Computing Two-Parameter Bifurcation Diagrams for Oscillating Circuits and Systems

Abstract

Various oscillating (periodic and chaotic) circuits and systems show interesting responses whose nature changes with varying parameters. It often happens that a change of one element (i.e. resistor) of a circuit or system may cause a simultaneous change of two (or more) coefficients in the underlying mathematical model (i.e. a system of nonlinear ordinary differential equations, or ODEs). In this paper we present two-parameter bifurcation diagrams of such circuits and systems, obtained when two parameters vary simultaneously. Four different numerical techniques are applied to two selected dynamical systems (an active oscillating circuit with a memristive element and an electric arc circuit). The focus of this paper is on the computationally intensive calculations rather than on analytical analysis of the oscillatory responses. Two-parameter bifurcation diagrams require solving systems of nonlinear ODEs several hundred thousand (or even a few million) times (depending on the assumed resolution), plus additional work to distinguish periodic solutions from chaotic ones. Our computations are done using various combinations of the C++, Fortran/Python and Julia environments with Runge-Kutta order-4 and order-5 numerical solvers and the 0-1 test for chaos. Several two-parameter bifurcation diagrams are presented.

Keywords

• Julia and Python programming
• oscillatory systems
• periodic and chaotic signals
• two-parameter bifurcation diagrams