Infinitely many homoclinic solutions for second order Hamiltonian systems

In this paper we study the existence of infinitely many homoclinic solutions for second order Hamiltonian systems , , where L(t) is unnecessarily positive definite for all t∈R, and W(t,u) is of subquadratic growth as |u|→∞.     

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.     

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

We improve Benci and Rabinowitz's Linking theorem for strongly indefinite functionals, giving estimates for a suitably defined relative Morse index of critical points. Such abstract result is applied to the existence problem of periodic orbits and homoclinic solutions of first order Hamiltonian systems in cases where the Palais-Smale condition does not hold. Received January 27, 1999 / Accepted January 14, 2000 / Published online July 20, 2000     

Infinitely many homoclinic solutions for a second‐order Hamiltonian system

In this paper, we study the homoclinic solutions of the following second‐order Hamiltonian system where , and . Applying the symmetric Mountain Pass Theorem, we establish a couple of sufficient conditions on the existence of infinitely many homoclinic solutions. Our results significantly generalize and improve related ones in the literature. For example, is not necessary to be uniformly positive definite or coercive; through is still assumed to be superquadratic near , it is not assumed to be superquadratic near .     

Homoclinic solutions for a class of the second order Hamiltonian systems

We study the existence of homoclinic orbits for the second order Hamiltonian system , where q∈Rⁿ and V∈C¹(R×Rⁿ,R), V(t,q)=-K(t,q)+W(t,q) is T-periodic in t. A map K satisfies the “pinching” condition b₁|q|²?K(t,q)?b₂|q|², W is superlinear at the infinity and f is sufficiently small in L²(R,Rⁿ). A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a certain sequence of the second order differential equations.     

Heteroclinic solutions for a class of the second order Hamiltonian systems

We shall be concerned with the existence of heteroclinic orbits for the second order Hamiltonian system , where q∈Rn and V∈C¹(R×Rn,R), V?0. We will assume that V and a certain subset M⊂Rn satisfy the following conditions. M is a set of isolated points and #M?2. For every sufficiently small ε>0 there exists δ>0 such that for all (t,z)∈R×Rn, if d(z,M)?ε then −V(t,z)?δ. The integrals , z∈M, are equi-bounded and −V(t,z)→∞, as |t|→∞, uniformly on compact subsets of Rn?M. Our result states that each point in M is joined to another point in M by a solution of our system.     

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.     

Relative Morse index and multiple brake orbits of asymptotically linear Hamiltonian systems in the presence of symmetries

In this paper, we define a relative Morse index for two continuous symmetric matrices paths in R2n satisfying condition (B1) and study its relation with the Maslov-type indices under brake orbit boundary value of these two symmetric matrices paths. As applications, using this relation we obtain a multiple existence of periodic brake orbit solutions of asymptotically linear Hamiltonian system in the presence of symmetries.     

Homoclinic orbits for a nonperiodic Hamiltonian system

In this paper we prove the existence and multiplicity of homoclinic orbits for first order Hamiltonian systems of the form

     

Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials

In this paper we study the following nonperiodic second order Hamiltonian systems

     

Multiplicity of homoclinics for a class of time recurrent second order Hamiltonian systems

We prove the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems in of the form , where we assume the existence of a sequence such that and as for any . Moreover, under a suitable non degeneracy condition, we prove that this class of systems admits multibump solutions. Received February 2, 1996 / In revised form July 5, 1996 / Accepted October 10, 1996     

A twist condition and periodic solutions of Hamiltonian systems

In this paper, we investigate existence of nontrivial periodic solutions to the Hamiltonian system

(HS)     

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.     

Multiple brake orbits in bounded convex symmetric domains

In this paper, we prove that there exist at least two geometrically distinct brake orbits in every bounded convex symmetric domain in .     

On superquadratic periodic systems with indefinite linear part

We consider a second order periodic system with an indefinite linear part and a potential function which is superquadratic but does not satisfy the AR-condition. Using Morse critical groups, we show that the system has at least one nontrivial solution.