# Asymptotic analysis of internal relaxation-oscillations in a conceptual climate model

## TL;DR

This paper develops a simplified climate model demonstrating that internal relaxation-oscillations, akin to ice-age cycles, can occur naturally without external forcing, supported by asymptotic analysis and paleoclimatic data comparison.

## Contribution

It introduces a new dynamical system based on the KCG climate model, analyzing its stability and oscillatory behavior, and derives a formula for oscillation periods.

## Key findings

- The model exhibits stable limit cycles with relaxation-oscillations.
- The period formula aligns with paleoclimatic data.
- Oscillations occur without external forcing across various parameters.

## Abstract

We construct a dynamical system based on the KCG (K\"all\'en, Crafoord, Ghil) conceptual climate model which includes the ice-albedo and precipitation-temperature feedbacks. Further, we classify the stability of various critical points of the system and identify a parameter which change generates a Hopf bifurcation. This gives rise to a stable limit cycle around a physically interesting critical point. Moreover, it follows from the general theory that the periodic orbit exhibits relaxation-oscillations which are a characteristic feature of the Pleistocene ice-ages. We provide an asymptotic analysis of their behaviour and derive a formula for the period along with several estimates. They, in turn, are in a decent agreement with paleoclimatic data and are independent of any parametrization used. Whence, our simple but robust model shows that a climate may exhibit internal relaxation-oscillations without any external forcing and for a wide range of parameters.