Rotating periodic solutions for asymptotically linear second‐order Hamiltonian systems with resonance at infinity

Abstract

In this paper, we consider a class of asymptotically linear second-order Hamiltonian system with resonance at infinity. We will use Morse theory combined with the technique of penalized functionals to obtain the existence of rotating periodic solutions.

Rotating periodic solutions for second order systems with Hartman-type nonlinearity

“…Here we modify this idea and gluing two C 1 paths of solutions together to obtain a continuous path of solutions, by which we can track along the trajectory to obtain the rotating periodic solutions (specially, quasi-periodic solutions) of (1.1). For some recent work on rotating periodic solutions of ODEs, one can see [16][17][18][19] and the references.…”

Learning Coupled Oscillators System with Reservoir Computing

“…In [24], the authors used the method of rotating periodic solutions to obtain the periodic synchronous solution of the above system. Rotating periodic solutions as a generalization of periodic solutions can be expressed by equations x(t + T) = Qx(t) with some orthogonal matrix Q [27][28][29][30]. We will focus on the reconstruction of a ring of three unidirectional coupled Lorenz oscillators.…”

Rotating periodic integrable solutions for second-order differential systems with nonresonance condition

“…Liu, Li and Yang [10] studied the asymptotically linear second-order Hamiltonian system, and by Morse theory and the technique of penalized functionals, the authors obtained the existence of rotating periodic solutions for system satisfying the resonance condition at infinity. Then in [11] and [12], using Morse theory and critical point theorems, they continued to study the existence of rotating periodic solutions for superlinear Hamiltonian systems and multiplicity of rotating periodic solutions for Hamiltonian systems with resonant conditions.…”