Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach

Abstract

We are concerned with non-autonomous radially symmetric systems with a singularity, which are T -periodic in time. By the use of topological degree theory, we prove the existence of large-amplitude periodic solutions whose minimal period is an integer multiple of T . Precise estimates are then given in the case of Keplerian-like systems, showing some resemblance between the orbits of those solutions and the circular orbits of the corresponding classical autonomous system.

Odd Periodic Solutions of Fully Second-Order Ordinary Differential Equations with Superlinear Nonlinearities

“…Many theorems and methods of nonlinear functional analysis have been applied to the periodic problems of (2), such as the upper and lower solutions method and monotone iterative technique [1][2][3][4], the continuation method of topological degree [5][6][7][8][9], variational method and critical point theory [10][11][12][13][14], method of phase-plane analysis [15][16][17][18][19], the Krasnoselskii's type fixed point theorem in cone [20][21][22][23], and the theory of fixed point index [24][25][26].…”

Periodic solutions of radially symmetric perturbations of Newtonian systems

“…As we have already proved in [5], in this case solutions with large-amplitude orbits cannot be periodic. We therefore look for periodic solutions with smaller amplitude.…”

“…We will show that if γ ≥ 2, even in this case there are infinitely many periodic solutions of (1). Hence, combining this result with the ones in [5,7], we can finally state an existence theorem for periodic solutions of (1), valid for any choice of the T -periodic forcing e(t). Theorem 1.…”

“…In [7], the above result was refined by removing the restriction that γ < 4. It should be noticed that, besides the periodic solutions, a whole family of quasiperiodic solutions was detected in [5,7]. Indeed, all those solutions have a Tperiodic oscillating radial component, while their mean angular velocity covers a whole interval of positive values, among which one can find the sub-multiples of 2π which correspond to the periodic (subharmonic) solutions.…”

“…In [5], we have already proved that if 1 < γ < 4 andē ≤ 0, then equation (1) has infinitely many periodic solutions, whose periods are large multiples of T , rotating around the origin on large-amplitude orbits. In [7], the above result was refined by removing the restriction that γ < 4.…”