Computational Bifurcation Analysis

TL;DR

This paper discusses computational methods for bifurcation analysis in dynamical systems, illustrating how continuation techniques and bifurcation diagrams help understand solution behaviors as parameters vary.

Contribution

It introduces a computational framework combining theoretical principles and practical algorithms for bifurcation analysis of equilibria and periodic orbits.

Findings

Illustrates the use of continuation techniques in bifurcation analysis

Provides general theoretical principles for bifurcation analysis

Demonstrates the approach on common dynamical system examples

Abstract

Bifurcation analysis collects techniques for characterizing the dependence of certain classes of solutions of a dynamical system on variations in problem parameters. Common solution classes of interest include equilibria and periodic orbits, the number and stability of which may vary as parameters vary. Continuation techniques generate continuous families of such solutions in the combined state and parameter space, e.g., curves (branches) of periodic orbits or surfaces of equilibria. Their advantage over simulation-based approaches is the ability to map out such families independently of the dynamic stability of the equilibria or periodic orbits. Bifurcation diagrams represent families of equilibria and periodic orbits as curves or surfaces in appropriate coordinate systems. Special points, such as bifurcations, are often highlighted in such diagrams. This article provides an illustration of this paradigm of synergy between theoretical derivations and computational analysis for several characteristic examples of bifurcation analysis in commonly encountered classes of problems. General theoretical principles are deduced from these illustrations and collected for the reader's subsequent reference.