Date: Wednesday 22nd January 2025, 1pm

Stability and Chaos in Hamiltonian Systems with Many Degrees of Freedom

Abstract:

In this talk, we delve into the complex dynamics of Hamiltonian systems with multiple degrees of freedom (DOF), exploring phase transitions from regular to chaotic behavior. We begin with an analysis of Hamiltonian systems with two DOF, using the Hénon-Heiles model as a case study to illustrate the onset of chaos at varying energy levels. Moving to higher-dimensional systems with N DOF, such as the Bose-Einstein Condensate (BEC) and the Fermi-Pasta-Ulam (FPU) lattice, we investigate the stability of simple periodic orbits (SPOs) and their critical role in the emergence of chaos.

Through linear stability analysis and Lyapunov spectra, we demonstrate how chaotic dynamics intensify with increasing energy and DOF. We will also discuss practical tools, such as the Smaller Alignment Index (SALI), for distinguishing between ordered and chaotic motion in these systems.

In the final part of the talk, I will highlight connections to recent work with Dr Christodoulidi, as published in Physica D (2024), examining the implications for statistical mechanics. This research provides valuable insights into the behaviour of complex systems, with applications spanning statistical mechanics, novel materials, and condensed matter physics.

Dr. Chris Antonopoulos is a lecturer at the School of Mathematics, Statistics and Actuarial Science (SMSAS), University of Essex.