Search

In this article, we study the existence of periodic solutions to second order Hamiltonian systems. Our goal is twofold. When the nonlinear term satisfies a strictly monotone condition, we show that, for any T > 0 , there exists a T-periodic solution with minimal period T. When the nonlinear term satisfies a non-decreasing condition, using a perturbation technique, we prove a similar result. In the latter case, the periodic solution corresponds to a critical point which minimizes the variational functional on the Nehari manifold which is not homeomorphic to the unit sphere. [ABSTRACT FROM AUTHOR]

Global bifurcation and spatio-temporal patterns of periodic solutions (with prescribed period) to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer/Leray–Schauder O (2) × Γ × ℤ 2 -equivariant degree theory. Here, O (2) is related to the reversal symmetry combined with the autonomous form of the system, Γ reflects extra spacial symmetries of the system, and ℤ 2 is related to the oddness of the right-hand side. Abstract results are supported by a concrete example with Γ = D 6 — the dihedral group of order 12. [ABSTRACT FROM AUTHOR]