Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories

TL;DR

This paper investigates hyperbolic chaos in mechanical triple linkage systems, analyzing their chaotic regimes and manifold tangencies, and identifies conditions under which hyperbolicity is maintained or violated.

Contribution

It provides a numerical analysis of hyperbolic chaos in triple linkage models, including systems with holonomic constraints and potential interactions, and tests for manifold tangencies.

Findings

Chaotic regimes exhibit hyperbolic attractors at low supercriticality.

Hyperbolicity is compromised in systems with potential interactions at higher excitation levels.

Numerical analysis of stable and unstable manifold intersections supports hyperbolic chaos presence.

Abstract

Dynamical equations are formulated and a numerical study is provided for self-oscillatory model systems based on the triple linkage hinge mechanism of Thurston -- Weeks -- Hunt -- MacKay. We consider systems with holonomic mechanical constraint of three rotators as well as systems, where three rotators interact by potential forces. We present and discuss some quantitative characteristics of the chaotic regimes (Lyapunov exponents, power spectrum). Chaotic dynamics of the models we consider are associated with hyperbolic attractors, at least, at relatively small supercriticality of the self-oscillating modes; that follows from numerical analysis of the distribution for angles of intersection of stable and unstable manifolds of phase trajectories on the attractors. In systems based on rotators with interacting potential the hyperbolicity is violated starting from a certain level of excitation.