Arnold Diffusion, Quantitative Estimates, and Stochastic Behavior in the Three‐Body Problem

Abstract

We consider a class of autonomous Hamiltonian systems subject to small, time‐periodic perturbations. When the perturbation parameter is set to zero, the energy of the system is preserved. This is no longer the case when the perturbation parameter is non‐zero. We describe a topological method to establish orbits which diffuse in energy for every suitably small perturbation parameter ε>0. The method yields quantitative estimates: (i)the existence of orbits along which the energy drifts by an amount independent of ε; the time required by such orbits to drift is O()1/ε; (ii)the existence of orbits along which the energy makes chaotic excursions; (iii)explicit estimates for the Hausdorff dimension of the set of such chaotic orbits; (iv)the existence of orbits along which the time evolution of energy approaches a stopped diffusion process (Brownian motion with drift), as ε tends to 0. For each ε fixed, the set of initial conditions of the orbits that yield the diffusion process has positive Lebesgue measure, and in the limit the measure of these sets approaches 0. Moreover, we can obtain any desired values of the drift and variance for the limiting Brownian motion for appropriate sets of initial conditions. A key feature of our topological method is that it can be implemented in computer‐assisted proofs. We give an application to a concrete model of the planar elliptic restricted three‐body problem, on the motion of an infinitesimal body relative to the Neptune‐Triton system. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Computer Assisted Proof of Drift Orbits Along Normally Hyperbolic Manifolds II: Application to the Restricted Three Body Problem

“…From (74), (75), ( 76) and (78) we obtain (7). Similarly, from (73), ( 75), ( 76) and (77) we have (8). We have thus shown that Γ is a well defined homoclinic channel.…”

“…The shadowing of orbits is then automatically taken care of by [4], and we do not need to carry out the explicit construction of windows as in [8]. Our results are weaker though than [8].…”

“…The difference is that in the current paper we only need to establish the intersections of the manifolds for the unperturbed system, and check some explicit conditions from [6]. The shadowing of orbits is then automatically taken care of by [4], and we do not need to carry out the explicit construction of windows as in [8]. Our results are weaker though than [8].…”

“…We now turn to proving (7)(8). For a fixed I let us take x = x 0 (X) where X is such that H 0 (L X ) = I.…”

“…A recent result which works with explicit masses, explicit energy and explicit interval of eccentricities that starts from zero and reaches a physical value is given in [8]. This is based on a computer assisted construction of 'correctly aligned windows'.…”