Definition of the Subject
Catastrophe theory is concerned with the mathematical modeling of sudden changes – so called “catastrophes” – in the behavior of natural systems, which can appear as a consequence of continuous changes of the system parameters. While in common speech the word catastrophe has a negative connotation, in mathematics it is neutral.
You can approach catastrophe theory from the point of view of differentiable maps or from the point of view of dynamical systems, that is, differential equations. We use the first case, where the theory is developed in the mathematical language of maps and functions (maps with range ℝ). We are interested in those points of the domain of differentiable functions where their gradient vanishes. Such points are called the “singularities” of the differentiable functions.
Assume that a system's behavior can be described by a potential function (see Sect. “Glossary”). Then the singularities of this function characterize the equilibrium points...
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- Singularity:
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Let \( { f\colon \mathbb{R}^{n} \to \mathbb{R}^{m} } \) be a differentiable map defined in some open neighborhood of the point \( { p \in \mathbb{R}^{n} } \) and \( { J_{p}f } \) its Jacobian matrix at p, consisting of the partial derivatives of all components of f with respect to all variables. f is called singular in p if rank \( { J_{p}f < \min{\{}n,m{\}} } \). If rank \( { J_{p}f = \min{\{}m,n{\}} } \), then f is called regular in p. For \( { m = 1 } \) (we call the map \( { f\colon \mathbb{R}^{n} \to \mathbb{R} } \) a differentiable function) the definition implies: A differentiable function \( { f\colon \mathbb{R}^{n} \to \mathbb{R} } \) is singular in \( { p \in \mathbb{R}^{n} } \), if \( { \operatorname{grad}\,f(p) = 0 } \). The point p, where the function is singular, is called a singularity of the function. Often the name “singularity” is used for the function itself if it is singular at a point p. A point where a function is singular is also called critical point. A critical point p of a function f is called a degenerated critical point if the Hessian (a quadratic matrix containing the second order partial derivatives) is singular at p; that means its determinant is zero at p.
- Diffeomorphism:
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A diffeomorphism is a bijective differentiable map between open sets of \( { \mathbb{R}^{n} } \) whose inverse is differentiable, too.
- Map germ:
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Two continuous maps \( { f\colon U \to \mathbb{R}^{k} } \) and \( { g\colon V \to \mathbb{R}^{k} } \), defined on neighborhoods U and V of \( { p \in \mathbb{R}^{n} } \), are called equivalent as germs at p if there exists a neighborhood \( { W \subset U \cap V } \) on which both coincide. Maps or functions, respectively, that are equivalent as germs can be considered to be equal regarding local features. The equivalence classes of this equivalence relation are called germs. The set of all germs of differentiable maps \( { \mathbb{R}^{n} \to \mathbb{R}^{k} } \) at a point p is named \( { \varepsilon_{p}(n,k) } \) . If p is the origin of \( { \mathbb{R}^{n} } \), one simply writes \( { \varepsilon(n,k) } \) instead of \( { \varepsilon_{0}(n,k) } \). Further, if \( { k = 1 } \), we write \( { \varepsilon(n) } \) instead of \( { \varepsilon(n,1) } \) and speak of function germs (also simplified as “germs”) at the origin of \( { \mathbb{R}^{n} } \). \( { \varepsilon(n,k) } \) is a vector space and \( { \varepsilon(n) } \) is an algebra , that is, a vector space with a structure of a ring. The ring \( { \varepsilon(n) } \) contains a unique maximal ideal \( { \mu(n)={\{}f \in \varepsilon(n)\vert f(0)=0{\}} } \). The ideal \( { \mu(n) } \) is generated by the germs of the coordinate functions \( { x_{1},\ldots,x_{n} } \). We use the form \( { \mu(n) = \langle x_{1},\ldots,x_{n}\rangle\varepsilon(n) } \) to emphasize, that this ideal is generated over the ring \( { \varepsilon(n) } \). So a function germ in \( { \mu(n) } \) is of the form \( \sum_{i=1}^n a_i(x) \cdot x_i \), with certain function germs \( a_{i}(x) \in \varepsilon(n) \).
- r-Equivalence:
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Two function germs \( { f,g \in \varepsilon(n) } \) are called r-equivalent if there exists a germ of a local diffeomorphism \( { h\colon \mathbb{R}^{n} \to \mathbb{R}^{n} } \) at the origin, such that \( { g = f \circ h } \).
- Unfolding:
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An unfolding of a differentiable function germ \( { f \in \mu(n) } \) is a germ \( { F \in \mu(n+r) } \) with \( { F\vert\mathbb{R}^{n} = f } \) (here ∣ means the restriction. The number r is called an unfolding dimension of f. An unfolding F of a germ f is called universal if every other unfolding of f can be received by suitable coordinate transformations, “morphisms of unfoldings ” from F, and the number of unfolding parameters of F is minimal (see “codimension”).
- Unfolding morphism:
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Suppose \( f \in \varepsilon(n) \) and \( F\colon \mathbb{R}^{n} \times \mathbb{R}^{k} \to \mathbb{R} \) and \( { G\colon \mathbb{R}^{n} \times \mathbb{R}^{r} \to \mathbb{R} } \) be unfoldings of f. A right‐morphism from F to G, also called unfolding morphism , is a pair (\( { \Phi,\alpha } \)), with \( { \Phi \in \varepsilon(n+k,n+r) } \) and \( { \alpha \in \mu(k) } \), such that:
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1.
\( { \Phi \vert \mathbb{R}^{n} = {\text{id}}(\mathbb{R}^{n}) } \), that is \( { \Phi(x,0) = (x,0) } \),
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2.
If \( { \Phi = (\phi,\psi) } \), with \( \phi \in \varepsilon(n+k,n) \), \( \psi \in \varepsilon(n+k,r) \), then \( \psi \in \varepsilon(k,r) \),
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3.
For all \( (x,u) \in \mathbb{R}^{n} \times \mathbb{R}^{k} \) we get \( F(x,u) = G(\Phi(x,u)) + \alpha(u) \).
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1.
- Catastrophe:
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A catastrophe is a universal unfolding of a singular function germ . The singular function germs are called organization centers of the catastrophes.
- Codimension:
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The codimension of a singularity f is given by \( { {\text{codim}}(f) = {\text{dim}}_{\mathbb{R}}\mu(n)/\langle\partial_{x}f\rangle } \) (quotient space). Here \( { \langle\partial_{x}f\rangle } \) is the Jacobian ideal generated by the partial derivatives of f and \( { \mu(n)={\{}f \in \varepsilon(n)\vert f(0) = 0{\}} } \) . The codimension of a singularity gives of the minimal number of unfolding parameters needed for the universal unfolding of the singularity.
- Potential function:
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Let \( { f\colon \mathbb{R}^{n} \to \mathbb{R}^{n} } \) be a differentiable map (that is, a differentiable vector field). If there exists a function \( { \varphi\colon \mathbb{R}^{n} \to \mathbb{R} } \) with the property that \( { \operatorname{grad}\varphi = f } \), then f is called a gradient vector field and φ is called a potential function of f.
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Sanns, W. (2009). Catastrophe Theory. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_47
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