Periodic perturbations of Hamiltonian systems

Abstract

We prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincaré–Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems

Multiplicity of clines for systems of indefinite differential equations arising from a multilocus population genetics model

“…Indeed, their flexibility is one of the advantages of the application of topological methods. The employment of topological tools to generalize existence results for ODEs to systems of differential equations obtained by suitable coupling of the original equation has been recently applied in various frameworks, for instance in [6,19,20,21]. We remark that such results are not necessarily restricted to small perturbations: indeed, as in our case, they apply also to larger suitable couplings.…”

Ground states and multiple solutions for Hamiltonian elliptic system with gradient term

“…Moreover, the existence of nontrivial solutions, ground state solutions, multiple solutions and semiclassical solutions were obtained in these works by using various variational arguments, such as dual methods, reduction methods, generalized mountain pass theorem, generalized linking theorem and many others. For further related topics including the Hamiltonian systems, we refer the reader to [3,11,12,21] and their references. When b ≠ , as we all know, there are a few works devoted to the existence and multiplicity of solutions of system (1.1), see [15,30,32,36,40].…”

Existence and multiplicity of solutions for second-order Hamiltonian systems satisfying generalized periodic boundary value conditions at resonance

“…Solutions of Hamiltonian systems are very important in applications. In recent years, the existence and multiplicity of solutions for Hamiltonian systems via critical point theory have been studied by many authors (see [2,[5][6][7][8][9][10][12][13][14][15][16][17][18][19][20][21][22]). In particular, by means of critical point theory, the least action principle, and the minimax method, the existence and multiplicity of periodic solutions for second-order Hamiltonian systems with periodic boundary conditions were extensively studied in the cases where the gradient of the nonlinearity is bounded sublinearly and linearly, and many interesting results are given in [5,9,10,[13][14][15][16][17][18][19]22].…”