Homoclinic orbits for a nonperiodic Hamiltonian system

Homoclinic Orbits of Nonperiodic Superquadratic Hamiltonian System

In this paper, we study the following first-order nonperiodic Hamiltonian system ż = JHz(t, z), where H ∈ C1(R× R ,R) is the form H(t, z) = 1 2 L(t)z · z + R(t, z). Under weak superquadratic condition on the nonlinearitiy. By applying the generalized Nehari manifold method developed recently by Szulkin and Weth, we prove the existence of homoclinic orbits, which are ground state solutions for a...

Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems with Some Twisted Conditions

and Applied Analysis 3 Remark 5. Lemma 1 shows that σ(A), the spectrum of A, consists of eigenvalues numbered by (counted in their multiplicities): ⋅ ⋅ ⋅ ≤ λ −2 ≤ λ −1 ≤ 0 < λ 1 ≤ λ 2 ≤ ⋅ ⋅ ⋅ (14) with λ ±k → ±∞ as k → ∞. Let B 0 (t) ≡ B 0 and B ∞ (t) ≡ B ∞ , with the constants B 0 , B ∞ satisfying λ l < B 0 < λ l+1 , and λ l+i < B ∞ < λ l+i+1 for some l ∈ Z and i ≥ 1 (or i ≤ −1). Define R (t, ...

Orbits homoclinic to resonances: the Hamiltonian case

In this paper we develop methods to show the existence of orbits homoclinic or heteroclinic to periodic orbits, hyperbolic fixed points or combinations of hyperbolic fixed points and/or periodic orbits in a class of two-degree-offreedom, integrable Hamiltonian systems subject to arbitrary Hamiltonian perturbations. Our methods differ from previous methods in that the invariant sets (periodic or...

Homoclinic Orbits for a Class of Noncoercive Discrete Hamiltonian Systems

Homoclinic orbits for discrete Hamiltonian systems with subquadratic potential

where n ∈ Z, u ∈ RN , u(n) = u(n + ) – u(n) is the forward difference operator, p,L : Z→ RN×N and W : Z× RN → R. As usual, we say that a solution u(n) of system (.) is homoclinic (to ) if u(n)→  as n→±∞. In addition, if u(n) ≡ , then u(n) is called a nontrivial homoclinic solution. In general, system (.) may be regarded as a discrete analogue of the following second order Hamiltonian sy...