Cusp catastrophe

The examined physical system always satisfies stipulated additional conditions (an additional stationary point or suitable symmetry) and then the described catastrophe may appear in such systems in a structurally stable way. If the examined system does not fulfil these conditions, the description of the system is inadequate and a possibility of the description of a catastrophe of higher codimension should be considered. For example, such a structurally stable extension of the pitchfork bifurcation is the cusp catastrophe described in Section 5.5.3.2. 

We shall consider dynamical systems in which the vector field f depends on two control parameters 

It can be shown that there exist five nonequivalent sensitive states of codimension two. Their standard forms will be discussed below. The catastrophes have common names we shall also introduce nomenclature (reflecting the type of a sensitive point). 

The following standard form corresponds to this catastrophe  

This is a generalization of the catastrophe of a saddle node type or, more specifically, of a pitchfork catastrophe. The sensitive state corresponds to one zero eigenvalue. The catastrophe is easy to describe, being of a gradient 

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A topological study ofthe electron transfer in Li + CI2 system and of the three-electron hond created throngh this transfer has been achieved, based on the topological concepts of Bonding Evolntion Theory. Our results suggest that the dual cusp catastrophe characterizes the diahatic surface crossings which are subjacent in the classical adiabatic analysis ofthe overall reaction path. 

For all molecules the cis conformation is the absolute minimum, whereas the trans conformation is either a secondary minimum (oxalyl fluoride) or a maximum (oxalyl chloride and oxalyl bromide). Oxalyl bromide presents a shallow minimum corresponding to a gauche conformer. Substituting F by Cl yields a cusp catastrophe which changes the two maxima at tt/2 and the minimum at 0 into a maximum at 0. The substitution of Cl by Br is responsible for a dual-fold catastrophe in which two wandering points near 27t/3 give rise to a new minimum (gauche conformation) and a new maximum. 

Another example is provided by pressure-induced phase transitions in which a cusp catastrophe transforms a double-well of the potential energy curve into a single-well and vice versa. In a wide pressure range, between 10 and 100 GPa, stishovite is the stable modification of silica. At about 100 GPa, a phase transition from PA2/mnm. (rutile) to Pmnm(CaCl2) occurs which corresponds to the twinning of the initial tetragonal cell... 

There is one last way to plot the results, which may appeal to you if you like to picture things in three dimensions. This method of presentation contains all of the others as cross sections or projections. If we plot the fixed points jt above the (r,/z) plane, we get the cusp catastrophe surface shown in Figure 3.6.5. The surface folds over on itself in certain places. The projection of these folds onto the (r,h) plane yields the bifurcation curves shown in Figure 3.6.2. A cross section at fixed h yields Figure 3.6.3, and a cross section at fixed r yields Figure 3.6.4. 

The stability diagram is very similar to Figure 3.6.2. It too can be regarded as the projection of a cusp catastrophe surface, as schematically illustrated in Figure 3.7.6. You are hereby challenged to graph the surface accurately ... 

Resonance curves and cusp catastrophe) In this exercise you are asked to determine how the equilibrium amplitude of the driven oscillations depends on the other parameters. 

Fig. 2 The geometry of the cusp catastrophe. The lower part of the figure is the control surface in u and V and its solution is drawn as the manifold (complex surface) above the control plane. The movement across the control surface (from left to right) is projected onto the manifold...

Fig. 3 A The face - chalice ambiguous figure drawn in B as a series of altered figures that map onto a cusp catastrophe. The potential energy of each ( , v) point as projected onto the manifold...

Fig. 19. Catastrophe surface M3, singularity set I3 and bifurcation set B3 of the cusp catastrophe (/t3).

Let us now return to the van der Waals equation (3.14). The equation is non-linear in V. The presence of third power of V and the requirement of vanishing of the first and second derivatives suggest that in some circumstances a cusp catastrophe (/43) may appear in the system. Hence, a change of variables should be performed in equation (3.14) to reduce, if possible, the equation to the form resulting from a structurally stable potential function of the cusp catastrophe V= x4 + ax2 + bx. 

Fig. 39. The liquid-vapour phase transition on the surface of the cusp catastrophe (A3).

In summary, the van der Waals equation may be derived from the condition of a minimum of a potential function corresponding to the cusp catastrophe (A3) in which the state variable x is the fluid (liquid or vapour) density, x = VJV— 1, while the parameters a, b are functions of the temperature T and pressure p (see equation (3.21)). 

The first important conclusion which can be drawn from the above construction is a structural stability of phenomena described by the van der Waals equation. The conclusion is consistent with an observation that the reduced van der Waals equation (3.18) describes the critical transition for a number of gases. The liquid-vapour phase transition may thus be describred on the surface of the cusp catastrophe, see Fig. 39, in which a projection on the control parameters plane and the isotherm T = const are also shown. 

Fig. 40. Section of the cusp catastrophe surface for the liquid-vapour system.

For example, the diffraction integral (3.25) containing the potential function of a cusp catastrophe (zf3), F(x c) = x4 + ax2 + bx, describes ligth scattering on a two-dimensional diffraction grating, see Fig. 48. A function defined by such an integral is called the Pearcey function. 

Fig. 48. Diffraction cusp catastrophe M3) (a) diffraction grating (b) oscillation integral Ai.

The slow surface (3.78) corresponds to the catastrophe surface M3 of a cusp catastrophe (A3), see Fig. 54 in which the values of the control parameter which cannot be exceeded in a living organism are marked on the b-axis (recall that b is related to a biochemical state of the heart muscle). The four fundamental states of dynamics of the heartbeat may now be described in terms of the value of parameter a (tension of the cardiac muscle). Since the parameter a is constant for a given state, the states of the system lie on the sections a = const of the cusp catastrophe surface. The four states are conveniently represented in the (x, b) plane, Fig. 55, the parameter xn having to have such a value that the linearized system (3.76) does not have a stable stationary point. The four states shown in Fig. 55 have the following interpretation ... 

Zeeman has given a system of differential equations, describing time evolution of the variables x, a, b, for which a slow surface is the cusp catastrophe surface (3.78). The system of Zeeman equations has an attracting and stable stationary point E. The process of nerve impulse transmission represented on the catastrophe surface M3 is shown in Fig. 57. 

The sensitive state corresponds to zero value of A. From this condition as well as from equation (6.40) for A and from (6.37) we obtain the condition for an occurrence of the sensitive state of the cusp catastrophe ... 

Wang L.G Miao X.X 2006. Study of Mechanism of Destabilization of the Mine PillarBased on a Cusp Catastrophic Model. Journal of Mining Safety Engineering, 23(2) 137-139. 

The crossing of the vertical branches of the inversion line results in a completely different phenomenon called catastrophic inversion (172,197) because it can be modeled as a cusp catastrophe transition as pointed out by Dickinson (201) and fiirther discussed by others (195—203). 

E Dickinson. Interpretation of emulsion phase inver sion as a cusp catastrophe. J Colloid Interface Sci84 284,1981. 

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