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Two Layer Model via Non Quasi Periodic Normal Form Theory

Gabriella Pinzari, Benedetto Scoppola, Matteo Veglianti

TL;DR

The paper addresses resonance capture in a dissipative, non-Hamiltonian two-layer spin-orbit model and develops a non-quasi-periodic (NQP) normal form theory to analyze dumped oscillations under friction. By a Lagrangian formulation and a sequence of near-identity coordinate changes, the authors obtain a normal form with an exponentially small remainder and derive an explicit trapping time bound $T = {\rm const}^{-1} \frac{\varepsilon}{\varepsilon_*\mu_1} e^{\frac{\gamma_1}{\mu_1}}$, guaranteeing proximity to dumped oscillations for times $|t| < T$. The work combines a detailed functional-analytic NFT framework with concrete estimates on the linearized system and its perturbations, yielding rigorous control of resonance trapping in a nonautonomous, dissipative setting. These results have potential implications for understanding spin–orbit resonance phenomena in planetary contexts (e.g., Mercury) and icy satellites where viscous friction and interlayer coupling play a role.

Abstract

The two layer model is a 2+1/2 degrees of freedom non autonomous dynamical system whose lower order expansion exhibits capture in resonance, numerically detected in a previous paper by the authors. In this paper, we reframe the model along the lines of a suitable version of (which we refer to as non quasi periodic) normal form theory and provide an explicit amount of the resonance trapping time, which is estimated as exponentially long, in terms of the small parameters of the system. Key words: capture into resonance; non quasi periodic normal form theory; friction. MSC 2020: 37J40; 70F40.

Two Layer Model via Non Quasi Periodic Normal Form Theory

TL;DR

The paper addresses resonance capture in a dissipative, non-Hamiltonian two-layer spin-orbit model and develops a non-quasi-periodic (NQP) normal form theory to analyze dumped oscillations under friction. By a Lagrangian formulation and a sequence of near-identity coordinate changes, the authors obtain a normal form with an exponentially small remainder and derive an explicit trapping time bound , guaranteeing proximity to dumped oscillations for times . The work combines a detailed functional-analytic NFT framework with concrete estimates on the linearized system and its perturbations, yielding rigorous control of resonance trapping in a nonautonomous, dissipative setting. These results have potential implications for understanding spin–orbit resonance phenomena in planetary contexts (e.g., Mercury) and icy satellites where viscous friction and interlayer coupling play a role.

Abstract

The two layer model is a 2+1/2 degrees of freedom non autonomous dynamical system whose lower order expansion exhibits capture in resonance, numerically detected in a previous paper by the authors. In this paper, we reframe the model along the lines of a suitable version of (which we refer to as non quasi periodic) normal form theory and provide an explicit amount of the resonance trapping time, which is estimated as exponentially long, in terms of the small parameters of the system. Key words: capture into resonance; non quasi periodic normal form theory; friction. MSC 2020: 37J40; 70F40.
Paper Structure (10 sections, 13 theorems, 194 equations)

This paper contains 10 sections, 13 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Lagrangian set--up and result
  3. The model pinzari2024spin
  4. The linearized system and result
  5. Proof of Theorem \ref{['main']} via NQP Normal Form Theory
  6. Proof of Theorems \ref{['iteration lemmaOLD']} and \ref{['normal form lemma']}
  7. Proof of Theorem \ref{['iteration lemmaOLD']}
  8. Proof of Theorem \ref{['normal form lemma']}
  9. Proof of Proposition \ref{['prop: asymptotically stable']}
  10. Proof of Proposition \ref{['prop: Lie operator']}

Key Result

Proposition 2.1

Under an open and generic condition (i.e., if the resolvent of the characteristic polynomial of $L$ does not vanish), if the inequality in friction coefficients and hold, then $L$ admits two distinct complex--conjugated couples of eigenvalues $\lambda_j$, with strictly negative real part and non--vanishing imaginary part. More precisely, the following bound holds

Theorems & Definitions (20)

  • Proposition 2.1
  • Theorem 2.1
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.1
  • Theorem 3.2
  • Definition 4.1
  • Proposition 4.1
  • Lemma 4.1: Cauchy Inequalities
  • Lemma 4.2
  • ...and 10 more