Generalized Pseudo-Hopf Bifurcation: Limit Cycle Position and Period

TL;DR

The paper introduces a generalized pseudo-Hopf bifurcation for planar piecewise-smooth vector fields by translating one side of the discontinuity set, yielding a one-parameter family $Z_b$ and a topological bifurcation mechanism that produces a crossing limit cycle near the origin. It then refines the position and period of the bifurcating cycle under a hierarchy of hypotheses, from differentiability to Dulac-type and quasi-homogeneous analyses, covering a wide array of local and non-local configurations including folds, centers/foci (elementary and nilpotent), cusps, periodic orbits, and hyperbolic polycycles. The core contributions are (i) explicit leading-term asymptotics for the limit cycle position $x(b)$, (ii) detailed half-monodromic maps and flight times for numerous configurations, and (iii) a comprehensive table summarizing the leading terms of both position and period across all pairings of interacting objects. These results extend the classical pseudo-Hopf framework beyond invisible folds and monodromic singularities to encompass broader geometric scenarios, enabling precise temporal predictions (via the period function) in both local and global sliding dynamics with potential applications in neuroscience, locomotion, and other nonlinear systems with discontinuities.

Abstract

We investigate planar piecewise-smooth vector fields with a discontinuity line, focusing on the bifurcation of crossing limit cycles that arise when one of the vector fields is translated along the discontinuity set. We establish topological conditions under which such bifurcations occur and, under additional generic hypotheses, derive precise asymptotic expressions for both the position and the period of the resulting limit cycle in terms of the perturbation parameter. Our results extend the classical pseudo-Hopf bifurcation: they are not restricted to invisible folds or elementary monodromic singularities, but also apply to nilpotent centers/foci, half-monodromic singularities such as cusps, periodic orbits, and hyperbolic polycycles, thereby encompassing both local and non-local configurations. We show that the period function exhibits distinct asymptotic behaviors depending on the interacting objects. In particular, we provide a comprehensive table summarizing the leading terms of the period and position of the limit cycle for each possible configuration.

Figures (11)

• Figure 1: Illustration of a classical pseudo-Hopf bifurcation. Here $b^+$ (resp. $b^-$) represent the invisible fold of the vector field acting above (resp. below) the switching set. Double arrows represent the flow of sliding vector field. Objects colored in blue (resp. red) are stable (resp. unstable). Colors available on the online version.
• Figure 2: Illustration of a generalized pseudo-Hopf composed by a periodic orbit and a tangential hyperbolic polycycle. Double arrows represent the sliding vector field. Objects colored in blue (resp. red) are stable (resp. unstable). Colors available on the online version.
• Figure 3: Illustration of $\varphi^\pm$ and $\Delta_0$, with $\delta=1$ and $\sigma=1$.
• Figure 4: Illustration of $\Delta(x,b)$ with $\delta=1$ and $\sigma=-1$.
• Figure 5: Illustration of $(a)$ the half-return maps $\varphi^\pm$ of $Z$ and $(b)$ its inverses $\zeta^\pm$. Observe $(c)$ that $\zeta^\pm$ are the half-return maps of $\mathcal{Z}$

Theorems & Definitions (28)

• Theorem A: Topological pseudo-Hopf
• proof
• Proposition 1: Differentiable pseudo-Hopf
• Theorem B: Smooth pseudo-Hopf
• Definition 1: Finitely flat functions
• Definition 2: Dulac-type functions
• Theorem C: Dulac-type pseudo-Hopf
• proof : Proof of Proposition \ref{['M2']}
• proof : Proof of Theorem \ref{['M3']}
• proof : Proof of Theorem \ref{['M4']}