Periodic solutions of singular radially symmetric systems with superlinear growth

Abstract

We prove the existence of infinitely many periodic solutions for periodically forced radially symmetric systems of second-order ODE's, with a singularity of repulsive type, where the nonlinearity has a superlinear growth at infinity. These solutions have periods, which are large integer multiples of the period of the forcing, and rotate exactly once around the origin in their period time, while having a fast oscillating radial component. Analogous results hold in the case of an annular potential well

Periodic orbits of a singular superlinear planar system

“…In particular, the case of an attractive singularity was studied in [8,10] and the case of a repulsive singularity was studied in [9]. In [11], a singular planar system admitting a superlinear growth was studied. As an illustrating example, the following system (i) If ω ≥ 1, then there exists a constant k 1 ≥ 1 such that for any integer k ≥ k 1 , (1.3) has a family of periodic solutions x k with minimal period kT and if μ k denotes the angular momentum associated to x k (t), then lim k→∞ μ k = 0.…”

“…Following the ideas in [8,9,11,12], we split the system into its radial and angular component and consider the angular momentum as a parameter. To do so, we write the solutions of (1.1) in polar coordinates…”

“…have been provided in [8][9][10][11][12] in a systematic way. In particular, the case of an attractive singularity was studied in [8,10] and the case of a repulsive singularity was studied in [9].…”

Periodic solutions of singular second order equations at resonance

“…Besides these applications, it has been found that such singular equations play important roles in studying of the Lyapunov stability of periodic solutions of Lagrangian equations [5], and the dynamics of radially symmetric Keplerian-like systems [11][12][13][14].…”

Periodic Impact Motions at Resonance of a Particle Bouncing on Spheres and Cylinders

“…The existence of periodic solutions of large period kT was previously studied by Fonda and Toader in [20] in the setting of a Keplerian-like system where the planet is viewed as a point (see also [18,21] for other situations). Problems modeling the motion of a particle hitting some surfaces have been widely studied in literature in different situations, see e.g.…”