scholarly journals Existence of Homoclinic Orbits for Hamiltonian Systems with Superquadratic Potentials

2009 ◽  
Vol 2009 ◽  
pp. 1-15
Author(s):  
Jian Ding ◽  
Junxiang Xu ◽  
Fubao Zhang
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Symmetric Matrix ◽  
Homoclinic Orbits ◽  
Linking Theorem

This paper concerns solutions for the Hamiltonian system:z˙=𝒥Hz(t,z). HereH(t,z)=(1/2)z⋅Lz+W(t,z),Lis a2N×2Nsymmetric matrix, andW∈C1(ℝ×ℝ2N,ℝ). We consider the case that0∈σc(−(𝒥(d/dt)+L))andWsatisfies some superquadratic condition different from the type of Ambrosetti-Rabinowitz. We study this problem by virtue of some weak linking theorem recently developed and prove the existence of homoclinic orbits.

scholarly journals Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Wenping Qin ◽  
Jian Zhang ◽  
Fukun Zhao
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Point Spectrum ◽  
Homoclinic Orbits ◽  
Hamiltonian Operator ◽  
Quadratic Nonlinearity ◽  
Linking Theorem ◽  
Strongly Indefinite Functionals ◽  
Superquadratic Nonlinearity

We study the following nonperiodic Hamiltonian systemż=JHz(t,z), whereH∈C1(R×R2N,R)is the formH(t,z)=(1/2)B(t)z⋅z+R(t,z). We introduce a new assumption onB(t)and prove that the corresponding Hamiltonian operator has only point spectrum. Moreover, by applying a generalized linking theorem for strongly indefinite functionals, we establish the existence of homoclinic orbits for asymptotically quadratic nonlinearity as well as the existence of infinitely many homoclinic orbits for superquadratic nonlinearity.

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 Periodic Solutions for Second Order Hamiltonian Systems with Impulses via the Local Linking Theorem

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Longsheng Bao ◽  
Binxiang Dai
Keyword(s):  
Periodic Solution ◽  
Hamiltonian System ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Second Order ◽  
Linking Theorem ◽  
Local Linking ◽  
Second Order Hamiltonian Systems ◽  
New Criterion

A class of second order impulsive Hamiltonian systems are considered. By applying a local linking theorem, we establish the new criterion to guarantee that this impulsive Hamiltonian system has at least one nontrivial T-periodic solution under local superquadratic condition. This result generalizes and improves some existing results in the known literature.

The shadowing chain lemma for singular Hamiltonian systems involving strong forces

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Marek Izydorek ◽  
Joanna Janczewska
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Homoclinic Orbits ◽  
Global Maximum ◽  
Infinite Depth ◽  
Strong Force ◽  
Autonomous Hamiltonian System ◽  
Existence And Multiplicity ◽  
Single Well ◽  
Singular Hamiltonian Systems

AbstractWe consider a planar autonomous Hamiltonian system :q+∇V(q) = 0, where the potential V: ℝ2 \{ζ}→ ℝ has a single well of infinite depth at some point ζ and a strict global maximum 0at two distinct points a and b. Under a strong force condition around the singularity ζ we will prove a lemma on the existence and multiplicity of heteroclinic and homoclinic orbits — the shadowing chain lemma — via minimization of action integrals and using simple geometrical arguments.

Homoclinic orbits for a class of Hamiltonian systems

1990 ◽  
Vol 114 (1-2) ◽  
pp. 33-38 ◽  
Author(s):  
Paul H. Rabinowitz
Keyword(s):  
Hamiltonian System ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Homoclinic Orbit ◽  
Mountain Pass Theorem ◽  
Local Maximum ◽  
Homoclinic Orbits ◽  
Order Hamiltonian System ◽  
Image Position ◽  
Second Order Hamiltonian System

SynopsisConsider the second order Hamiltonian system:where q ∊ ℝn and V ∊ C1 (ℝ ×ℝn ℝ) is T periodic in t. Suppose Vq (t, 0) = 0, 0 is a local maximum for V(t,.) and V(t, x) | x| → ∞ Under these and some additional technical assumptions we prove that (HS) has a homoclinic orbit q emanating from 0. The orbit q is obtained as the limit as k → ∞ of 2kT periodic solutions (i.e. subharmonics) qk of (HS). The subharmonics qk are obtained in turn via the Mountain Pass Theorem.

Odd Homoclinic Orbits for a Second Order Hamiltonian System

2012 ◽  
Vol 12 (1) ◽  
Author(s):  
Liliane A. Maia ◽  
Olimpio H. Miyagaki ◽  
Sergio H. M. Soares
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Second Order ◽  
Concentration Compactness ◽  
Concentration Compactness Principle ◽  
Order Hamiltonian System ◽  
Compactness Principle ◽  
Second Order Hamiltonian System

AbstractThe aim of this paper is to find an odd homoclinic orbit for a class of reversible Hamiltonian systems. The proof is variational and it employs a version of the concentration compactness principle of P. L. Lions in a lemma due to Struwe.

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.

FAMILIES OF TRANSVERSE POINCARÉ HOMOCLINIC ORBITS IN 2N-DIMENSIONAL HAMILTONIAN SYSTEMS CLOSE TO THE SYSTEM WITH A LOOP TO A SADDLE-CENTER

1996 ◽  
Vol 06 (06) ◽  
pp. 991-1006 ◽  
Author(s):  
O. YU. KOLTSOVA ◽  
L. M. LERMAN
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Homoclinic Orbit ◽  
Degrees Of Freedom ◽  
Homoclinic Orbits ◽  
New Criterion

We prove the theorem: if an n-degrees-of-freedom Hamiltonian system has an equilibrium of the saddle-center type (there is a pair of simple eigenvalues ±iω; the rest of the spectrum consists of eigenvalues with nonzero real parts) with a homoclinic orbit to it then this system, and all those close to it, have transversal Poincaré homoclinic orbits to Lyapunov periodic orbits if some genericity conditions are satisfied. These conditions are pointed out explicitly. Thus a new criterion of nonintegrability has been obtained.

Homoclinic orbits for a class of singular second order Hamiltonian systems in ℝ3

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Joanna Janczewska ◽  
Jakub Maksymiuk
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Homoclinic Orbits ◽  
Second Order ◽  
Singular Points ◽  
Global Maximum ◽  
Strong Force ◽  
Second Order Hamiltonian Systems ◽  
Order Hamiltonian System ◽  
Second Order Hamiltonian System

AbstractWe consider a conservative second order Hamiltonian system $$\ddot q + \nabla V(q) = 0$$ in ℝ3 with a potential V having a global maximum at the origin and a line l ∩ {0} = ϑ as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system.

Sign in / Sign up

Export Citation Format

Share Document