[Paper Review] Diffusion time and splitting of separatrices for nearly integrable isochronous Hamiltonian systems

Massimiliano Berti, Philippe Bolle|ArXiv.org|Sep 19, 2000

Quantum chaos and dynamical systems4 references4 citations

TL;DR

This paper establishes improved estimates for Arnold's diffusion time in nearly integrable isochronous Hamiltonian systems using a variational shadowing approach, and rigorously justifies the Poincaré-Melnikov approximation for the splitting of separatrices in three-time-scale systems. It proves that the splitting is correctly predicted by the Poincaré-Melnikov function under generic conditions, with explicit error bounds in the perturbation parameter and analyticity strip width.

ABSTRACT

We consider the problem of Arnold's diffusion for nearly integrable isochronous Hamiltonian systems. We prove a shadowing theorem which improves the known estimates for the diffusion time. We also justify for three time scales systems that the splitting of the separatrices is correctly predicted by the Poincare'-Melnikov function.

Motivation & Objective

• To improve existing estimates for Arnold's diffusion time in nearly integrable isochronous Hamiltonian systems.
• To establish the validity of the Poincaré-Melnikov function in predicting the splitting of separatrices for three-time-scale systems.
• To provide a variational framework that yields sharper bounds on diffusion time compared to prior geometric and Mather-theoretic methods.
• To justify the asymptotic behavior of the splitting in systems with weak coupling between fast, intermediate, and slow oscillators.

Proposed method

• A variational shadowing theorem is developed using a functional $ F_{ u}(A,\theta) $ defined as the action integral over pseudo-heteroclinic solutions of the quasi-periodically forced pendulum equation.
• The homoclinic function $ G_{\nu}(A) $ is derived as $ F_{\nu}(A,0) $, representing the difference in generating functions of stable and unstable manifolds at $ q = \pi $.
• Analytic continuation of the action functional into complex strips allows estimation of Fourier coefficients of $ \widetilde{G}_{\nu} $, the transformed homoclinic function.
• The Poincaré-Melnikov primitive $ \Gamma(A) $ is used as an approximation, with explicit formula $ \Gamma_k = f_k \frac{2\pi(k\cdot\omega)}{\sinh(k\cdot\omega \pi/2)} $.
• Error bounds are derived via complex analysis: $ |\widetilde{G}_k - \mu\Gamma_k| \leq \frac{C\mu^2||f||^2}{\delta^4} \exp\left(-\sum a_i|k_i|\right) \exp\left(-|k\cdot\omega|(\pi/2 - \delta)\right) $.
• For three-time-scale systems, the method is applied with $ \omega_\varepsilon = (\varepsilon^{-1/2}, \varepsilon^a) $, and asymptotic expansions of $ \widetilde{G}_\nu $ are derived in terms of $ \mu \Gamma_0, \mu \Gamma_1 $, and error terms.

Experimental results

Research questions

• RQ1 Can the diffusion time in nearly integrable isochronous Hamiltonian systems be estimated more sharply using a variational approach?
• RQ2 To what extent does the Poincaré-Melnikov function accurately predict the splitting of separatrices in three-time-scale systems?
• RQ3 What conditions on the perturbation $ f $ ensure the existence of diffusion orbits via the shadowing mechanism?
• RQ4 How do the Fourier coefficients of the splitting function behave under analytic continuation into complex strips?
• RQ5 What is the quantitative relationship between the perturbation strength $ \mu $, the analyticity width $ \delta $, and the error in the Poincaré-Melnikov approximation?

Key findings

• The variational shadowing theorem improves known diffusion time estimates by providing tighter bounds than previous geometric and Mather-theoretic methods.
• The splitting of separatrices in three-time-scale systems is rigorously justified by the Poincaré-Melnikov function, with error bounds of order $ \mu^2||f||^2 \delta^{-4} \exp(-|k\cdot\omega|(\pi/2 - \delta)) $.
• For three-time-scale systems, the homoclinic function $ \widetilde{G}_\nu $ admits an asymptotic expansion: $ \widetilde{G}_\nu = Const + \mu\Gamma_0 + 2\mathrm{Re}(\mu\Gamma_1 e^{iA_1}) + O(\mu\varepsilon^{-1/2}||f||\exp(-\pi/\sqrt{\varepsilon})) $.
• The error in the $ \Gamma_1 $ term is bounded by $ O(\mu^2||f||^2 \varepsilon^{-2} \exp(-\pi/(2\sqrt{\varepsilon}))) $, showing exponential decay in the slow frequency scale.
• Under generic conditions on $ f $, such as non-vanishing Fourier modes $ f_{0,l}, f_{1,m} \neq 0 $, the splitting condition in Theorem 2.1 is satisfied, ensuring the existence of diffusion orbits.
• The result holds in any dimension $ n \geq 2 $, extending prior results limited to $ n=2 $, and improves on [17] by requiring only $ \mu\varepsilon^{-3/2} \to 0 $, not $ \mu = \varepsilon^p $ with $ p > 2+a $.