# Instantaneous Pairing of Lyapunov Exponents in Chaotic Hamiltonian Dynamics and the 2017 Ian Snook Prize

## TL;DR

This paper investigates the instantaneous pairing of Lyapunov exponents in chaotic Hamiltonian systems, providing insights into local instability measures and their relation to phase volume conservation, with applications to simple nonlinear models.

## Contribution

It introduces a detailed analysis of local Lyapunov exponents in Hamiltonian chaos, extending understanding of phase space dynamics beyond long-time averages.

## Key findings

- Instantaneous Lyapunov exponents show pairing behavior in Hamiltonian systems.
- Analysis of small models like double pendulum and $\phi^4$ chain illustrates local instability.
- Provides pedagogical insights into phase volume conservation and chaos mechanisms.

## Abstract

The time-averaged Lyapunov exponents support a mechanistic description of the chaos generated in and by nonlinear dynamical systems. The exponents are ordered from largest to smallest with the largest one describing the exponential growth rate of the (small) distance between two neighboring phase-space trajectories. Two exponents describe the rate for areas defined by three nearby trajectories. The sum of the first three exponents is the rate for volumes defined by four nearby trajectories, and so on. Lyapunov exponents for Hamiltonian systems are symmetric. The time-reversibility of the motion equations links the growth and decay rates together in pairs. This pairing provides a more detailed explanation than Liouville's for the conservation of phase volume in Hamiltonian mechanics. Although correct for long-time averages, the dependence of trajectories on their past is responsible for the observed lack of detailed pairing for the instantaneous "local" exponents, $\{\ \lambda_i(t) \ \}$ . The 2017 Ian Snook Prizes will be awarded to the author(s) of an accessible and pedagogical discussion of local Lyapunov instability in small systems. We desire that this discussion build on the two nonlinear models described here, a double pendulum with Hooke's-Law links and a periodic chain of Hooke's-Law particles tethered to their lattice sites. The latter system is the $\phi^4$ model popularized by Aoki and Kusnezov. A four-particle version is small enough for comprehensive numerical work and large enough to illustrate ideas of general validity.