Normal forms for saddle-node bifurcations: Takens' coefficient and applications in climate models

TL;DR

This paper develops explicit extended normal forms for saddle-node bifurcations, including Takens' coefficient, providing detailed local and global conjugacy results and applications to climate models.

Contribution

It introduces explicit formulas for extended normal forms of saddle-node bifurcations, including Takens' coefficient, with applications to climate models and detailed conjugacy analysis.

Findings

Explicit formulas for functions in the normal form.

Differentiable conjugacy extends beyond the bifurcation point.

Applications demonstrated in climate science models.

Abstract

We show that a one-dimensional differential equation depending on a parameter $\mu$ with a saddle-node bifurcation at $\mu =0$ can be modelled by an extended normal form $\dot y = \nu (\mu )-y^2+a(\mu )y^3$, where the functions $\nu$ and $a$ are solutions to equations that can be written down explicitly. The equivalence to the original equations is a local differentiable conjugacy on the basins of attraction and repulsion of stationary points in the parameter region for which these exist, and is a differentiable conjugacy on the whole local interval otherwise. (Recall that in standard approaches local equivalence is topological rather than differentiable.) The value $a(0)$ is Takens' coefficient from normal form theory. The results explain the sense in which normal forms extend away from the bifurcation point and provide a new and more detailed characterisation of the saddle-node bifurcation. The one-dimensional system can be derived from higher dimensional equations using centre manifold theory. We illustrate this using two examples from climate science and show how the functions $\nu$ and $a$ can be determined analytically in some settings and numerically in others.