[Paper Review] Computer Assisted Proofs of Attracting Invariant Tori for ODEs

Maciej J. Capiński, Emmanuel Fleurantin+1 more|arXiv (Cornell University)|May 20, 2019

Quantum chaos and dynamical systems43 references4 citations

TL;DR

This paper presents a computer-assisted proof method for verifying the existence and regularity of two-dimensional attracting invariant tori in three-dimensional dissipative ODEs, using validated numerical integration and cone conditions. It establishes verifiable lower bounds on regularity ($C^k$ for rotational tori, $C^0$ for resonant tori) in non-perturbative regimes, with successful implementations for both periodic and autonomous systems, including a 6-minute proof for a resonant torus at $\alpha = 0.85$.

ABSTRACT

This work studies existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We obtain verifiable lower bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We consider separately two important cases of rotational and resonant tori. In the rotational case we obtain $C^k$ lower bounds on the regularity of the embedding. In the resonant case we verify the existence of tori which are only $C^0$ and neither star-shaped nor Lipschitz

Motivation & Objective

• To develop a rigorous computer-assisted method for proving the existence of attracting invariant tori in three-dimensional dissipative ODEs beyond the perturbative regime.
• To establish verifiable lower bounds on the regularity of such tori, distinguishing between rotational and resonant cases.
• To address the challenge of non-uniform contraction and strong twist dynamics that hinder traditional validation techniques.
• To implement and demonstrate the method on explicit examples, including a periodic perturbation and a Neimark-Sacker bifurcation scenario.
• To provide a framework that verifies tori as $C^k$ (rotational) or $C^0$ (resonant), even when they are not star-shaped or Lipschitz.

Proposed method

• Use validated $C^k$ integrators (Wilczak and Zgliczyński) to rigorously compute solutions and variational equations of the ODEs and their Poincaré maps.
• For rotational tori, apply an outer approximation via polygon coverings and verify cone conditions to confirm the existence of a $C^k$ invariant torus.
• For resonant tori, use an inner approximation by constructing the torus from invariant pieces and verifying their union forms a torus via trapping and contraction.
• Implement a Poincaré map-based analysis to reduce the problem to discrete dynamics, enabling the use of rigorous numerical tools.
• Validate contraction and invariance of neighborhoods using interval arithmetic and verified integration, ensuring $f(U) \subset U$ for a set $U$ enclosing the torus.
• Apply cone condition verification to confirm the existence of a well-defined stable bundle, even in the presence of strong twist dynamics.

Experimental results

Research questions

• RQ1 Can computer-assisted proofs verify the existence of attracting invariant tori in 3D dissipative ODEs outside the perturbative regime?
• RQ2 What verifiable regularity bounds can be established for such tori, particularly in the rotational and resonant cases?
• RQ3 How can validated numerical integration and cone conditions be combined to rigorously prove the existence of a torus when contraction is non-uniform?
• RQ4 What are the computational and theoretical limitations when the torus exhibits strong twist dynamics or near-bifurcation parameters?
• RQ5 Can the method distinguish between $C^k$ rotational tori and $C^0$ resonant tori that are neither star-shaped nor Lipschitz?

Key findings

• A computer-assisted proof for a resonant torus at $\alpha = 0.85$ was completed in under 6 minutes on a single 3GHz Intel i7 processor.
• For $\alpha = 0.75$, a validated invariant set was found in $U$ with 10,000 cubes, but cone conditions could not be verified due to insufficient derivative bounds and strong twist.
• The method fails for $\alpha = 0.75$ when using $f = P^{16}$ due to long integration times and poor derivative estimates, despite $f(U) \subset U$ being validated.
• The presence of large twist parameters ($|\delta| \gg 1$) in the derivative matrix forces the use of very narrow cones ($a \leq |\delta|^{-1}$), making cone condition validation extremely difficult.
• The method breaks down near the Neimark-Sacker bifurcation due to weak contraction requiring many iterates, leading to poor derivative bounds and unstable cone validation.
• The paper demonstrates that $C^k$ regularity bounds can be rigorously verified for rotational tori, while resonant tori are shown to be only $C^0$, even when not star-shaped or Lipschitz.