Catastrophe, Gestalt and Thom’s Natural Philosophy of 3-D Space as Underlying All Abstract Forms – Thought of the Day 157.0

The main result of mathematical catastrophe theory consists in the classification of unfoldings (= evolutions around the center (the germ) of a dynamic system after its destabilization). The classification depends on two sorts of variables:

(a) The set of internal variables (= variables already contained in the germ of the dynamic system). The cardinal of this set is called corank,

(b) the set of external variables (= variables governing the evolution of the system). Its cardinal is called codimension.

The table below shows the elementary catastrophes for Thom:

The A-unfoldings are called cuspoids, the D-unfoldings umbilics. Applications of the E-unfoldings have only been considered in A geometric model of anorexia and its treatment. By loosening the condition for topological equivalence of unfoldings, we can enlarge the list, taking in the family of double cusps (X₉) which has codimension 8. The unfoldings A₃(the cusp) and A₅ (the butterfly) have a positive and a negative variant A₊₃, A_-3, A₊₅, A_-5.

We obtain Thorn’s original list of seven “catastrophes” if we consider only unfoldings up to codimension 4 and without the negative variants of A₃ and A₅.

Thom argues that “gestalts” are locally con­stituted by maximally four disjoint constituents which have a common point of equilibrium, a common origin. This restriction is ultimately founded in Gibb’s law of phases, which states that in three-dimensional space maximally four independent systems can be in equilibrium. In Thom’s natural philosophy, three-dimensional space is underlying all abstract forms. He, therefore, presumes that the restriction to four constituents in a “gestalt” is a kind of cognitive universal. In spite of the plausibility of Thom’s arguments there is a weaker assumption that the number of constituents in a gestalt should be finite and small. All unfoldings with codimension (i.e. number of external variables) smaller than or equal to 5 have simple germs. The unfoldings with corank (i.e. number of internal variables) greater than two have moduli. As a matter of fact the most prominent semantic archetypes will come from those unfoldings considered by René Thom in his sketch of catastrophe theoretic semantics.

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Symmetrical – Asymmetrical Dialectics Within Catastrophical Dynamics. Thought of the Day 148.0

Catastrophe theory has been developed as a deterministic theory for systems that may respond to continuous changes in control variables by a discontinuous change from one equilibrium state to another. A key idea is that system under study is driven towards an equilibrium state. The behavior of the dynamical systems under study is completely determined by a so-called potential function, which depends on behavioral and control variables. The behavioral, or state variable describes the state of the system, while control variables determine the behavior of the system. The dynamics under catastrophe models can become extremely complex, and according to the classification theory of Thom, there are seven different families based on the number of control and dependent variables.

Let us suppose that the process y_t evolves over t = 1,…, T as

dy_t = -dV(y_t; α, β)dt/dy_t —– (1)

where V (y_t; α, β) is the potential function describing the dynamics of the state variable y_t controlled by parameters α and β determining the system. When the right-hand side of (1) equals zero, −dV (y_t; α, β)/dy_t = 0, the system is in equilibrium. If the system is at a non-equilibrium point, it will move back to its equilibrium where the potential function takes the minimum values with respect to y_t. While the concept of potential function is very general, i.e. it can be quadratic yielding equilibrium of a simple flat response surface, one of the most applied potential functions in behavioral sciences, a cusp potential function is defined as

−V(y_t; α, β) = −1/4y_t⁴ + 1/2βy_t² + αy_t —– (2)

with equilibria at

-dV(y_t; α, β)dt/dy_t = -y_t³ + βy_t + α —– (3)

being equal to zero. The two dimensions of the control space, α and β, further depend on realizations from i = 1 . . . , n independent variables x_i,t. Thus it is convenient to think about them as functions

α_x = α₀ +α₁x_1,t +…+ α_nx_n,t —– (4)

β_x = β₀ + β₁x_1,t +…+ β_nx_n,t —– (5)

The control functions α_x and β_x are called normal and splitting factors, or asymmetry and bifurcation factors, respectively and they determine the predicted values of y_t given x_i,t. This means that for each combination of values of independent variables there might be up to three predicted values of the state variable given by roots of

-dV(y_t; α_x, β_x)dt/dy_t = -y_t³ + βy_t + α = 0 —– (6)

This equation has one solution if

δ_x = 1/4α_x² − 1/27β_x³ —– (7)

is greater than zero, δ_x > 0 and three solutions if δ_x < 0. This construction can serve as a statistic for bimodality, one of the catastrophe flags. The set of values for which the discriminant is equal to zero, δ_x = 0 is the bifurcation set which determines the set of singularity points in the system. In the case of three roots, the central root is called an “anti-prediction” and is least probable state of the system. Inside the bifurcation, when δ_x < 0, the surface predicts two possible values of the state variable which means that the state variable is bimodal in this case.

Most of the systems in behavioral sciences are subject to noise stemming from measurement errors or inherent stochastic nature of the system under study. Thus for a real-world applications, it is necessary to add non-deterministic behavior into the system. As catastrophe theory has primarily been developed to describe deterministic systems, it may not be obvious how to extend the theory to stochastic systems. An important bridge has been provided by the Itô stochastic differential equations to establish a link between the potential function of a deterministic catastrophe system and the stationary probability density function of the corresponding stochastic process. Adding a stochastic Gaussian white noise term to the system

dy_t = -dV(y_t; α_x, β_x)dt/dy_t + σ_ytdW_t —– (8)

where -dV(y_t; α_x, β_x)dt/dy_t is the deterministic term, or drift function representing the equilibrium state of the cusp catastrophe, σ_yt is the diffusion function and W_t is a Wiener process. When the diffusion function is constant, σ_yt = σ, and the current measurement scale is not to be nonlinearly transformed, the stochastic potential function is proportional to deterministic potential function and probability distribution function corresponding to the solution from (8) converges to a probability distribution function of a limiting stationary stochastic process as dynamics of y_t are assumed to be much faster than changes in x_i,t. The probability density that describes the distribution of the system’s states at any t is then

f_s(y|x) = ψ exp((−1/4)y⁴ + (β_x/2)y² + α_xy)/σ —– (9)

The constant ψ normalizes the probability distribution function so its integral over the entire range equals to one. As bifurcation factor β_x changes from negative to positive, the f_s(y|x) changes its shape from unimodal to bimodal. On the other hand, α_x causes asymmetry in f_s(y|x).

Catastrophe Revisited. Note Quote.

Transversality and structural stability are the topics of Thom’s important transversality and isotopy theorems; the first one says that transversality is a stable property, the second one that transverse crossings are themselves stable. These theorems can be extended to families of functions: If f: Rⁿ x R^r–>R is equivalent to any family f + p: Rⁿ x R^r–>R, where p is a sufficiently small family Rⁿ x R^r–> R, then f is structurally stable. There may be individual functions with degenerate critical points in such a family, but these exceptions from the rule are in a sense “checked” by the other family members. Such families can be obtained e.g. by parametrizing the original function with one or several extra variables. Thom’s classification theorem, comes in at this level.

So, in a given state function, catastrophe theory separates between two kinds of functions: one “Morse” piece, containing the nondegenerate critical points, and one piece, where the (parametrized) family contains at least one degenerate critical point. The second piece has two sets of variables; the state variables (denoted x, y…) responsible for the critical points, and the control variables or parameters (denoted a, b, c…), capable of stabilizing a degenerate critical point or steering away from it to nondegenerate members of the same function family. Each control parameter can control the degenerate point only in one direction; the more degenerate a singular point is (the number of independent directions equal to the corank), the more control parameters will be needed. The number of control parameters needed to stabilize a degenerate point (“the universal unfolding of the singularity” with the same dimension as the number of control parameters) is called the codimension of the system. With these considerations in mind, keeping close to surveyable, four-dimensional spacetime, Thom defined an “elementary catastrophe theory” with seven elementary catastrophes, where the number of state variables is one or two: x, y, and the number of control parameters, equal to the codimension, at most four: a, b, c, d. (With five parameters there will be eleven catastrophes). The tool used here is the above mentioned classification theorem, which lists all possible organizing centers (quadratic, cubic forms etc.) in which there are stable unfoldings (by means of control parameters acting on state variables). 

Two elementary catastrophes: fold and cusp

1. In the first place the classification theorem points out the simple potential function y = x³ as a candidate for study. It has a degenerate critical point at {0, 0} and is always declining (with minus sign), needing an addition from the outside in order to grow locally. All possible perturbations of this function are essentially of type x³ + x or type x³ – x (more generally x³ + ax); which means that the critical point (y, x = 0) is of codimension one. Fig. below shows the universal unfolding of the organizing centre y = x³, the fold:

This catastrophe, says Thom, can be interpreted as “the start of something” or “the end of something”, in other words as a “limit”, temporal or spatial. In this particular case (and only in this case) the complete graph in internal (x) and external space (y) with the control parameter a running from positive to negative values can be shown in a three-dimensional graph (Fig. below); it is evident why this catastrophe is called “fold”:

One point should be stressed already at this stage, it will be repeated again later on. In “Topological models…”, Thom remarks on the “informational content” of the degenerate critical point: 

This notion of universal unfolding plays a central role in our biological models. To some extent, it replaces the vague and misused term of ‘information’, so frequently found in the writings of geneticists and molecular biologists. The ‘information’ symbolized by the degenerate singularity V(x) is ‘transcribed’, ‘decoded’ or ‘unfolded’ into the morphology appearing in the space of external variables which span the universal unfolding family of the singularity V(x). 

2. Next let us as organizing centre pick the second potential function pointed out by the classification theorem: y = x⁴. It has a unique minimum (0, 0), but it is not generic , since nearby potentials can be of a different qualitative type, e.g. they can have two minima. But the two-parameter function x⁴ + ax² + bx is generic and contains all possible unfoldings of y = x⁴. The graph of this function, with four variables: y, x, a, b, can not be shown, the display must be restricted to three dimensions. The obvious way out is to study the derivative f'(x) = 4x³ + 2ax + b for y = 0 and in the proximity of x = 0. It turns out, that this derivative has the qualities of the fold, shown in the Fig. below; the catastrophes are like Chinese boxes, one contained within the next of the hierarchy. 

Finally we look for the position of the degenerate critical points projected on (a,b)-space, this projection has given the catastrophe its name: the “cusp” (Fig. below). (An arrowhead or a spearhead is a cusp). The edges of the cusp, the bifurcation set, point out the catastrophe zone, above the area between these limits the potential has two Morse minima and one maximum, outside the cusp limits there is one single Morse minimum. With the given configuration (the parameter a perpendicular to the axis of the cusp) a is called the normal factor – since x will increase continuously with a if b < 0, while b is called the splitting factor because the fold surface is split into two sheets if b > 0. If the control axes are instead located on either side of the cusp (A = b + a and B = b – a) A and B are called conflicting factors; A tries to push the result to the upper sheet (attractor), B to the lower sheet of the fold. (Here is an “inlet” for truly external factors; it is well-known how e.g. shadow or excessive light affects the morphogenetic process of plants. 

Thom states: the cusp is a pleat, a fault, its temporal interpretation is “to separate, to unite, to capture, to generate, to change”. Countless attempts to model bimodal distributions are connected with the cusp, it is the most used (and maybe the most misused) of the elementary catastrophes. 

Zeeman has treated stock exchange and currency behaviour from one and the same model, namely what he terms the cusp catastrophe with a slow feedback. Here the rate of change of indexes (or currencies) is considered as dependent variable, while different buying patterns (“fundamental” /N in fig. below and “chartist” /S in fig. below) serve as normal and splitting parameters. Zeeman argues: the response time of X to changes in N and S is much faster than the feedback of X on N and S, so the flow lines will be almost vertical everywhere. If we fix N and S, X will seek a stable equilibrium position, an attractor surface (or: two attractor surfaces, separated by a repellor sheet and “connected” by catastrophes; one sheet is inflation/bull market, one sheet deflation/bear market, one catastrophe collapse of market or currency. Note that the second catastrophe is absent with the given flow direction. This is important, it tells us that the whole pattern can be manipulated, “adapted” by means of feedbacks/flow directions). Close to the attractor surface, N and S become increasingly important; there will be two horizontal components, representing the (slow) feedback effects of N and S on X. The whole sheet (the fold) is given by the equation X³ – (S – S_o)X – N = 0, the edge of the cusp by 3X² + S_o = S, which gives the equation 4(S – S_o)³ = 27 N² for the bifurcation curve. 

Figure: “Cusp with a slow feedback”, according to Zeeman (1977). X, the state variable, measures the rate of change of an index, N = normal parameter, S = splitting parameter, the catastrophic behaviour begins at S_o. On the back part of the upper sheet, N is assumed constant and dS/dt positive, on the fore part dN/dT is assumed to be negative and dS/dt positive; this gives the flow direction of the feedback. On the fore part of the lower sheet both dN/dt and dS/dt are assumed to be negative, on the back part dN/dt is assumed to be positive and dS/dt still negative, this gives the flow direction of feedback on this sheet. The cusp projection on the {N,S}-plane is shaded grey, the visible part of the repellor sheet black. (The reductionist character of these models must always be kept in mind; here two obvious key parameters are considered, while others of a weaker or more ephemeral kind – e.g. interest levels – are ignored.)