Chaotic Dynamics of the Kepler Problem with Oscillating Singularity

Alessandro Margheri,Pedro J Torres

doi:10.1515/ans-2015-5026

Alessandro Margheri, Pedro J Torres

Open Access

https://doi.org/10.1515/ans-2015-5026

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Abstract

Abstract We prove the presence of chaotic dynamics for the classical two-body Kepler problem with a time-periodic gravitational coefficient oscillating between two fixed values. The set of chaotic solutions we detect is coded by the number of revolutions in each period. The chaotic dynamics is obtained for large period T as well as for small angular momentum μ. In particular, we provide an explicit lower bound on T and explicit upper bound on μ which guarantee the existence of complex dynamics. We get our results by applying a simple and well-known topological method, the stretching along the path technique. Our results are robust with respect to small perturbations of the gravitational coefficient and to the addition of a small friction term.

Highlights

The motivation of this paper is to study some aspects of the dynamics of the system u = −h(t) u |u|3 (1.1)where h(t) is a T -periodic function
If the sailing capacity is significant, there is an interplay between the attractive gravitational force and the repulsive force due to the solar radiation pressure exerted by the star, in such a way that the motion of the body is ruled by system (1.1) with h(t) σ(t)L(t) 4πc
Our objective is to provide sufficient conditions such that (1.2), and (1.1) has chaotic dynamics, with periodic orbits of any period, when the restriction of h(t) to the interval [0, T ] is a piecewise constant function of the form h(t) = h1 h2 if t ∈ [0, T1] if t ∈]T1, T1 + T2 =: T ]

Summary

Introduction

The motivation of this paper is to study some aspects of the dynamics of the system u. Our objective is to provide sufficient conditions such that (1.2), and (1.1) has chaotic dynamics, with periodic orbits of any period, when the restriction of h(t) to the interval [0, T ] is a piecewise constant function of the form h(t) = h1 h2 if t ∈ [0, T1] if t ∈]T1, T1 + T2 =: T ] This particular choice of h(t), which, does not allow to tackle the general model, may seem just of mathematical interest. If the twist on the boundaries of the annuli is sufficiently strong, the SAP method guarantees the existence of complex dynamics for φ, with periodic points of any period, in the topological rectangles which realize the link This is the general framework which we will use to obtain our results.

Definition of chaos and SAP method

Analysis of phase portraits

Chaotic solutions of the Kepler problem with small angular momentum