Attractivity of closed sets proved by using a family of Liapunov functions

Corne, J.L; Rouche, N.

doi:10.1016/0022-0396(73)90016-8

Cited by 15 publications

(1 citation statement)

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“…Corne and Rouche [8] and Diacu [9] were perhaps the first contributors towards the development of a qualitative theory of the Kepler problem with drag (1.1) for general families of non-constant drag forces δ = δ(u, u), depending on u and u. [8] considered (1.1) with δ(u, u) = k(| u|)/| u| and showed, under some additional assumptions on the function k, that all solutions go to the singularity (potentially in finite time). On the other hand, [9] analyzed the qualitative dynamics of the dissipative Kepler problem within a generalized class of Stokes drag; this family includes the important Poynting-Robertson case.…”

Section: Introductionmentioning

confidence: 99%

Revisiting the Kepler problem with linear drag using the blowup method and normal form theory

Kristiansen

2024

Nonlinearity

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In this paper, we revisit the Kepler problem with linear drag. With dissipation, the energy and the angular momentum are both decreasing, but in Margheri et al (2017 Celest. Mech. Dyn. Astron. 127 35–48) it was shown that the eccentricity vector has a well-defined limit in the case of linear drag. This limiting eccentricity vector defines a conserved quantity, and in the present paper, we prove that the corresponding invariant sets are smooth manifolds. These results rely on normal form theory and a blowup transformation, which reveals that the invariant manifolds are (nonhyperbolic) stable sets of (limiting) periodic orbits. Moreover, we identify a separate invariant manifold which corresponds to a zero limiting eccentricity vector. This manifold is obtained as a generalized center manifold over the zero eigenspace of a zero-Hopf point. Finally, we present a detailed blowup analysis, which provides a geometric picture of the dynamics. We believe that our approach and results will have general interest in problems with blowup dynamics, including the Kepler problem with generalized nonlinear drag.

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