Energy-second-moment map analysis as an approach to quantify the irregularity of Hamiltonian systems

TL;DR

The paper addresses how to quantify irregularity in Hamiltonian trajectories by introducing the energy-second-moment map for systems with $H(\mathbf{q},\mathbf{p},t)=\frac{1}{2}\mathbf{p}^2+V(\mathbf{q},t)$. This map yields a linear relation between the instantaneous energy and second moments and their initial values, represented by a $3\times3$ transfer matrix $\Xi(t)$ with $\det\Xi(t)=1$ that arises from a linear third-order extension of the canonical equations; its adjoint leads to simple Lyapunov function expressions, enabling a direct regularity test via long-time limits. The regularity analysis leverages Lyapunov theory for nonautonomous linear systems, including a Perron-based triangulation to define $\lambda_k(t)$ and the irregularity coefficient $\sigma_{\Lambda}$, and provides explicit expressions $\lambda_1(t)=\frac{1}{2t}\ln\frac{a_2}{a_1}$, $\lambda_2(t)=\frac{1}{2t}\ln a_1$, $\lambda_3(t)=-\frac{1}{2t}\ln a_2$ in terms of auxiliary functions $a_1,a_2$ derived from the fundamental solutions $\xi_k(t)$. The method is demonstrated on three systems—the Hénon–Heiles oscillator, a nonlinear two-dimensional oscillator, and the circular restricted three-body problem—showing that convergent $\lambda_k(t)$ track regular or quasi-regular motion, while non-convergent $\lambda_k(t)$ indicate irregular or chaotic dynamics, with the added ability to identify transient quasi-regular intervals in chaotic trajectories. Overall, the energy-second-moment map provides a practical, analytic, and complementary tool to conventional stability-matrix analysis for characterizing irregularity in Hamiltonian dynamics.

Abstract

A different approach will be presented that aims to scrutinize the phase-space trajectories of a general class of Hamiltonian systems with regard to their regular or irregular behavior. The approach is based on the `energy-second-moment map' that can be constructed for all Hamiltonian systems of the generic form $H=p^{2}/2+V(q,t)$. With a three-component vector $s$ consisting of the system's energy $H$ and second moments $qp$, $q^{2}$, this map linearly relates the vector $s(t)$ at time $t$ with the vector's initial state $s(0)$ at $t=0$. It will turn out that this map is directly obtained from the solution of a linear third-order equation that establishes an extension of the set of canonical equations. The Lyapunov functions of the energy-second-moment map will be shown to have simple analytical representations in terms of the solutions of this linear third-order equation. Applying Lyapunov's regularity analysis for linear systems, we will show that the Lyapunov functions of the energy-second-moment map yields information on the irregularity of the particular phase-space trajectory. Our results will be illustrated by means of numerical examples.

Figures (9)

• Figure 1: Numerical errors $\Delta I_1=h|_{t=0}-h_0$, $\Delta I_2=-\frac{1}{2}\bm{q}\bm{p}|_{t=0}+\frac{1}{2}\bm{q}_0\bm{p}_0$, and $\Delta I_3=\frac{1}{4}\bm{q}^{2}|_{t=0}-\frac{1}{4}\bm{q}_0^{2}$ of the invariants $I_1=h|_{t=0}$, $I_2=-\frac{1}{2}\bm{q}\bm{p}|_{t=0}$, and $I_3=\frac{1}{4}\bm{q}^{2}|_{t=0}$, calculated with quadruple precision.
• Figure 2: Left: $y,p_{y}$-Poincaré surface-of-section representation of an irregular trajectory in the Hénon-Heiles oscillator (\ref{['hhdef']}) with the limiting energy $h=1/6$ for the initial condition $(x_{0},p_{x,0},y_{0},p_{y,0})=(0,0.5367,-0.2,0)$ and $C=1$. Right: Lyapunov function $\lambda_{1}(t)$ from Eq. (\ref{['lyapfunct1_num']}) for this trajectory.
• Figure 3: Left: real-space projection of a regular trajectory of the Hénon-Heiles oscillator (\ref{['hhdef']}) with the limiting energy $h=1/6$ as obtained for the initial condition $(x_{0},p_{x,0},y_{0},p_{y,0})=(0,0.3765,0.55,0)$ and $C=1$. Right: Lyapunov function $\lambda_{1}(t)$ from Eq. (\ref{['lyapfunct1_num']}) for this trajectory.
• Figure 4: Left: $y,p_{y}$-Poincaré surface-of-section representation of a regular trajectory near the boundary of the regular region with the limiting energy $h=1/6$ as obtained for the initial condition $(x_{0},p_{x,0},y_{0},p_{y,0})=(0,0.3420,0.60,0.02)$ and $C=1$. Right: Lyapunov function $\lambda_{1}(t)$ from Eq. (\ref{['lyapfunct1_num']}) for this trajectory.
• Figure 5: Right: enlarged view of the Lyapunov function $\lambda_{1}(t)$ from Fig. \ref{['f1']} for the time interval $75\times 10^{3}\le t\le 80.5\times 10^{3}$. Left: subset of the $y,p_{y}$-Poincaré-section points of Fig. \ref{['f1']} that emerges during that particular time interval.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Theorem 3