Disconnected large bifurcation supports and Cartesian products of bifurcations

TL;DR

The work develops a rigorous framework for decomposing multi-parameter bifurcations of vector fields on $S^2$ into Cartesian products, via the notion of large bifurcation support (LBS) and the extension triviality property (ET/SET). It proves that for a glocal family with a disconnected LBS and suitable local SET/topological-distinguishability conditions, the unfolding is equivalent to a Cartesian product of bifurcations near each LBS component, capturing a form of independent bifurcation behavior. The authors also present a Banach-manifold formulation and provide stabilization and splitting techniques, along with a detailed analysis of two-parameter degeneracies and bifurcation diagrams, showing how non-Andronov elements constrain possible interactions between components. The results underscore the necessity of stability-type hypotheses, and the stabilization/splitting framework offers a robust path to identifying Cartesian-product unfoldings in higher-parameter settings with potential applications to global bifurcation analysis on the sphere.

Abstract

A bifurcation that occurs in a multiparameter family is a Cartesian product if it splits into two factors in the sense that one bifurcation takes place in one part of the phase portrait, another one -- in another part, and they are in a sense independent, do not interact with each other. To understand how a family bifurcates, it is sufficient to study it in a neighborhood of the so-called large bifurcation support. Given a family of vector fields on $S^2$ that unfolds a field $v_0$, the respective large bifurcation support is a closed $v_0$-invariant subset of the sphere indicating parts of the phase portrait of $v_0$ affected by bifurcations. One should consider disconnected large bifurcation supports in order to obtain Cartesian products for sure. We prove that, if the large bifurcation support is disconnected and the restriction of the original family to some neighborhood of each connected component is structurally stable (plus some mild extra conditions), then the original family is a Cartesian product of the bifurcations that occur near the components of the large bifurcation support. We also show that the structural stability requirement cannot be omitted.

Figures (3)

• Figure 1: Two parabolic cycles (thick closed orbits). There are also a hyperbolic repeller (a star) and a hyperbolic attractor (a hexagon). The field has no other closed orbits and singular points. The $\mathrm{LBS}$ of any family unfolding this field consists of two thick cycles. Thinner arrows correspond to usual orbits and demonstrate that cycles are semi-stable.
• Figure 2: A parabolic cycle (PC) and a saddle-node (SN). There are also two hyperbolic repellers (stars), a hyperbolic attractor (a hexagon) and a hyperbolic saddle (S) with separatrices. The field has no other closed orbits and singular points. A generic unfolding $V$ of this field has the $\mathrm{LBS}$ that consists of the saddle S with two unstable separatrices, the parabolic cycle PC and the saddle-node SN with the parabolic sector (colored with dark gray), while the $\mathrm{Acc}{(V)}$ contains also the whole interior domain of the parabolic cycle PC (shown by the chess filling), two stable separatrices of the saddle S and two hyperbolic repellers (stars).
• Figure 3: An example of the bifurcation diagram that $W$ might have.

Theorems & Definitions (76)

• Definition 1: Glocal families
• Remark 1
• Remark 2
• Definition 2
• Conjecture 1
• Conjecture 2
• Conjecture 3
• Remark 3
• Definition 3: Weak and strong equivalences
• Definition 4: Moderate equivalence