Singularities, Bifurcations and Catastrophes

Abstract

Suitable for advanced undergraduates, postgraduates and researchers, this self-contained textbook provides an introduction to the mathematics lying at the foundations of bifurcation theory. The theory is built up gradually, beginning with the well-developed approach to singularity theory through right-equivalence. The text proceeds with contact equivalence of map-germs and finally presents the path formulation of bifurcation theory. This formulation, developed partly by the author, is more general and more flexible than the original one dating from the 1980s. A series of appendices discuss standard background material, such as calculus of several variables, existence and uniqueness theorems for ODEs, and some basic material on rings and modules. Based on the author's own teaching experience, the book contains numerous examples and illustrations. The wealth of end-of-chapter problems develop and reinforce understanding of the key ideas and techniques: solutions to a selection are provided.

Climate response and sensitivity: time scales and late tipping points

“…Due to the functional forms of temperature-dependent albedo and emissivity, the model (3.1) can have one, two or three stable equilibria depending on the parameter values. This is organized by a fifth order ‘butterfly’ singularity [44]: see the electronic supplementary material [45] for a verification and in-depth analysis of the bifurcation structure of this model. Nonetheless, for the parameter values given in table 1, the model is bistable for a certain range of values of the parameter μ: in this bistable region, the model supports a stable cold ‘icehouse’ and a warm ‘hothouse’ climate state (see figure 3).…”

The 1:1 resonance in Hamiltonian systems

“…Restricted to the 3-sphere {H 2 = 1}, the projection mapping involved is a Hopf mapping so the reduced phase space is a 2-sphere. Then we apply equivariant singularity theory to the map germ (H, H 2 ) and find a universal unfolding subject to non-degeneracy conditions on the coefficients in the higher order terms of H. By the nature of our method, we can not hope for more than local results and we exploit this fact by switching to germs, see [3,26,28]. Very briefly: a map germ is the collection of mappings equal to one another on an arbitrary small neighbourhood of a given point, say 0.…”

Catastrophe conditions for vector fields in Rn

“…Given a map F : R n ×R p → R n and a particular point (x * , α * ) ∈ R n ×R p , singularity theory provides the means to classify any singularity that may occur there with great generality, see e.g. [3,7,16,17,23]. Our aim here is essentially to turn this around, to provide readily solvable conditions that can be solved to find the point (x * , α * ) at which some suspected singularity or bifurcation occurs.…”