Feature-rich bifurcations in a simple electronic circuit

TL;DR

This paper investigates a simple electronic circuit with a voltage-controlled current source, revealing complex bifurcation behaviors such as node to spiral, Hopf, and homoclinic bifurcations, with implications for oscillator design.

Contribution

It identifies and analyzes multiple local and global bifurcations in a basic circuit, providing insights into system stability and potential applications in oscillator design.

Findings

Demonstrates node to spiral bifurcation in the circuit

Identifies Hopf bifurcation leading to oscillations

Shows global homoclinic bifurcation affecting stability

Abstract

A simple electronic circuit with a voltage controlled current source is investigated. The circuit exhibits rich dynamics upon varying the circuit elements such as L,C and R, and the control factor of the current source. Among several other interesting features, the circuit demonstrates two local bifurcations, namely, node to spiral and Hopf bifurcation, and a global homoclinic bifurcation. Phase-portraits corresponding to these bifurcations are presented and the implications of these bifurcations on system stability are discussed. In particular, the circuit parameters corresponding to the onset of Hopf bifurcation may be exploited to design an oscillator with stable frequency and amplitude. The circuit may be easily implemented with nonlinear resistive elements such as diodes or transistors in saturation and a gyrator block as the voltage controlled current source.