Periodic, quasi-periodic and unbounded solutions of radially symmetric systems with repulsive singularities at resonance

Abstract

In this paper, we are concerned with periodic solutions, quasiperiodic solutions and unbounded solutions for radially symmetric systems with singularities at resonance, which are 2π-periodic in time. The method is based on the qualitative analysis of Poincaré map with action-angle variables. The existence of infinitely many periodic and quasiperiodic solutions or unbounded motions depends on the oscillatory properties of a certain function.

Rotating periodic solutions for second‐order dynamical systems with singularities of repulsive type

“…Hence, our result applies to both the strong and weak force cases. In addition, our results imply the existence of nonplanar collisionless periodic and quasi‐periodic solutions, while in , only periodic solutions and quasi‐periodic solutions in the plane were obtained. Specially, we can obtain the existence of periodic solutions of if Q k = I n for some , which have been obtained in .…”

“…Moreover, in , Fonda and Ureña obtained a connected set of subharmonic and quasi‐periodic solutions of the planar system . In , Liu, Torres, and Qian applied the qualitative analysis of Poincaré map to obtain the existence of periodic, quasi‐periodic, and unbounded solutions in the plane for the repulsive singular system . In , the authors proved that system with admits collisionless rotating periodic solutions when the Habets–Sanchez‐type strong force condition holds and ∇ g is a coercive function.…”

Unbounded Solutions to Systems of Differential Equations at Resonance

“…In particular, according to [1,Proposition 3.1], any solution of (1.3) is unbounded both in the past and in the future whenever Both results were proved via an abstract method, based on the use of Lyapunov-like functions, developed in the same paper. Generalizations of this kind of results to more general situations (like asymmetric oscillators and planar Hamiltonian systems) were then obtained by many authors (see, among others, [2,[6][7][8][9][10]16,18,20,21,23] and the references therein). Yet, as far as we know, the boundedness problem for system (1.1) is rather unexplored.…”

Stability and exact periodic solutions of indefinite equations arising from the Kepler problem on the sphere

“…(1.1) has been widely studied by using various methods such as the shooting argument [2,9], critical point theory [1,3], coincidence degree theory [4,[6][7][8], averaging method [15] and geometric approach [16,20]. Here, the weight function h is continuous and periodic, β is a positive parameter and g : (A, B) → (0, +∞) is a continuous function maybe having singularities at u = A and u = B.…”