Systematic experimental exploration of bifurcations with non-invasive control

TL;DR

This paper introduces a systematic method for exploring bifurcations in nonlinear experiments, overcoming limitations of existing control schemes to accurately track equilibria and periodic orbits, demonstrated on a mechanical oscillator.

Contribution

The authors develop a general approach to overcome the odd-number limitation in non-invasive control, enabling detailed bifurcation analysis in physical experiments.

Findings

Successfully traced resonance surfaces near bifurcations

Overcame odd-number limitation in control schemes

Demonstrated method on a mechanical nonlinear oscillator

Abstract

We present a general method for systematically investigating the dynamics and bifurcations of a physical nonlinear experiment. In particular, we show how the odd-number limitation inherent in popular non-invasive control schemes, such as (Pyragas) time-delayed or washout-filtered feedback control, can be overcome for tracking equilibria or forced periodic orbits in experiments. To demonstrate the use of our non-invasive control, we trace out experimentally the resonance surface of a periodically forced mechanical nonlinear oscillator near the onset of instability, around two saddle-node bifurcations (folds) and a cusp bifurcation.