Oscillations in a Slow-Fast Paleoclimate Model for Glacial Cycles

TL;DR

This paper analyzes a simplified climate model for glacial cycles, identifying bifurcations and limit cycles through analytical and numerical methods, enhancing understanding of climate oscillations.

Contribution

It introduces a reduced planar model with bifurcation analysis and explicit construction of homoclinic orbits, advancing the mathematical understanding of glacial cycle dynamics.

Findings

Identification of Hopf and Bautin bifurcations

Construction of homoclinic orbits using Melnikov's method

Numerical validation of analytical results

Abstract

This paper investigates a deterministic variant of the Saltzman-Maasch model for Pleistocene glacial cycles, formulated as a three-dimensional dynamical system with cubic feedback in the atmospheric carbon dioxide equation. After reducing the model to a planar system on a critical manifold, we perform a detailed bifurcation analysis and identify both Hopf and Bautin (generalized Hopf) bifurcations, which govern the emergence of stable and unstable limit cycles. To analyze global transitions, we perform a rescaling of time and variables to derive a leading-order Hamiltonian system. This reduction enables the explicit construction of homoclinic orbits and the application of Melnikov's method to assess their persistence under perturbations. The analytical findings are further corroborated by numerical simulations.