Amplitude death phenomena in delay--coupled Hamiltonian systems

TL;DR

This paper investigates how delay coupling in Hamiltonian systems leads to amplitude death, stabilizing point attractors and stopping oscillations, with analysis on harmonic and anharmonic oscillators including the Hénon-Heiles system.

Contribution

It introduces the phenomenon of amplitude death in delay-coupled Hamiltonian systems and analyzes its dynamics and phenomenology.

Findings

Amplitude death occurs for sufficiently strong delay coupling.

Phase-flip phenomena accompany the transition to amplitude death.

The study applies the analysis to harmonic and anharmonic oscillators, including Hénon-Heiles.

Abstract

Hamiltonian systems, when coupled {\it via} time--delayed interactions, do not remain conservative. In the uncoupled system, the motion can typically be periodic, quasiperiodic or chaotic. This changes drastically when delay coupling is introduced since now attractors can be created in the phase space. In particular for sufficiently strong coupling there can be amplitude death (AD), namely the stabilization of point attractors and the cessation of oscillatory motion. The approach to the state of AD or oscillation death is also accompanied by a phase--flip in the transient dynamics. A discussion and analysis of the phenomenology is made through an application to the specific cases of harmonic as well as anharmoniccoupled oscillators, in particular the H\'enon-Heiles system.