Computational Bifurcation Analysis

TL;DR

This work surveys computational bifurcation analysis by intertwining analytical insights with numerical continuation, using the CSTR model as a primary illustration and extending to symmetry-rich oscillator networks. It demonstrates how continuation identifies equilibria, Hopf, saddle-node, and Bogdanov–Takens bifurcations, and how periodic orbits are continued from Hopf points, including Hopf bubbles and homoclinic connections. The text also discusses defining systems, discretization via boundary-value problems, and the adaptive strategies required for robust tracking, while highlighting non-generic and symmetry-induced phenomena and their remedies. The study provides practical guidance and tools (e.g., coco) for modeling, automating, and visualizing bifurcation structures, with broad applicability to ODEs and PDE-inspired systems, and includes data and scripts for reproducibility.

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.

Figures (12)

• Figure 1: When $\beta=0$, the number of equilibria equals three for $\delta<1/e^2$ and $\sigma\in \left(e^{-y_*}y_*^2, e^{-y^*}y^{*2}\right)$, where $(1-y_*)e^{-y_*}=(1-y^*)e^{-y^*}=-\delta$. Two equilbria are found along the curve $y\mapsto(\sigma,\delta)=\left(y^2,y-1\right)e^{-y}$.
• Figure 2: For $x_{\mathrm{BT},1}<x<x_{\mathrm{BT},2}\Leftrightarrow p(x)<0$, the $\ell_1=0$ level set consists of degenerate Hopf bifurcations. In the figure, the $p(x)=0$ level set consists of families of Bogdanov-Takens points that coincide for $\gamma=4/27$. Supercritical Hopf bifurcations occur for $(x,\gamma)$ inside the right-most region enclosed by the $\ell_1=0$ level set and below $p(x)=0$.
• Figure 3: (a) Branch of equilibria in the CSTR problem \ref{['eq:cstr:model']}, obtained by fixing $y$ while varying $\delta$ and $\sigma$. Line style indicates number of unstable eigenvalues (USTAB). Hopf (red diamond) and saddle-node (green diamond) bifurcations are indicated along the branch. (b) Curves of saddle-node and Hopf bifurcations of equilibria in the $(\sigma,\delta)$-plane. Along the curves cusp, Bogdanov-Takens, and degenerate Hopf bifurcation points are indicated. A curve where the saddle is neutral starts at the Bogdanov-Takens points and contains a branch point of the defining problem. Other parameters: $\beta=0$, $\gamma=0.1$.
• Figure 4: Branches of periodic orbits connecting Hopf bifurcations (Hopf bubbles) under variations in $\sigma$ and for different values of $\delta\geq0.15$. For $\delta$ below the value associated with the degenerate Hopf bifurcation, such bubbles include a saddle-node bifurcation of periodic orbits. Other parameters: $\beta=0$, $\gamma=0.1$.
• Figure 5: Branches of periodic orbits emanating from Hopf bifurcations at $(\sigma,\delta)=(0.52,0.11)$ (shown in Fig. \ref{['fig:equilibria']}(a)) and $(\sigma,\delta)=(0.44,0.091)$, for fixed $\delta$ and varying $\sigma$. (a,b) Bifurcation diagrams in $(\sigma,\max(x)-\min(x))$-plane and $(\sigma,T)$-plane, where $T$ denotes the period. (c,d) Phase portrait and time profiles of periodic orbits with period $20$ close to the homoclinic and SNIC bifurcation, respectively. Other parameters: $\beta=0$, $\gamma=0.1$.