Bistable boundary conditions implying codimension 2 bifurcations

TL;DR

The paper proves a global topological constraint: for a 2-parameter smooth dynamical-systems family with finite restpoints, a boundary $S/Z$-curve composed of two opposing folds and no other boundary bifurcations, combined with the absence of a fold-Hopf bifurcation inside the parameter domain, implies at least one interior cusp bifurcation (an odd number of cusps on the associated catastrophe manifold boundary). The authors develop a global framework using the catastrophe manifold $\mathcal{M}$, fold curves, saddle connections, and a center-manifold bundle to derive a parity argument for cusp points, and extend the result to gradient catastrophe contexts via a corollary for $C^3$ function families $f_\theta$. The approach does not require a bound on the number of equilibria or the phase-space dimension, highlighting a robust global-condition-to-local-bifurcation link with potential applications in data-driven bifurcation analysis. The methods, including the construction of $\varepsilon$-tunnels and the fiber-orientation parity argument, establish a template for proving codimension-2 bifurcation presence from boundary conditions and lay groundwork for future extensions to broader MS-family settings.

Abstract

We consider generic families $X_\param$ of smooth dynamical systems depending on parameters $\param\in P$ where $P$ is a 2-dimensional simply connected domain and assume that each $X_\param$ only has a finite number of restpoints and periodic orbits. We prove that if over the boundary of $P$ there is a S or Z shaped bifurcation graph containing two opposing fold bifurcation points while over the rest of the boundary there are no other bifurcation points, then, if there is no fold-Hopf bifurcation in $P$ then there is at least one cusp in the interior of $P$.

Figures (3)

• Figure 1: A. Over the boundary $\partial P$ of the parameter space there are no bifurcation points except the two folds in the S/Z curve over $\partial P$ (red and purple curve). The folds are opposed in the sense defined in the text, a concept that formalises the notion of a S/Z curve. Theorem 1 asserts that in this case there are an odd number of cusps in $P$. The figure shows the simplest case, where there is one. The coloured folded surface is the catastrophe manifold for this example. Note how the dynamics on the centre manifold of the fold point switches as the black fold curve goes through the cusp.
• Figure 2: A. Schematic diagram of how a tunnel $T$ connects two fold circles in $\mathcal{M}_S$. B. Schematic diagram of the curve $\bar{\mathcal{C}}$ and how it is constructed using $\mathcal{C}$ the fold circles and tunnels. C. A schematic illustration of why flips in orientation caused by the transitions from saddle to fold points in a tunnel occur in pairs. The flips occur at the marked transitions from the tunnel. Since there are no cusps near the tunnel ends the orientations at each of the tunnel ends agree. Consequently, there is either zero or two flips in orientation as the two sides are traversed.
• Figure 3: Schematic of neighbourhoods and approximating curves. A. Looping curve (red) and nudging curve (blue). B. Disposition of the neighbourhoods $U_i$, $U_{i \mathtt{ mod } m}$ and $N_i$. C. Showing how an approximating curve for an attractor goes around, first, around a standard cusp and then a dual cusp. D. Explaining the loop extension.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Lemma 1
• Corollary 1
• Proposition 1
• Proposition 2
• Lemma 2
• Lemma 3
• Theorem 3
• Lemma 4