Related Experiment Videos
Lyapunov exponents from unstable periodic orbits.
Roberto Franzosi1, Pietro Poggi, Monica Cerruti-Sola
1Dipartimento di Fisica, Università di Pisa, via Buonarroti 2, I-56127 Pisa, Italy. Roberto.Franzosi@df.unipi.it
Summary
We present a new method to analytically calculate the largest Lyapunov exponent in chaotic systems using unstable periodic orbits (UPOs). This approach accurately predicts chaotic instability for the Fermi-Pasta-Ulam-beta model.
Area of Science:
- Dynamical Systems and Chaos Theory
- Statistical Mechanics
- Computational Physics
Background:
- Hamiltonian chaotic systems exhibit complex dynamics.
- Lyapunov exponents quantify the rate of chaos.
- Unstable periodic orbits (UPOs) are fundamental to understanding system behavior.
Purpose of the Study:
- To develop an analytical method for computing the largest Lyapunov exponent.
- To validate the method using the Fermi-Pasta-Ulam-beta (FPU-beta) model.
- To demonstrate the role of UPOs as probes of chaotic dynamics.
Main Methods:
- Analytical computation of the largest Lyapunov exponent.
- Utilizing a recently developed theory for Hamiltonian chaos.
- Calculating time averages of metric tensor curvature and fluctuations along UPOs.
Main Results:
- Successfully re-derived the analytic value of the largest Lyapunov exponent for the FPU-beta model.
- Achieved excellent agreement with Lyapunov exponents from standard numerical simulations.
- Confirmed the efficacy of UPOs in characterizing chaotic properties.
Conclusions:
- Unstable periodic orbits (UPOs) serve as efficient probes for dynamical properties in chaotic systems.
- The proposed analytical method provides a reliable alternative to numerical simulations for determining Lyapunov exponents.
- This work advances the theoretical understanding of chaos in Hamiltonian systems.