Quasi-periodic perturbations within the reversible context 2 in KAM theory

The paper consists of two sections. In Section 1, we give a short review of KAM theory with an emphasis on Whitney smooth families of invariant tori in typical Hamiltonian and reversible systems. In Section 2, we prove a KAM-type result for non-autonomous reversible systems (depending quasi-periodically on time) within the almost unexplored reversible context 2. This context refers to the situation where dim Fix G < (1/2) codim T, here Fix G is the fixed point manifold of the reversing involution G and T is the invariant torus one deals with.Comment: 15 page

Periodic solutions of second order Hamiltonian systems bifurcating from infinity

The goal of this article is to study closed connected sets of periodic solutions, of autonomous second order Hamiltonian systems, emanating from infinity. The main idea is to apply the degree for SO(2)-equivariant gradient operators defined by the second author. Using the results due to Rabier we show that we cannot apply the Leray-Schauder degree to prove the main results of this article. It is worth pointing out that since we study connected sets of solutions, we also cannot use the Conley index technique and the Morse theory.Comment: 24 page

On the behavior of periodic solutions of planar autonomous Hamiltonian systems with multivalued periodic perturbations

Aim of the paper is to provide a method to analyze the behavior of $T$-periodic solutions x_\eps, \eps>0, of a perturbed planar Hamiltonian system near a cycle $x_0$, of smallest period $T$, of the unperturbed system. The perturbation is represented by a $T$-periodic multivalued map which vanishes as \eps\to0. In several problems from nonsmooth mechanical systems this multivalued perturbation comes from the Filippov regularization of a nonlinear discontinuous $T$-periodic term. \noindent Through the paper, assuming the existence of a $T$-periodic solution x_\eps for \eps>0 small, under the condition that $x_0$ is a nondegenerate cycle of the linearized unperturbed Hamiltonian system we provide a formula for the distance between any point $x_0(t)$ and the trajectories x_\eps([0,T]) along a transversal direction to $x_0(t).

New existence and multiplicity of homoclinic solutions for second order non-autonomous systems

In this paper, we study the second order non-autonomous system \begin{eqnarray*} \ddot{u}(t)+A\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0, \ \ \forall t\in\mathbb{R}, \end{eqnarray*} where $A$ is an antisymmetric $N\times N$ constant matrix, $L\in C(\mathbb{R},\mathbb{R}^{N\times N})$ may not be uniformly positive definite for all $t\in\mathbb{R}$, and $W(t,u)$ is allowed to be sign-changing and local superquadratic. Under some simple assumptions on $A$, $L$ and $W$, we establish some existence criteria to guarantee that the above system has at least one homoclinic solution or infinitely many homoclinic solutions by using mountain pass theorem or fountain theorem, respectively. Recent results in the literature are generalized and significantly improved

Periodic solutions of second-order systems with subquadratic convex potential

In this paper, we investigate the existence of periodic solutions for the second order systems at resonance: \begin{equation} \begin{cases} \ddot u(t)+m^2\omega^2u(t)+\nabla F(t,u(t))=0\qquad \mbox{a.e. }t\in [0,T],\\ u(0)-u(T)=\dot u(0)-\dot u(T)=0, \end{cases} \end{equation} where $m>0$, the potential $F(t,x)$ is convex in $x$ and satisfies some general subquadratic conditions. The main results generalize and improve Theorem 3.7 in J. Mawhin and M. Willem [Critical point theory and Hamiltonian systems, Springer-Verlag, New York, 1989]