Catastrophe Theory in Mathematics Explained Clearly

• Catastrophe theory is a branch of bifurcation theory in the study of dynamical structures in mathematics. It is also a part of the overall field of geometry's singularity theory.

• In a dynamic system, catastrophe refers to the loss of equilibrium. This theory's main approach is to divide dynamic variables into two categories: slow and fast. Then, as a result of the dynamics of slow variables, the stability features of fast variables can slowly shift.

• Catastrophe theory began in the 1960s with the work of French mathematician Rene Thom and gained popularity in the 1970s, thanks to Christopher Zeeman's efforts. 

• It considers the unique situation in which the long-run stable equilibrium can be described as the minimum of a smooth, well-defined potential function known as the Lyapunov function.

• The change in the shape of an arched bridge as the load on it is steadily increased is a clear example of the behaviour studied by catastrophe theory. The bridge deforms in a reasonably uniform fashion until the load exceeds a critical value, at which point the bridge's shape shifts abruptly and the bridge collapses. While the word catastrophe conjures up images of a catastrophic incident, many of the state transitions so-called are not. 

• The techniques of catastrophe theory have been used to investigate the reflection or refraction of light by or through flowing water, as well as a variety of other optical phenomena.

• Small changes in a nonlinear system's parameters may cause equilibria to appear or vanish, or change from attracting to repelling and vice versa, resulting in significant and abrupt changes in the system's behaviour. 

• However, catastrophe theory shows that such bifurcation points appear to occur as part of well-defined qualitative geometrical structures when viewed in larger parameter space.

• Catastrophe theory examines degenerate critical points of the potential function, where not only the first derivative but one or higher derivatives are also zero. These are known as the germs of the catastrophe geometries. 

• By extending the potential function as a Taylor series in small parameter perturbations, the degeneracy of these critical points can be shown.

• When degenerate points are not only coincidental but structurally stable, they serve as organising centres for lower-degeneracy geometric structures, with important features in the parameter space surrounding them.

• There are only seven generic structures for these bifurcation geometries, with corresponding standard forms into which the Taylor series around the catastrophe germs can be transformed by diffeomorphism if the potential function depends on two or fewer active variables and four or fewer active parameters. These seven basic forms are now presented, along with Thom's names for them.

Catastrophe Theory For Potential functions of One Active Variable

Dynamical structures that explain the evolution of a state variable x over time t are studied in catastrophe theory:

\[\overline{x}=\frac{dx}{dt}=-\frac{dV(u,x)}{dx}\]

V is the potential function in the above equation, and u is usually a vector or a scalar that parametrizes the potential function. u is also known as the control variable since its value will change over time.

Fold Catastrophe

The potential function for fold catastrophe is 

V = x3 + ax

The potential function V has two extrema when a < 0 is zero: one stable and one unstable. The system will obey the stable minimum point if the parameter is gradually increased. At a = 0, however, the stable and unstable extrema collide and destroy each other. This is where the path splits. There is no longer a stable solution when a > 0. When a physical system is followed through a fold bifurcation, it is discovered that as an approaches 0, the consistency of a < 0 solutions is abruptly lost, and the system abruptly switches to a new, very different behaviour. The tipping point refers to the bifurcation value of parameter a.

The below image gives the stable and unstable pair of extrema to disappear at a fold bifurcation.

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Cusp Catastrophe

The potential function for cusp catastrophe is

V = x4 + ax2 + bx

When investigating what happens to a fold bifurcation when a second parameter, b, is added to the control space, the cusp geometry is formed. When the parameters are changed, a curve of points in (a, b) space appears, indicating that equilibrium has been lost and the stable solution will immediately leap to a different outcome. The bifurcation curve in a cusp geometry loop back on itself, resulting in a second branch where this alternative solution loses stability and returns to the original solution set. 

Hysteresis loops can be observed by increasing ‘b’ and then decreasing it repeatedly, as the mechanism alternately follows one solution, jumps to the other, follows the other back, and then jumps back to the first. This, however, is only possible in the parameter space a < 0. The hysteresis loops get smaller and smaller as an is increased until they vanish entirely above a = 0 (the cusp catastrophe), and there is only one stable solution.

Consider what happens if ‘b’ is kept constant when ‘a’ is changed. As ‘a’ is reduced in the symmetrical case b = 0, one stable solution splits into two stable solutions and one unstable solution as the physical structure moves through the cusp point, resulting in a pitchfork bifurcation (0,0). There is no abrupt shift in the physical solution being followed away from the cusp point: all that happens when going through the curve of fold bifurcations is that an alternative second solution becomes visible.

The relationship between local and external stresses can be used to assess the catastrophic failure of a complex system with simultaneous redundancy. The cusp catastrophe behaviour is close to the structural fracture mechanics model. A complex system's reserve potential is predicted by the model. Other applications include outer sphere electron transfer, which is common in chemical and biological systems, and real estate price forecasting.

The most critical functional implications of catastrophe theory are fold bifurcations and cusp geometry. They are trends that show up in physics, engineering, and mathematical modelling time and time again. They trigger intense gravitational lensing events, which give astronomers one of the methods for detecting black holes and dark matter in the universe: gravitational lensing, which produces several images of distant quasars.

In the below diagram curves of the brown and red colour of x satisfying dV/dx = 0 for parameters (a, b) drawn for parameter b continuously varied, for many values of parameter a. There is only one extremising value of x for each point (a, b) in parameter space outside the cusp locus of bifurcations (blue). Two different values of x give local minima of V(x) for each (a, b) within the cusp, separated by a value of x giving a local limit.

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Swallowtail Catastrophe

The potential function for swallowtail catastrophe is

V = x5 + ax3 + bx2 + cx

There are three dimensions to the control parameter space. In parameter space, the bifurcation set is made up of threefold bifurcation surfaces that meet in two lines of cusp bifurcations, which meet at a single swallowtail bifurcation point. One minimum and one maximum of the potential function vanish when the parameters pass through the surface of fold bifurcations. Two minima and one maximum are replaced by one minimum at the cusp bifurcations, and the fold bifurcations vanish beyond them. Two minima and two maxima all converge at a single x value at the swallowtail stage.

Depending on the values of b and c, there is either one maximum-minimum pair or none outside the swallowtail for a > 0. At the swallowtail point, two of the surfaces of fold bifurcations, as well as the two lines of cusp bifurcations where they intersect for a < 0, vanish, leaving only a single surface of fold bifurcations. 

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Butterfly Catastrophe

The potential function for butterfly catastrophe is

V = x5 + ax3 + bx2 + cx

The potential function can have three, two, or one local minima, separated by fold bifurcation loci, depending on the parameter values. When a > 0, the different 3-surfaces of fold bifurcations, the 2-surfaces of cusp bifurcations, and the lines of swallowtail bifurcations all meet and disappear at the butterfly stage, leaving only a single cusp structure.

Catastrophe Theory For Potential Functions of Two Active Variables

Corank 2 catastrophes include umbilic catastrophes. They are closely related to the geometry of nearly spherical surfaces and can be seen in optics in the focal surfaces formed by light reflecting off a surface in three dimensions: umbilical point. The hyperbolic umbilic catastrophe, according to Thom, modelled the splitting of a wave, while the elliptical umbilic catastrophe modelled the formation of hair-like structures.

Hyperbolic Umbilic Catastrophe

Three control factors and two behaviour axes can all lead to hyperbolic umbilic catastrophe. The universal unfolding of the function germ V = x3+y3+axy+bx+cy is the hyperbolic umbilic. Since the codimension of f is 3, the universal unfolding F of f has three unfolding parameters.

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Elliptic Umbilic Catastrophe

Three control factors and two behaviour axes can all lead to an elliptic umbilic catastrophe. The elliptical umbilic is a three-dimensional catastrophe with the f equation

\[V=\frac{x^{3}}{3}-xy^{2}+a(x^{2}+y^{2})+bx+cy\]

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Parabolic Umbilic Catastrophe

A catastrophe that can happen due to four control factors and two behavioural axes will give parabolic umbilic catastrophe. The unfolding V = x2y + y4 + ax2 + by2 + cx + dy gives the parabolic umbilic catastrophe.

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Applications of Catastrophe Theory

• The catastrophe theory was used in a lake environment to see whether introducing more predatory fish would dramatically change water conditions.

• To investigate soil fertility self-development in the Zhuxi watershed of Changting County in China's red soil hilly area, a new framework was developed using catastrophe theory, laboratory experiment, fieldwork, and 3S (geographic information system, global positioning system, and remote sensing).

• The effect of commercial grazing and soil water availability on the soil stability index (SSI) was also assessed using catastrophe theory, suggesting that the landscape became more vulnerable to erosion events as a result of multiple droughts and grazing.

• For assessing land ecological protection in Shanghai, China, a catastrophe model was set up that combined multiple assessment indices of land eco-security based on inherent contradictions and relative importance of indices without calculating weights.

Conclusion

A set of methods used to analyse and describe how a system can experience sudden significant changes in behaviour when one or more of the variables that govern it are modified continuously is known as catastrophe theory. Since the variables and resulting behaviours are usefully represented as curves or surfaces, catastrophe theory is commonly considered a branch of geometry, and the systematic formulation of the theory is attributed primarily to the French topologist Rene Thom.