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.

scholarly journals Nonexistence of Homoclinic Orbits for a Class of Hamiltonian Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Xiaoyan Lin ◽  
Qi-Ming Zhang ◽  
X. H. Tang
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Satisfying Condition ◽  
Sufficient Conditions ◽  
Homoclinic Orbits ◽  
First Order ◽  
Nonlinear Hamiltonian System

We give several sufficient conditions under which the first-order nonlinear Hamiltonian systemx'(t)=α(t)x(t)+f(t,y(t)),  y'(t)=-g(t,x(t))-α(t)y(t)has no solution(x(t),y(t))satisfying condition0<∫-∞+∞[|x(t)|ν+(1+β(t))|y(t)|μ]dt<+∞‍,whereμ,ν>1and(1/μ)+(1/ν)=1,0≤xf(t,x)≤β(t)|x|μ,xg(t,x)≤γ0(t)|x|ν,β(t),γ0(t)≥0, andα(t)are locally Lebesgue integrable real-valued functions defined onℝ.

scholarly journals Homoclinic Orbits for a Class of Noncoercive Discrete Hamiltonian Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
Long Yuhua
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Homoclinic Orbits ◽  
Existence Results ◽  
Mountain Pass ◽  
First Order ◽  
Discrete Hamiltonian System ◽  
Generalized Mountain Pass Theorem

A class of first-order noncoercive discrete Hamiltonian systems are considered. Based on a generalized mountain pass theorem, some existence results of homoclinic orbits are obtained when the discrete Hamiltonian system is not periodical and need not satisfy the global Ambrosetti-Rabinowitz condition.

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.

Limit Cycle Bifurcations from an Order-3 Nilpotent Center of Cubic Hamiltonian Systems Perturbed by Cubic Polynomials

2020 ◽  
Vol 30 (09) ◽  
pp. 2050126
Author(s):  
Li Zhang ◽  
Chenchen Wang ◽  
Zhaoping Hu
Keyword(s):  
Hamiltonian System ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Melnikov Function ◽  
First Order ◽  
Cubic Polynomials ◽  
Nilpotent Center

From [Han et al., 2009a] we know that the highest order of the nilpotent center of cubic Hamiltonian system is [Formula: see text]. In this paper, perturbing the Hamiltonian system which has a nilpotent center of order [Formula: see text] at the origin by cubic polynomials, we study the number of limit cycles of the corresponding cubic near-Hamiltonian systems near the origin. We prove that we can find seven and at most seven limit cycles near the origin by the first-order Melnikov function.

Limit Cycle Bifurcations by Perturbing a Hamiltonian System with a 3-Polycycle Having a Cusp of Order One or Two

2018 ◽  
Vol 28 (03) ◽  
pp. 1850038
Author(s):  
Marzieh Mousavi ◽  
Hamid R. Z. Zangeneh
Keyword(s):  
Asymptotic Expansion ◽  
Hamiltonian System ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Perturbed Systems

In this paper, we study the asymptotic expansion of the first order Melnikov function near a 3-polycycle connecting a cusp (of order one or two) to two hyperbolic saddles for a near-Hamiltonian system in the plane. The formulas for the first coefficients of the expansion are given as well as the method of bifurcation of limit cycles. Then we use the results to study two Hamiltonian systems with this 3-polycycle and determine the number and distribution of limit cycles that can bifurcate from the perturbed systems. Moreover, a sharp upper bound for the number of limit cycles bifurcated from the whole periodic annulus is found when there is a cusp of order one.

LOCALIZING BOUNDS FOR COMPACT INVARIANT SETS OF NONLINEAR SYSTEMS POSSESSING FIRST INTEGRALS WITH APPLICATIONS TO HAMILTONIAN SYSTEMS

2010 ◽  
Vol 20 (05) ◽  
pp. 1477-1483 ◽  
Author(s):  
KONSTANTIN E. STARKOV
Keyword(s):  
Hamiltonian System ◽  
Nonlinear Systems ◽  
Hamiltonian Systems ◽  
Positive Definite ◽  
First Integrals ◽  
Invariant Sets ◽  
Yang Mills ◽  
First Order ◽  
Special Case ◽  
Extremum Conditions

In this paper, we study the localization problem of compact invariant sets of nonlinear systems possessing first integrals by using the first order extremum conditions and positive definite polynomials. In the case of natural polynomial Hamiltonian systems, our results include those in [Starkov, 2008] as a special case. This paper discusses the application to studies of the generalized Yang–Mills Hamiltonian system and the Hamiltonian system describing dynamics of hydrogenic atoms in external fields.

Connecting orbits for a reversible Hamiltonian system

2000 ◽  
Vol 20 (6) ◽  
pp. 1767-1784 ◽  
Author(s):  
PAUL H. RABINOWITZ
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Homoclinic Solutions ◽  
Connecting Orbits

The existence of heteroclinic and homoclinic solutions which shadow corresponding chains of such solutions is established for a class of reversible Hamiltonian systems. The proof involves elementary minimization arguments.

Limit Cycle Bifurcations by Perturbing a Hamiltonian System with a Cuspidal Loop of Order m

2015 ◽  
Vol 25 (06) ◽  
pp. 1550083 ◽  
Author(s):  
Yanqing Xiong
Keyword(s):  
Hamiltonian System ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Liénard System ◽  
First Order ◽  
Lienard System

This paper is concerned with the expansion of the first-order Melnikov function for general Hamiltonian systems with a cuspidal loop having order m. Some criteria and formulas are derived, which can be used to obtain first-order coefficients in the expansion. In particular, we deduce the first-order coefficients for the case m = 3 and give the corresponding conditions of existing several limit cycles. As an application, we study a Liénard system of type (n, 9) and prove that it can have 14 limit cycles near a cuspidal loop of order 3 for n = 8.

Quasilinear Hamiltonian systems

1987 ◽  
Vol 106 (3-4) ◽  
pp. 195-204
Author(s):  
Jan S. Rogulski
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Symplectic Form ◽  
Scalar Product ◽  
Sufficient Conditions ◽  
Hamiltonian Form ◽  
Quasilinear Systems ◽  
Necessary And Sufficient ◽  
Evolution Systems ◽  
General Method

SynopsisWe consider quasilinear systems of 2N partial differential equations with 2N unknown functions depending on n + 1 variables as evolution systems on the space L2(Rn, RN) × L2(Rns, RN) endowed with a symplectic form induced by the standard scalar product on L2(Rn, RN). The necessary and sufficient conditions for such a system to be a Hamiltonian system are derived. The main purpose of this paper is to propose a straightforward link between the symplectic approach formulated by Chernoff, Hughes and Marsden and the multisymplectic formulations of evolution systems created by Kijowski and developed by Gawedzki and Kondracki. A general method of constructing the multisymplectic form and the Hamiltonian form for these systems is given.

scholarly journals Second Order Melnikov Functions of Piecewise Hamiltonian Systems

2020 ◽  
Vol 30 (01) ◽  
pp. 2050016
Author(s):  
Peixing Yang ◽  
Jean-Pierre Françoise ◽  
Jiang Yu
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Second Order ◽  
Quadratic Polynomial ◽  
First Order ◽  
Melnikov Functions

In this paper, we consider the general perturbations of piecewise Hamiltonian systems. A formula for the second order Melnikov functions is derived when the first order Melnikov functions vanish. As an application, we can improve an upper bound of the number of bifurcated limit cycles of a piecewise Hamiltonian system with quadratic polynomial perturbations.

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