Experimental observation of topological transition in linear and non-linear parametric oscillators

TL;DR

This paper investigates topological transitions in parametric oscillators caused by perturbations, demonstrating both theoretically and experimentally how stable states change abruptly at critical defect parameters, with implications for understanding nonlinear dynamics.

Contribution

It introduces a topological framework using winding numbers to interpret state transitions in linear and nonlinear parametric oscillators, supported by experimental validation.

Findings

Abrupt state changes at critical defect parameters

Topological interpretation via winding number

Experimental observation using vibrated fluid surface

Abstract

Parametric oscillators are examples of externally driven systems that can exhibit two stable states with opposite phase depending on the initial conditions. In this work, we propose to study what happens when the external forcing is perturbed by a continuously parametrized defect. Initially in one of its stable state, the oscillator will be perturbed by the defect and finally reach another stable state, which can be its initial one or the other one. For some critical value of the defect parameter, the final state changes abruptly. We investigate theoretically and experimentally such transition both in the linear and non-linear case, and the effect of non-linearities is discussed. A topological interpretation in terms of winding number is proposed, and we show that winding changes correspond to singularities in the temporal dynamics. An experimental observation of such transition is performed using parametric Faraday instability at the surface of a vibrated fluid.