Hopf–Bautin and Homoclinic Bifurcations in a Saltzman–Maasch Model with Cubic Feedback

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 analytically 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 predictions are further validated through numerical continuation and simulations, providing a rigorous foundation for previously reported numerical observations and establishing, in particular, the analytical existence of Bautin bifurcations, homoclinic connections via Melnikov analysis, and a systematic slow–fast geometric reduction of the model.