Catastrophe conditions for vector fields in Rn

Author:

Jeffrey Mike RORCID

Abstract

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 R n , we broaden the concept of Thom’s catastrophes to find bifurcation points of (non-gradient) vector fields of any dimension. We introduce a family of determinants , such that a codimension r bifurcation point is found by solving the system , subject to certain non-degeneracy conditions. The determinants generalize the derivatives j x j F ( x ) 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 partial differential equation.

Publisher

IOP Publishing

Subject

General Physics and Astronomy,Mathematical Physics,Modeling and Simulation,Statistics and Probability,Statistical and Nonlinear Physics

Cited by 3 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Elementary catastrophes underlying bifurcations of vector fields and PDEs;Nonlinearity;2024-06-20

2. Underlying catastrophes: umbilics and pattern formation;São Paulo Journal of Mathematical Sciences;2024-05-02

3. Wave-Pinned Patterns for Cell Polarity—A Catastrophe Theory Explanation;SIAM Journal on Applied Dynamical Systems;2024-02-26

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