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Improved stability estimates at elliptic equilibria of Hamiltonian systems

Massimiliano Guzzo, Chiara Caracciolo, Gabriella Pinzari

Abstract

This paper deals with an improvement of the "a-priori stability bounds" on the variation of the action variables and on the stability time obtained from a given Birkhoff normal form around the elliptic equilibrium point of an Hamiltonian system satisfying a non-resonance condition of finite order N. In particular, we improve the standard a-priori lower bound on the stability time from a purely linear dependence on the inverse of the polynomial norm of the remainder of the normal form to the sum of a linear term (which is still present but with a different constant coefficient) and a quadratic one. The prevalence between the linear and the quadratic term depends on the resonance properties of all the monomials in the remainder of the normal form with degree from N to a finite order M. We also provide a comparative example of the new estimates and the traditional a priori ones in the framework of computer-assisted proofs.

Improved stability estimates at elliptic equilibria of Hamiltonian systems

Abstract

This paper deals with an improvement of the "a-priori stability bounds" on the variation of the action variables and on the stability time obtained from a given Birkhoff normal form around the elliptic equilibrium point of an Hamiltonian system satisfying a non-resonance condition of finite order N. In particular, we improve the standard a-priori lower bound on the stability time from a purely linear dependence on the inverse of the polynomial norm of the remainder of the normal form to the sum of a linear term (which is still present but with a different constant coefficient) and a quadratic one. The prevalence between the linear and the quadratic term depends on the resonance properties of all the monomials in the remainder of the normal form with degree from N to a finite order M. We also provide a comparative example of the new estimates and the traditional a priori ones in the framework of computer-assisted proofs.
Paper Structure (24 sections, 12 theorems, 129 equations, 3 figures, 1 table)

This paper contains 24 sections, 12 theorems, 129 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The statements
  3. Proof of Theorems \ref{['main simplified']}, \ref{['main']} and \ref{['main middle']}
  4. Proof of Lemma \ref{['lemmafirstint']}
  5. Some preliminary lemmas.
  6. Proof of Lemma \ref{['derivatives']}
  7. Proof of Lemma \ref{['Poisson brackets']}
  8. Proof of Lemma \ref{['bound on actions']}
  9. Proof of Theorem \ref{['main']}
  10. Proof of Theorem \ref{['main middle']}
  11. Proof of Theorem \ref{['main simplified']}
  12. Rigorous numerical application to a 3 d.o.f. case
  13. The 3 d.o.f. FPU $\alpha$-model and its Birkhoff normal forms.
  14. The classical estimate of the stability time.
  15. Comparison of the stability times.
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $H$, as in Hamiltonian, be in Birkhoff normal form of order $N$ and analytic in $D(R_0)$. Let $M\in \mathbb N$, with $M\ge N{+1}$, $\alpha\in(0, {1/2}]$ and $a>0$. Then, for any $0<R\le R_0/2$ verifying any solution $(w(t), z(t))$ of the Hamilton equation of $H$ with initial datum in $D(R)$ satisfying reality at time zero does not leave ${{D(R(1+\alpha))}}$ for all $t$ such that $|t|\le T_0$,

Figures (3)

  • Figure 1: Logscale plot of the function $T_1(R)$ and $T_c(R)$ with $N=9$ and $M=18$ (left), $M=27$ (right). The lines highlight the decrease rates: $T_{c}\simeq R^{1-N}$; $T_1\simeq\min\{c_1 R^{1-2N},c_2 R^{1-M}\}$. In the plot on the right one can identify two regimes for $T_1$: for small $R$, it goes as $R^{-17}$, while after some time term $R^{-26}$ becomes predominant. The two estimates come to coincide when $\|f^*\|\ge \|f_0+f_*\|$.
  • Figure 2: Logscale plot of the estimated actions variation with the classical (violet squares) and new (green circles) estimate, for different times $T$. The initial condition satisfies $I_j(0)<R^2$ with $R=5.$e-4.
  • Figure 3: Representation of the cardinality $\#\Lambda_{a, K}{(\Omega_0)}$ for the vector $\Omega_0\in{\errmessage{\Bbb\space allowed only in math mode} R}^3$ defined in Section 4, Eq. (\ref{['Omega0FPU']}) (left panel), and of the Logscale plot of the relative cardinality of $\#\Lambda_{a, K}{(\Omega_0)}$ with respect to the set of all the integer vectors $\nu$ with $\left |\nu\right |=K$.

Theorems & Definitions (22)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • ...and 12 more