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

Oleg Makarenkov, Esther Widiasih

TL;DR

The paper addresses how attracting limit cycles can bifurcate from a degenerate fold-fold singularity on the switching surface $L$ in a non-smooth glacial-cycle model. It develops a rigorous analysis of a three-variable Filippov system, employing a Poincaré map and time-map expansions in a small parameter to prove the existence and uniqueness of an attracting limit cycle near the singular point, with leading-order period scaling $T = O( abla^{1/2})$. The results provide a mechanistic link between non-smooth bifurcation structure and periodic glacial cycles, and offer a pathway to connect this mathematical mechanism with the Mid Pleistocene Transition when orbital forcing is included. This work thus advances understanding of how nonlinear feedbacks in conceptual climate models can generate realistic, periodically repeating climate regimes via a degenerate fold-fold bifurcation.

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.

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

TL;DR

The paper addresses how attracting limit cycles can bifurcate from a degenerate fold-fold singularity on the switching surface in a non-smooth glacial-cycle model. It develops a rigorous analysis of a three-variable Filippov system, employing a Poincaré map and time-map expansions in a small parameter to prove the existence and uniqueness of an attracting limit cycle near the singular point, with leading-order period scaling . The results provide a mechanistic link between non-smooth bifurcation structure and periodic glacial cycles, and offer a pathway to connect this mathematical mechanism with the Mid Pleistocene Transition when orbital forcing is included. This work thus advances understanding of how nonlinear feedbacks in conceptual climate models can generate realistic, periodically repeating climate regimes via a degenerate fold-fold bifurcation.

Abstract

We study the occurrence of limit cycles from a point on the discontinuity hyperplane 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.

Paper Structure

This paper contains 7 sections, 1 theorem, 69 equations, 1 figure.

Table of Contents

  1. Background
  2. Model
  3. A Brief Recap of the Model Development and Analysis
  4. Nonsmooth System of the Glacial Cycles
  5. Analysis
  6. Numerical computations
  7. Closing Remarks

Key Result

Theorem 3.1

Let be such a point that and Assume that Finally, assume that Then, for all $\varepsilon>0$ sufficiently small, system (sys1) admits a unique attractive limit cycle in a small neighborhood of $(x_0,y_0,z_0)$ that shrinks to $(x_0,y_0,z_0)$ as $\varepsilon\to 0.$ The period of the cycle equals

Figures (1)

  • Figure 1: Simulation of the attracting cycle of system (\ref{['sys0']}) with the parameters of Section \ref{['sec-comp']} and $\varepsilon=0.001.$

Theorems & Definitions (1)

  • Theorem 3.1