Catastrophe conditions for vector fields in Rn

Abstract

Practical conditions are given here for finding and classifying high codimension intersection points of n hypersurfaces in n dimensions. By interpreting those hypersurfaces as the nullclines of a vector field in Rn, we broaden the concept of Thom’s catastrophes to find bi- furcation points of (non-gradient) vector fields of any dimension. We introduce a family of determinants Bj , such that a codimension r bifurcation point is found by solving the system B1 = ... = Br = 0, subject to certain non-degeneracy conditions. The determinants Bj generalize the derivatives ∂^jF(x)/∂x^j that vanish at a catastrophe of a scalar function F(x). We do not extend catastrophe theory or singularity theory themselves, but provide a means to apply them more readily to the multi-dimensional dynamical models that appear, for example, in the study of various engineered or living systems. For illustration we apply our conditions to locate butterfly and star catastrophes in a second order PDE.

Elementary catastrophes underlying bifurcations of vector fields and PDEs

“…These days catastrophes have been largely absorbed into the broader theory of bifurcations, but here we will argue that a wider application of Thom's original concept is possible, using the idea of underlying catastrophes introduced in [12]. This concept attempts to apply the elementary catastrophes to much wider classes of systems than they were intended to encompass, basically to any multidimensional systems with general spatial and temporal variations.…”

“…A more direct extension of catastrophes to such problems is made possible by a recent suggestion from [12], that any bifurcation actually has an elementary catastrophe underlying it. The underlying catastophe was described in [12] as 'a zero of a vector field encountering a singularity', and I will make this identification precise here.…”

“…A more direct extension of catastrophes to such problems is made possible by a recent suggestion from [12], that any bifurcation actually has an elementary catastrophe underlying it. The underlying catastophe was described in [12] as 'a zero of a vector field encountering a singularity', and I will make this identification precise here. I also give an example of a reaction-diffusion problem where diffusion terms trigger an underlying catastrophe, separating a system's steady state concentrations into different mixing regimes, the boundaries between which are clearly recognisable as a butterfly catastrophe.…”

“…= B r to be solvable, and they determine the values of the G r,k1..kr−1 to be defined below. The expression (1) is called the primary form of the catastrophe in [12], and is similar to Arnold's expressions for the principle family of class A r+1 for singularities of vector fields from [2]. The extent to which this constitutes a local model or 'normal form' of a codimension r underlying catastrophe, such that F can be transformed into such an expression locally, will be discussed in a forthcoming work [5].…”

“…We will first give a suggestion of the practical possibilities of these conditions with an example applying underlying catastrophes to a reaction-diffusion equation in section 2. We then give more formal statements and general expressions for these 'B-G' determinants in section 3, recapping from [12]. To help understand how underlying catastrophes relate to established bifurcation theory, we will show that we can locate the conditions B i = 0 within the Thom-Boardman classification of singularities, treating the function F as a general mapping rather than specifically a vector field.…”