Bifurcation of Limit Cycles from a Fold-Fold Singularity in a Glacial Cycles Model

TL;DR

This paper investigates how a degenerate fold-fold singularity in a glacial cycles model leads to the emergence of an attracting limit cycle, revealing a key dynamical mechanism behind glacial oscillations.

Contribution

It demonstrates that a parameter passing through a fold-fold singularity causes the creation of an attracting limit cycle in a glacial cycles model.

Findings

Degenerate fold-fold singularity induces limit cycle formation.

Limit cycle appears as parameter crosses a critical value.

Mechanism explains oscillatory behavior in glacial models.

Abstract

We study the occurrence of limit cycles from a point on the discontinuity hyperplane $L$ between two smooth vector fields where the two vector fields both point towards one another. Generically, such a point (called switched equilibrium in control) is asymptotically stable, but we consider the situation where the two vector fields become tangent to L at the switched equilibrium under varying parameter making a degenerate fold-fold singularity. We prove that moving the parameter past such a singular value leads to the occurrence of an attracting limit cycle, which is exactly the dynamical mechanism we then discover in a conceptual model of glacial cycles.