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

2025/11/04 by Makarenkov, Oleg, Widiasih, Esther
#34A36 #37G15 #86A08 #86A08 34A36 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.02143

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.

Citations

• Bifurcation of limit cycles from a switched equilibrium in planar switched systems and its application to power converters
• Bifurcation of limit cycles from a fold-fold singularity in planar switched systems
• Dynamics of the Budyko Energy Balance Model

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