Homoclinic solutions for eventually autonomous high-dimensional Hamiltonian systems

Holmes and Stuart [Z. Angew. Math. Phys. 43 (1992) 598-625] have investigated homoclinic solutions for eventually autonomous planar flows by analysing the geometry of the stable and unstable manifolds. We extend their discussion to higher-dimensional systems of Hamiltonian type by formulating the problem as the existence of intersection points of two Lagrangian manifolds. Their various assumptions can be restated and interpreted as ensuring some complexity of the generating function of one of the Lagrangian manifold with respect to symplectic coordinates that trivialise the second Lagrangian manifold. The critical points thus obtained correspond to homoclinic solutions. The main new feature in high-dimensions is that twice as many homoclinic solutions are found as for planar flows, in analogy with results obtained for autonomous Lagrangian systems by Ambrosetti and Coti Zelati [Rend. Sem. Mat. Univ. Padova 89 (1993) 177-194].     

Homoclinic orbits for Hamiltonian systems induced by impulses

In this paper, we consider a class of impulsive Hamiltonian systems with a p‐Laplacian operator. Under certain conditions, we establish the existence of homoclinic orbits by means of the mountain pass theorem and an approximation technique. In some special cases, the homoclinic orbits are induced by the impulses in the sense that the associated non‐impulsive systems admit no non‐trivial homoclinic orbits. Copyright © 2015 John Wiley & Sons, Ltd.     

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 , where t∈R, u∈Rn and W₁,W₂∈C¹(R×Rn,R) and f∈C(R,Rn) are not necessary periodic in t. This result generalizes and improves some existing results in the literature.     

Homoclinic orbits for second-order Hamiltonian systems with subquadratic potentials

In this paper we consider a class of subquadratic second-order Hamiltonian systems and new results about the existence and multiplicity of homoclinic orbits are obtained by using the Minimizing Theorem and the Clark’s Theorem respectively and a new compact imbedding theorem is also proved.     

Stability sets of multiparameter Hamiltonian systems

A real linear Hamiltonian system with constant coefficients that depend on several real parameters is considered. A method is proposed for calculating the sets of all values of the parameters for which the stationary solution of this system is stable for fixed values of the parameters (that is, the stability sets). The application of the method is demonstrated for a gyroscopic problem described by a Hamiltonian system with four degrees of freedom and three parameters. Computer algebra, in particular, a Gröbner basis and a Power Geometry are used. It is shown that the four-parameter generalization of this problem does not contain fundamentally new difficulties.     

Homoclinic solutions for a class of second order Hamiltonian systems

In this paper, we study homoclinic solutions for the nonperiodic second order Hamiltonian systems where L is unnecessarily coercive or uniformly positively definite, and is only locally defined near the origin with respect to u. Under some general conditions on L and W, we show that the above system has infinitely many homoclinic solutions near the origin. Some related results in the literature are extended and generalized.     

Weakly invariant sets of hybrid systems

We consider a mathematical model of a hybrid system in which the continuous dynamics generated at any point in time by one of a given finite family of continuous systems alternates with discrete operations commanding either an instantaneous switching from one system to another, or an instantaneous passage from current coordinates to some other coordinates, or both operations simultaneously. As a special case, we consider a model of a linear switching system. For a hybrid system, we introduce the notion of a weakly invariant set and analyze its structure. We obtain a representation of a weakly invariant set as a union of sets of simpler structure. For the latter sets, we introduce special value functions, for which we obtain expressions by methods of convex analysis. For the same functions, we derive equations of the Hamilton-Jacobi-Bellman type, which permit one to pass from the problem of constructing weakly invariant sets to the control synthesis problem for a hybrid system.     

Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems

In this paper, we study the existence of infinitely many homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. By using the variant fountain theorem, we obtain a new criterion for guaranteeing that second-order Hamiltonian systems has infinitely many homoclinic solutions. Recent results from the literature are generalized and significantly improved. An example is also given in this paper to illustrate our main results.     

Homoclinic solutions for a class of asymptotically quadratic Hamiltonian systems

In this paper we are devoted to considering the existence of homoclinic solutions for some second order non-autonomous Hamiltonian systems with the asymptotically quadratic potential at infinity. The proof is based on a variant version of the Mountain Pass Theorem. Recent results in the literature are generalized and significantly improved.     

Homoclinic and heteroclinic orbits for a class of Hamiltonian systems

The existence of a rich structure of homoclinic and heteroclinic solutions is established for a family of Hamiltonian systems that serve as a simpler model for the multiple pendulum system. The proof is based on recently developed arguments from the calculus of variations that have proved useful in finding actual solutions of an equation near approximate solution.This research was sponsorted in part by the National Science Foundation under grant #MCS-8110556 and the U. S. Army Research Office under Contract #DAAL03-87-K-0043. Any reproduction for the purpose of the United States Government is permitted.     

Homoclinic orbits for the second-order Hamiltonian systems with obstacle item

This paper is concerned with the existence of homoclinic orbits for the second-order Hamiltonian system with obstacle item, ü(t)-A u(t) =▽F (t, u), where F (t, u) is T-periodic in t with ▽F (t, u) = L(t)u + ▽R(t,u). By using a generalized linking theorem for strongly indefinite functionals, we prove the existence of homoclinic orbits for both the super-quadratic case and the asymptotically linear one.