Parametric excitation and Hopf bifurcation analysis of a time delayed nonlinear feedback oscillator

TL;DR

This paper analyzes how parametric excitation influences nonlinear oscillators with time delay, focusing on stability, bifurcations, and resonance behaviors using analytical and bifurcation methods.

Contribution

It introduces a delay model for nonlinear feedback oscillators and characterizes resonance, antiresonance, and bifurcation phenomena with analytical and bifurcation analysis.

Findings

Identification of resonance and antiresonance behaviors.

Characterization of Hopf bifurcation and stability of periodic solutions.

Influence of delay, damping, and nonlinear terms on system dynamics.

Abstract

In this paper, an attempt has been made to understand the parametric excitation of a periodic orbit of nonlinear oscillator which can be a limit cycle, center or a slowly decaying center-type oscillation. For this a delay model is considered with nonlinear feedback oscillator defined in terms of Li\'enard oscillator description which can give rise to any one of the periodic orbits stated above. We have characterized the resonance and antiresonance behaviour for arbitrary nonlinear system from their stability and bifurcation analyses in reference to the standard delayed van der Pol system. An approximate analytical solution using Krylov--Bogoliubov (K-B) averaging method is utilised to recognize the sub-harmonic resonance and antiresonance, and average energy consumption per cycle. Direction of Hopf bifurcation and stability of the periodic solution bifurcating from the trivial fixed point are carried out using normal form and center manifold theory. The parametric excitation is also thoroughly investigated via bifurcation analysis to find the role of the control parameters like time delay, damping and nonlinear terms.