Hybrid Bifurcations: Periodicity from Eliminating a Line of Equilibria

TL;DR

This work introduces hybrid bifurcations, a mechanism by which a Hopf bifurcation can occur after eliminating a line of equilibria via a parallel drift in a stationary parameter $\mu$. By reducing to a three-dimensional center manifold and deriving a truncated cylindrical normal form, the authors prove the existence of a $C^1$-branch of periodic orbits emanating from the Hopf point $X_H$ with $y(t;\mu)=O(\sqrt{|\mu|})$, $z(t;\mu)=O(\mu)$, and period approaching $2\pi/\omega$ as $\mu\to0$, while classifying the bifurcation into Type-H, Type-ES, and Type-EU according to a discriminant. The key contributions include explicit stability criteria obtained via averaging and Liouville's formula, and a concrete application showing stable periodic coexistence of two predators in a Holling type II ecosystem without requiring proximity to extinction or conserved quantities. The results broaden bifurcation theory by enabling periodic solutions in systems with lines of equilibria and offer a new mechanism for oscillatory behavior in smooth dynamical systems and related PDE/functional-differential contexts.

Abstract

We describe a new mechanism that triggers periodic orbits in smooth dynamical systems. To this end, we introduce the concept of hybrid bifurcations: Such bifurcations occur when a line of equilibria with an exchange point of normal stability vanishes. Our main result is the existence and stability criteria of periodic orbits that bifurcate from breaking a line of equilibria. As an application, we obtain stable periodic coexistent solutions in an ecosystem for two competing predators with Holling's type II functional response.

Figures (3)

• Figure 1: Phase portraits in the $(r,z)$-plane of Hopf bifurcations without parameters described by the truncated cylindrical form \ref{['ToyModel']}. Any foliation transverse to the line of equilibria (dash-dotted) at the Hopf point (grey circle) is also transverse to the flow nearby; thus it is not flow-invariant. The Hopf point splits the line into center-unstable equilibria (hollow circles) and center-stable equilibria (solid circles). In the hyperbolic case, two invariant manifolds (thicker lines) with rotations in $\varphi$ form an invariant cone. In the elliptic case, the Hopf point is surrounded by heteroclinic orbits. Notice that exponentially small oscillations induced by the higher-order terms not in \ref{['ToyModel']} are not depicted here; see Li15.
• Figure 2: Three types of hybrid Hopf bifurcation. At $\mu = 0$, a Hopf bifurcation without parameters occurs. The Hopf point (grey circle) splits the line of equilibria (dash-dotted) into center-unstable equilibria (hollow circles) and center-stable equilibria (solid circles). We orient $\mu$ so that the flow is parallel for $\mu<0$ and a periodic orbit (two squares, up to a phase shift $\varphi \mapsto \varphi + \pi$) bifurcates from the Hopf point for $\mu>0$. Notice that at $\mu = 0$ this orientation leads to the appearance of opposite stability properties along the line between the hyperbolic type H and the elliptic type E. In a Type-H hybrid Hopf bifurcation, the periodic orbit is a saddle (invariant manifolds in thick). Near an elliptic Hopf point, we can find stable periodic orbits (Type-ES, solid squares) or fully unstable periodic orbits (Type-EU, hollow squares). Notice that for $\mu > 0$, Theorem \ref{['T:Bifurcation']} does not exclude the existence of other invariant subsets than the periodic orbit in the region outside the parallel drifts.
• Figure 3: Parameter continuation of periodic orbits that emerge from the Hopf point towards the boundary plane $Q_1$. The parameter values are $\delta_1=0.8$, $\delta_2=0.5$, $\alpha_1=0.1$, $\alpha_2=0.2$, $\lambda=0.4$. The Hopf point, with $\mu=0$, is given by $X_H=(0.2133,0.1667,0.4)$. The twelve periodic orbits are obtained from smaller (darker) to larger (lighter) by increasing $\mu > 0$. Our choice for $\mu$ to depict such global continuation is, respectively, $\mu = 0.0005, 0.001, 0.002, 0.005, 0.008, 0.011, 0.014, 0.017, 0.02, 0.025, 0.03, 0.05$.

Theorems & Definitions (18)

• Theorem 2.1: Hybrid Hopf bifurcations
• Remark 2.2: Shape of bifurcation branch
• Remark 2.3: Comparison with classical Hopf bifurcations
• Remark 3.1
• Lemma 3.2: All Hopf points are elliptic
• Theorem 3.3: Stable periodic coexistence
• Lemma 4.1: Unperturbed cylindrical form
• Remark 4.2
• proof : Proof of Lemma \ref{['L:UnpertNormalForm']}.
• Lemma 4.3: Perturbed cylindrical form