Limits of the Formal Integrals of Motion

TL;DR

The paper investigates the limits of formal (asymptotic) integrals of motion for a two-degree-of-freedom Hamiltonian with a cubic perturbation, contrasting irrational and resonant frequency ratios. For irrational ratios, a recursive construction yields a non-singular formal integral $\Phi$, while in rational (resonant) cases some divisors vanish, producing secular terms that must be canceled via zero-order integral refinements and coefficients $c_1,c_2,c_3$; the authors work out explicit forms for resonances $4/1$, $5/1$, $3/2$, and $4/3$ and analyze special low-order cases $3/1$, $2/1$, and $1/1$. The study shows that, despite elaborate cancellation schemes, strong perturbations typically lead to chaos through resonance overlap, limiting the applicability of the formal integral; however, in some regimes, especially near specific resonances or with symmetry ($\beta=\gamma=0$), invariant curves can still be well approximated and the dynamics resemble generalized Lissajous figures. The results illuminate when a third integral is meaningful in Galactic Dynamics and related nonlinear systems, and they relate to foundational work by Nekhoroshev, Moser, and Contopoulos on resonant structures and chaotic behavior.

Abstract

We consider a formal (approximate) integral of motion in Hamiltonians of the form $H=\frac{1}{2}(X^2+Y^2+ω_1^2x^2+ω_2^2y^2)+ε(ηxy^2+αx^3+βx^2y+γy^3)$ generalizing previous cases with $β=γ=0$. First we give the general form of this integral when $ω_1/ω_2$ is irrational and then we consider the case of commensurable frequencies. In particular we study the integrals for the resonances $ω_1/ω_2=4/1, 5/1, 3/2, 4/3, 3/1$ and $2/1$. We also calculate the invariant curves and the orbits in the cases $ω_1/ω_2=2/1$ and $1/1$ (with $β=γ=0$) and we compare the exact-numerical and the theoretical results predicted by the formal integral when $βγ\neq0$. In the special case $ω_1/ω_2=1/1$ we find an integral when $β=γ=0$ and $ηα\neq0$ or $η=α=0$ and $βγ\neq 0$, but this is not possible when $ηαβγ\neq 0$. However, we find that the invariant curves and the orbits can be approximated by a non-resonant integral with $ω_1/ω_2=5\sqrt{2}/7=1.010\dots$.

Figures (10)

• Figure 1: Case $\epsilon = 0.5$, $\eta=\alpha=1$, $\beta=\gamma=0$, $\omega_1=2$, $\omega_2=1$. (a) Invariant curves, theoretical (red) and exact (blue). The blue region contains chaotic orbits. (b) Lissajous orbits corresponding to the left part of (a) (black) and to the right part (blue). The red curve is the curve of zero velocity (CZV). c) The finite time Lyapunov characteristic number for an orbit in the chaotic sea of a) ($x(0)=-0.0676, y(0)=0.0, p_x(0)=1.36, p_y(0)=0.364,\delta x(0)=1,\delta y(0)=0,\delta p_x=0,\delta p_y=0, E=1$). We see that it saturates at the value $\chi\simeq 0.02$.
• Figure 2: Case $\epsilon = 0.6$, $\eta=\alpha=1$, $\beta=\gamma=0$, $\omega_1=2$, $\omega_2=1$. a) The corresponding Poincaré surface of section. The right part of a) contains mostly chaotic orbits. b) Two representative orbits. c) The finite time Lyapunov characteristic number for an orbit in the chaotic sea of a) ($x(0)=-0.25, y(0)=0.0, p_x(0)=0, p_y(0)=1.323,\delta x(0)=1,\delta y(0)=0,\delta p_x=0,\delta p_y=0, E=1$). We see that it saturates at the value $\chi\simeq 0.051$.
• Figure 3: Case $\epsilon = 0.62$, $\eta=\alpha=1$, $\beta=\gamma=0$, $\omega_1=2$, $\omega_2=1$. a) The corresponding Poincaré surface of section. b) Two characteristic orbits. Just above the energy of escape along $y^2$ many chaotic orbits in b) (blue) escape to $y=\pm\infty$ through the openings of the CZV.
• Figure 4: Case with $\epsilon = 0.1$, $\eta=\alpha=\beta=\gamma=1$, $\omega_1=2$, $\omega_2=1$. (a) The theoretical invariant curves (red) and the corresponding exact (numerical) curves (blue). (b) Two representative Lissajous orbits.
• Figure 5: Case with $\epsilon = 0.13$, $\eta=\alpha=\beta=\gamma=1$, $\omega_1=2$, $\omega_2=1$. a) The corresponding Poincaré surface of section. b) Two representative Lissajous orbits. Some chaotic orbits escape to $y =-\infty$ through the single opening of the CZV.