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

P. A. Glendinning, David Simpson

Research output: Contribution to journalArticlepeer-review


We show that a one-dimensional differential equation depending on a parameter μ with a saddle-node bifurcation at μ=0 can be modelled by an extended normal form y˙=ν(μ)-y2+a(μ)y3,where the functions ν 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 characterization 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 ν and a can be determined analytically in some settings and numerically in others.

Original languageEnglish
Article number20220548
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2267
Early online date30 Nov 2022
Publication statusPublished - 30 Nov 2022


  • climate models
  • normal form
  • saddle-node bifurcation
  • smooth conjugacy


Dive into the research topics of 'Normal forms for saddle-node bifurcations: Takens’ coefficient and applications in climate models'. Together they form a unique fingerprint.

Cite this