takens-Bogdanov points in Hamiltonian systems

The paper is devoted to studying the bifurcation of periodic and homoclinic orbits in a 2n-dimensional Hamiltonian system with 1 parameter from a TB-point (Hamiltonian saddle node). In addition to the proof of existence, the paper gives an expansion formula of the bifurcating homoclinic orbits. With the help of center manifold reduction and a blow up transformation, the problem is focused on studying a planar Hamiltonian system, the proof for the perturbed homoclinic and periodic orbits is elementary in the sense that it uses only implicit function arguments. Two applications to travelling waves in PDEs are shown.     

HOMOCLINIC ORBITS FOR A CLASS OF HAMILTONIAN SYSTEMS WITH POTENTIALS CHANGING SIGN

In this paper, we prove the existence of nontrivial homoclinic orbits for a class of Hamiltonian systems with potential changing sign. We use Mountain Pass Lemma.     

Symbolic dynamics and Arnold diffusion

We consider hyperbolic tori of three degrees of freedom initially hyperbolic Hamiltonian systems. We prove that if the stable and unstable manifold of a hyperbolic torus intersect transversaly, then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a Bernoulli shift. The proof is based on a new geometrico-dynamical feature of partially hyperbolic systems, the transversality-torsion phenomenon, which produces complete hyperbolicity from partial hyperbolicity. We deduce the existence of infinitely many hyperbolic periodic orbits near the given torus. The relevance of these results for the instability of near-integrable Hamiltonian systems is then discussed. For a given transition chain, we construct chain of hyperbolic periodic orbits. Then we easily prove the existence of periodic orbits of arbitrarily high period close to such chain using standard results on hyperbolic sets.     

具有小受迫项的哈密顿系统的同宿轨道

By Brezis-Nirenberg type Mountain Pass Theorem, the research has focused on the existence of nontrivial homoclinic orbits for a class of second order Hamiltonian systems with non-Ambrosetti-Rabinowitz type superquadratic potentials and small forced terms.