Dn Symmetric Hamiltonian System: A Network of Coupled Gyroscopes as a Case Study

TL;DR

This paper explores the dynamics of a high-dimensional Hamiltonian system with dihedral symmetry, using a ring of vibratory gyroscopes as a case study, revealing bifurcations and synchronization phenomena.

Contribution

It introduces a Hamiltonian framework for analyzing symmetric high-dimensional systems and derives normal forms to study bifurcations in coupled gyroscope networks.

Findings

Normal form analysis of gyroscope rings

Identification of bifurcations leading to synchronization

Effect of coupling schemes on system dynamics

Abstract

The evolution of a large class of biological, physical and engineering systems can be studied through both dynamical systems theory and Hamiltonian mechanics. The former theory, in particular its specialization to study systems with symmetry, is already well developed and has been used extensively on a wide variety of spatio-temporal systems. There are, however, fewer results on higher-dimensional Hamiltonian systems with symmetry. This lack of results has lead us to investigate the role of symmetry, in particular dihedral symmetry, on high-dimensional coupled Hamiltonian systems. As a representative example, we consider the model equations of a ring of vibratory gyroscopes. The equations are reformulated in a Hamiltonian structure and the corresponding normal forms are derived. Through a normal form analysis, we investigated the effects of various coupling schemes and unraveled the nature of the bifurcations that lead the ring of gyroscopes into and out of synchronization. The Hamiltonian approach is specially useful in investigating the collective behavior of small and large ring sizes and it can be readily extended to other symmetry-related systems.