[Paper Review] Exponential estimates of symplectic slow manifolds

Kristian Uldall Kristiansen, Claudia Wulff|arXiv (Cornell University)|Aug 21, 2012

Quantum chaos and dynamical systems22 references4 citations

TL;DR

This paper establishes the existence of an exponentially accurate symplectic slow manifold for analytic Hamiltonian slow-fast systems with finitely many slow degrees of freedom and infinitely many fast degrees of freedom, using a symplectic version of MacKay's iterative method. The key result is that the error in the slow manifold is exponentially small in the perturbation parameter $\epsilon$, without requiring normal hyperbolicity or resonance conditions.

ABSTRACT

In this paper we prove the existence of an almost invariant symplectic slow manifold for analytic Hamiltonian slow-fast systems with finitely many slow degrees of freedom for which the error field is exponentially small. We allow for infinitely many fast degrees of freedom. The method we use is motivated by a paper of MacKay from 2004. The method does not notice resonances, and therefore we do not pose any restrictions on the motion normal to the slow manifold other than it being fast and analytic. We also present a stability result and obtain a generalization of a result of Gelfreich and Lerman on an invariant slow manifold to (finitely) many fast degrees of freedom.

Motivation & Objective

• To establish the existence of an almost invariant symplectic slow manifold for analytic Hamiltonian systems with finitely many slow degrees of freedom and infinitely many fast degrees of freedom.
• To remove restrictions on normal hyperbolicity and resonance conditions, allowing for general fast dynamics including PDEs.
• To generalize a result by Gelfreich and Lerman on invariant slow manifolds to systems with finitely many fast degrees of freedom.
• To provide a stability result for the constructed symplectic slow manifold.
• To rigorously validate MacKay's iterative method for symplectic systems, showing exponential convergence without relying on hyperbolicity.

Proposed method

• Adapts MacKay's iterative method for improving slow manifolds to the symplectic setting, ensuring the transformation preserves the symplectic structure.
• Uses generating functions and symplectic transformations to iteratively correct the slow manifold approximation while maintaining the Hamiltonian structure.
• Applies an iterative lemma to control the size of corrections and error terms in the slow manifold, ensuring convergence.
• Employs majorant series and analytic estimates in complex domains to bound the growth of nonlinear terms and ensure convergence.
• Imposes conditions on the decay of the error field through iterative bounds involving $\delta_n$, $\xi_n$, and $\epsilon$, leading to exponential decay.
• Uses complex-analytic techniques to handle the infinite-dimensional fast subsystem, treating it as a semilinear evolution PDE.

Experimental results

Research questions

• RQ1 Can an exponentially accurate symplectic slow manifold be constructed for analytic Hamiltonian systems with finitely many slow and infinitely many fast degrees of freedom?
• RQ2 Does MacKay's iterative method for improving slow manifolds preserve the symplectic structure when applied to Hamiltonian systems?
• RQ3 Can the method be applied without requiring normal hyperbolicity or excluding resonant dynamics?
• RQ4 What is the quantitative rate of convergence of the iterative correction process for the slow manifold?
• RQ5 How does the error in the slow manifold approximation behave in terms of the small parameter $\epsilon$?

Key findings

• The constructed slow manifold has an error field that is exponentially small in $\epsilon$, specifically $\mathcal{O}(e^{-c/\epsilon})$ for some $c>0$, under analyticity assumptions.
• The method does not require normal hyperbolicity or resonance conditions, making it applicable to normally elliptic systems.
• The iterative correction process leads to a sequence of corrections $\delta_n$ satisfying $\delta_n \leq C \epsilon^{n+1}$, implying exponential convergence.
• The error in the slow manifold approximation after $n$ iterations is bounded by $\mathcal{O}(\epsilon^{n+1})$, with the full limit achieving exponential accuracy.
• The method generalizes a result of Gelfreich and Lerman by extending the existence of invariant slow manifolds to systems with finitely many fast degrees of freedom.
• A counterexample is provided showing that MacKay's conjecture about $\mathcal{O}((\epsilon \|W\|)^n)$ convergence is incorrect for $n>1$, as the error does not factorize in that way.