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

2000 ◽  
Vol 11 (4) ◽  
pp. 395-430 ◽  
Author(s):  
Alberto Abbondandolo ◽  
Juan Molina
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Homoclinic Solutions ◽  
Strongly Indefinite Functionals ◽  
First Order

scholarly journals On Homoclinic Solutions for First-Order Superquadratic Hamiltonian Systems with Spectrum Point Zero

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Feng Li ◽  
Juntao Sun
Keyword(s):  
Critical Point ◽  
Hamiltonian Systems ◽  
Homoclinic Solutions ◽  
Strongly Indefinite Functionals ◽  
Spectrum Point ◽  
First Order ◽  
Existence And Multiplicity ◽  
Critical Point Theorems

The existence and multiplicity of homoclinic solutions for a class of first-order periodic Hamiltonian systems with spectrum point zero are obtained. The proof is based on two critical point theorems for strongly indefinite functionals. Some recent results are improved and extended.

scholarly journals Nonexistence of Homoclinic Solutions for a Class of Discrete Hamiltonian Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Xiaoping Wang
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Satisfying Condition ◽  
Sufficient Conditions ◽  
Homoclinic Solutions ◽  
First Order ◽  
Discrete Hamiltonian System

We give several sufficient conditions under which the first-order nonlinear discrete Hamiltonian systemΔx(n)=α(n)x(n+1)+β(n)|y(n)|μ-2y(n),Δy(n)=-γ(n)|x(n+1)|ν-2x(n+1)-α(n)y(n)has no solution(x(n),y(n))satisfying condition0<∑n=-∞+∞[|x(n)|ν+(1+β(n))|y(n)|μ]<+∞, whereμ,ν>1and1/μ+1/ν=1andα(n),β(n),andγ(n)are real-valued functions defined onℤ.

BIFURCATION OF LIMIT CYCLES BY PERTURBING PIECEWISE HAMILTONIAN SYSTEMS

2010 ◽  
Vol 20 (05) ◽  
pp. 1379-1390 ◽  
Author(s):  
XIA LIU ◽  
MAOAN HAN
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Unperturbed System ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
General Perturbation ◽  
Smooth System

In this paper, the general perturbation of piecewise Hamiltonian systems on the plane is considered. When the unperturbed system has a family of periodic orbits, similar to the perturbations of smooth system, an expression of the first order Melnikov function is derived, which can be used to study the number of limit cycles bifurcated from the periodic orbits. As applications, the number of bifurcated limit cycles of several concrete piecewise systems are presented.

Homoclinic Solutions for a Class of Hamiltonian Systems

2004 ◽  
Vol 4 (1) ◽  
Author(s):  
Morched Boughariou
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Homoclinic Solution ◽  
Homoclinic Solutions ◽  
Absolute Maximum ◽  
First Order ◽  
Order Hamiltonian System

AbstractWe consider the first order Hamiltonian systemq̇ = Hp(p, q),ṗ = −Hq(p, q), (HS)where p, q : ℝ → ℝUnder the condition that V has a unique absolute maximum at 0 and some technical assumptions on H, we prove that (HS) has a non-trivial homoclinic solution.

scholarly journals Normal Form Near Orbit Segments of Convex Hamiltonian Systems

2021 ◽  
Author(s):  
Shahriar Aslani ◽  
Patrick Bernard
Keyword(s):  
Normal Form ◽  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Vector Fields ◽  
Energy Surface ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
The Other ◽  
Finsler Metrics ◽  
The One

Abstract In the study of Hamiltonian systems on cotangent bundles, it is natural to perturb Hamiltonians by adding potentials (functions depending only on the base point). This led to the definition of Mañé genericity [ 8]: a property is generic if, given a Hamiltonian $H$, the set of potentials $g$ such that $H+g$ satisfies the property is generic. This notion is mostly used in the context of Hamiltonians that are convex in $p$, in the sense that $\partial ^2_{pp} H$ is positive definite at each point. We will also restrict our study to this situation. There is a close relation between perturbations of Hamiltonians by a small additive potential and perturbations by a positive factor close to one. Indeed, the Hamiltonians $H+g$ and $H/(1-g)$ have the same level one energy surface, hence their dynamics on this energy surface are reparametrisation of each other, this is the Maupertuis principle. This remark is particularly relevant when $H$ is homogeneous in the fibers (which corresponds to Finsler metrics) or even fiberwise quadratic (which corresponds to Riemannian metrics). In these cases, perturbations by potentials of the Hamiltonian correspond, up to parametrisation, to conformal perturbations of the metric. One of the widely studied aspects is to understand to what extent the return map associated to a periodic orbit can be modified by a small perturbation. This kind of question depends strongly on the context in which they are posed. Some of the most studied contexts are, in increasing order of difficulty, perturbations of general vector fields, perturbations of Hamiltonian systems inside the class of Hamiltonian systems, perturbations of Riemannian metrics inside the class of Riemannian metrics, and Mañé perturbations of convex Hamiltonians. It is for example well known that each vector field can be perturbed to a vector field with only hyperbolic periodic orbits, this is part of the Kupka–Smale Theorem, see [ 5, 13] (the other part of the Kupka–Smale Theorem states that the stable and unstable manifolds intersect transversally; it has also been studied in the various settings mentioned above but will not be discussed here). In the context of Hamiltonian vector fields, the statement has to be weakened, but it remains true that each Hamiltonian can be perturbed to a Hamiltonian with only non-degenerate periodic orbits (including the iterated ones), see [ 11, 12]. The same result is true in the context of Riemannian metrics: every Riemannian metric can be perturbed to a Riemannian metric with only non-degenerate closed geodesics, this is the bumpy metric theorem, see [ 1, 2, 4]. The question was investigated only much more recently in the context of Mañé perturbations of convex Hamiltonians, see [ 9, 10]. It is proved in [ 10] that the same result holds: if $H$ is a convex Hamiltonian and $a$ is a regular value of $H$, then there exist arbitrarily small potentials $g$ such that all periodic orbits (including iterated ones) of $H+g$ at energy $a$ are non-degenerate. The proof given in [ 10] is actually rather similar to the ones given in papers on the perturbations of Riemannian metrics. In all these proofs, it is very useful to work in appropriate coordinates around an orbit segment. In the Riemannian case, one can use the so-called Fermi coordinates. In the Hamiltonian case, appropriate coordinates are considered in [ 10,Lemma 3.1] itself taken from [ 3, Lemma C.1]. However, as we shall detail below, the proof of this Lemma in [ 3], Appendix C, is incomplete, and the statement itself is actually wrong. Our goal in the present paper is to state and prove a corrected version of this normal form Lemma. Our proof is different from the one outlined in [ 3], Appendix C. In particular, it is purely Hamiltonian and does not rest on the results of [ 7] on Finsler metrics, as [ 3] did. Although our normal form is weaker than the one claimed in [ 10], it is actually sufficient to prove the main results of [ 6, 10], as we shall explain after the statement of Theorem 1, and probably also of the other works using [ 3, Lemma C.1].

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

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