[Paper Review] Arnold Diffusion, Quantitative Estimates and Stochastic Behavior in the Three-Body Problem

Maciej J. Capiński, Marian Gidea|arXiv (Cornell University)|Dec 10, 2018

Quantum chaos and dynamical systems49 references4 citations

TL;DR

This paper presents a topological method to rigorously establish Arnold diffusion in Hamiltonian systems under small time-periodic perturbations, proving the existence of orbits with energy drift independent of the perturbation size $\varepsilon$, chaotic energy excursions, and convergence to a Brownian motion with drift as $\varepsilon \to 0$. The method yields explicit quantitative estimates on diffusion time ($O(1/\varepsilon)$), Lebesgue measure of diffusing orbits, Hausdorff dimension of chaotic orbits, and stochastic limits, and is implementable in computer-assisted proofs for concrete models like the Neptune-Triton system in the planar elliptic restricted 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 $\varepsilon>0$. The method yields quantitative estimates: (i) The existence of orbits along which the energy drifts by an amount independent of $\varepsilon$. The time required by such orbits to drift is $O(1/\varepsilon)$; (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 $\varepsilon$ tends to $0$. For each $\varepsilon$ 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 zero. 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.

Motivation & Objective

• To establish the existence of orbits with large energy drift in Hamiltonian systems under arbitrarily small time-periodic perturbations.
• To provide explicit quantitative estimates on the diffusion time, measure of diffusing orbits, and Hausdorff dimension of chaotic orbits.
• To show that the time evolution of energy for certain orbits converges to a stopped diffusion process (Brownian motion with drift) as the perturbation parameter $\varepsilon \to 0$.
• To develop a method implementable in computer-assisted proofs using validated numerical computations for concrete physical models.
• To apply the framework to the planar elliptic restricted three-body problem, particularly the Neptune-Triton system, for physically relevant parameters.

Proposed method

• A topological approach based on the construction of transition chains using invariant manifolds and covering relations in phase space.
• Verification of finitely many explicit conditions up to finite precision to infer asymptotic behavior for an interval of $\varepsilon > 0$ including values arbitrarily close to zero.
• Use of symbolic dynamics and propagation of disks to track the evolution of sets under the flow and establish diffusion.
• Application of the method to a Hamiltonian system with action-angle variables, where $I$ is conserved in the unperturbed case ($\varepsilon = 0$).
• Estimation of the Hausdorff dimension of chaotic orbits via the construction of Cantor-like sets in the phase space.
• Derivation of stochastic limits by showing that the energy evolution of certain orbits approaches a Brownian motion with drift as $\varepsilon \to 0$, with explicit control over drift and variance.

Experimental results

Research questions

• RQ1 Can Arnold diffusion be rigorously established in concrete physical models, such as the three-body problem, under small perturbations?
• RQ2 What are the explicit quantitative bounds on the time required for energy to drift by a fixed amount independent of $\varepsilon$?
• RQ3 What is the Lebesgue measure of initial conditions leading to energy diffusion for each fixed $\varepsilon > 0$, and how does it behave as $\varepsilon \to 0$?
• RQ4 What is the Hausdorff dimension of the set of orbits exhibiting chaotic energy excursions?
• RQ5 Can the limiting stochastic process of energy evolution be explicitly described as $\varepsilon \to 0$, and can drift and variance be controlled via initial conditions?

Key findings

• Orbits exist along which the energy drifts by an amount independent of $\varepsilon$, and the time required for such drift is $O(1/\varepsilon)$.
• For each fixed $\varepsilon > 0$, the set of initial conditions leading to energy diffusion has positive Lebesgue measure, and this measure tends to zero as $\varepsilon \to 0$.
• The Hausdorff dimension of the set of orbits exhibiting chaotic energy excursions is explicitly estimated and bounded below by a positive constant independent of $\varepsilon$.
• The time evolution of energy for certain orbits converges to a stopped diffusion process (Brownian motion with drift) as $\varepsilon \to 0$, with drift and variance explicitly controllable via initial conditions.
• The method is implementable in computer-assisted proofs using validated numerical methods, enabling rigorous verification for concrete models like the Neptune-Triton system.
• The results apply to the planar elliptic restricted three-body problem with physically relevant parameters, including the observed mass ratio and eccentricity of the Neptune-Triton system.