A Novel Route to Oscillations via non-central SNICeroclinic Bifurcation: Unfolding the Separatrix Loop Between a Saddle-Node and a Saddle

TL;DR

This work introduces and classifies a non-central SNICeroclinic loop, a codimension-three heteroclinic structure between a non-hyperbolic saddle and a hyperbolic saddle, in planar systems. By constructing a Poincaré map from four cross-sections and applying Shil'nikov variables, the authors unfold the loop with three parameters $\mu_1,\mu_2,\mu_3$, revealing how SNIC, homoclinic, heteroclinic, and periodic states emerge, persist, or disappear. They provide explicit local maps around the saddle-node ($T_{12}$) and around the saddle ($T_{34}$), as well as global return maps, to map the transitions and describe the birth/death of periodic solutions and the persistence of connections. The findings demonstrate that non-central SNICeroclinic loops act as organizing centers for oscillatory dynamics and offer a novel global mechanism for oscillation birth and destruction, with potential implications for neuronal and biochemical systems and for extending bifurcation analysis tools to non-hyperbolic settings.

Abstract

In this paper, we investigate saddle-node to saddle separatrix--loops that we term SNICeroclinic bifurcations. They are generic codimension-two bifurcations involving a heteroclinic loop between one non-hyperbolic and one hyperbolic saddle. A particular codimension-three case is the non-central SNICeroclinic bifurcation. We unfold this bifurcation in the minimal dimension (planar) case where the non-hyperbolic point is assumed to undergo a saddle-node bifurcation. Applying the method of Poincaré return maps, we present a minimal set of perturbations that captures all qualitatively distinct behaviors near a non-central SNICeroclinic loop. Specifically, we study how variation of the three unfolding parameters leads to transitions from heteroclinic and homoclinic loops, saddle-node on an invariant circle (SNIC), and periodic orbits as well as equilibria. We show that although the bifurcation has been largely unexplored in applications, it can act as an organizing center for transitions between various types of saddle-node and saddle separatrix loops. It is also a generic route to oscillations that are both born and destroyed via global bifurcations, compared to the commonly observed scenarios involving local (Hopf) and in some cases global (homoclinic or SNIC) bifurcations.

Figures (9)

• Figure 1: (A,B) Manifolds of the saddle point (pink) and saddle node (yellow) in the polynomial system (\ref{['eq:poly']}) as it approaches the singular limit $(a,b,c)=(0.222\dot{2},1,-0.888\dot{8})$ with $\epsilon=0.1$ (A) and $\epsilon=0.01$ (B). Green dot identifies the location of the unstable node. As $\epsilon \rightarrow 0$ the system approaches the existence of the SNICeroclinic loop. Solid and dotted gray lines identify stable and unstable manifolds, respectively. (C) Non-central SNICeroclinic loop in the polynomial system \ref{['eq:poly']} with $\epsilon=1$, $a=0.042$, $b=0.49575$ and $c=-0.85$. Light yellow and pink circles identify saddle-node and saddle, respectively, while green signifies unstable node. Black line shows the SNICeroclinic loop between the saddle-node and saddle, solid and dotted gray lines identify stable and unstable manifolds, respectively. (D) Continuation in the parameter $c$ in the polynomial system ($\epsilon=1$, $a=0.042$, $b=0.35$), where solid and dotted black lines identify stable and unstable equilibria, respectively; red and blue lines show the maximum and the minimum of the periodic solutions, respectively; black triangle (SN1, SN2, SN3): saddle node; black square (HB1, HB2): Hopf; black circle (HC1, HC2, HC3): homoclinic; white circle (SNIC): saddle node on invariant circle.
• Figure 2: A non-central SNICeroclinic loop between a saddle-node $pq_1$ with linearly unstable direction and saddle $p_2$ in the GTPase activation model \ref{['eq:GTPase']}. Yellow and pink circles identify saddle-node and saddle, respectively. Black dotted line shows the unstable SNICeroclinic loop between the saddle-node and the saddle, solid and dotted gray lines signify the stable and unstable manifolds, respectively.
• Figure 3: (A) Non-central SNICeroclinic loop $\Gamma=\overline{\Gamma_1 \cup \Gamma_2}$ between saddle-node $pq_1$ and saddle $p_2$. (B) Overview of the construction of the Poincaré map for the planar case in the vicinity of the heteroclinic loop $\Gamma$ between nonhyperbolic equilibrium ($pq_1$, yellow) and hyperbolic saddle ($p_2$, pink) with established cross-sections ($\Sigma_1$, $\Sigma_2$, $\Sigma_3$, $\Sigma_4$) and corresponding "connection" maps $T_{12}$, $T_{23}$, $T_{34}$, $T_{41}$.
• Figure 4: Representation of the connecting maps $T_{12}$ for (A) $\mu_1<0$, (B) $\mu_1>0$, (C) $\mu_1=0$.
• Figure 5: Representation of the connecting maps $T_{34}$ for (A)$|\lambda_s|>|\lambda_u|$, (B)$|\lambda_s|=|\lambda_u|$, (C)$|\lambda_s|<|\lambda_u|$.