Catastrophe gradient system

Numerous rigorous results have been obtained for gradient systems. One natural method of investigation of gradient systems is elementary catastrophe theory the field of catastrophe theory dealing with an examination of gradient systems. In the case of the gradient system of equations (1.8), properties of a stationary state, that is the state invariant with time, may be readily studied... 

As explained in Section 1.2, the simplest field of applications of catastrophe theory are gradient systems (1.8). In the case of gradient systems, static catastrophes obeying the condition (1.9) can be studied by the methods of elementary catastrophe theory. Let us recall that a fundamental task of elementary catastrophe theory is the determination how properties of a set of critical points of potential function K(x c) depend on control parameters c. In other words, the problem involves an examination in what way properties of a set of critical points (denoted as M and called the... 

It follows from the examples examined above that in gradient systems catastrophes may occur only in a case when the system is described by a potential function having a degenerate critical point, for in this case the set I delimitating in M the functions of a various differential type is not an empty set. On exceeding by the system the set Z on the catastrophe surface M, the change in a local type of a potential function V(x c), i.e. a catastrophe, takes place at a continuous change of control parameters. 

From the standpoint of elementary catastrophe theory, the functions having degenerate critical points are most interesting. As follows from Section 2.2, in gradient systems catastrophes may happen only in a case when the system is described by a potential function having a degenerate critical point. 

From the Korzukhin theorem follows an important conclusion. Any dynamical systems of the form (4.58) may be regarded as those corresponding to slow dynamics of a standard kinetic system. In other words, the behaviour of dynamical systems can be modelled using chemical reactions. In particular, any of the gradient systems may be modelled in this way. As will be shown in Chapter 5, catastrophes occurring in complex dynamical systems are equivalent to catastrophes appearing in much simpler systems. The latter can be classified — these are so-called standard forms. The standard forms are of the form (4.58) and it follows from the Korzukhin theorem that they can be modelled by the standard equations of chemical kinetics (4.27), corresponding to a realistic mechanism of chemical reactions. 

Stationary states of a gradient system may be said to lie on the catastrophe surface M and degenerate stationary states to be placed on the singularity set I (see Chapter 2). 

The analysis of properties of gradient systems carried out in terms of elementary catastrophe theory (examination of critical points of the potential V) and of nongradient systems by means of singularity theory (examination of singularities of the vector function F) provides an incentive to investigate the relation between possible catastrophes and the eigenvalues of the stability matrix. 

The Hopf bifurcation is a dynamical catastrophe, since a stable stationary state bifurcates to a limit cycle hence, state variables change with time. Interestingly, the Hopf bifurcation may be visualized by elementary catastrophe theory despite the fact that the Hopf bifurcation may not appear in a gradient system and, furthermore, it is a dynamical catastrophe. 

This question has a certain mathematical interest, as gradient systems are relatively easy to deal with from the point of view of exotic phenomena (or, to put it in a more mathematical way, from the point of view of the qualitative theory of differential equations) in general (see, for example, Hirsch Smale, 1974) and from the point of view of catastrophe theory in particular (Thom, 1975, p. 55). 

For the one-component case the reaction-diffusion system can be considered as a gradient system, therefore the results of elementary catastrophe theory can be applied. Ebeling Malchow (1979) analysed bifurcations in (pseudo)-one-component systems by this technique. They showed that stable homogeneous stationary states are also stable in reaction-diffusion systems. 2. Ihe two-component model plays an important role in the analysis of... 

In the qualitative theory of differential equations, especially in the so called "catastrophe theory", it has turned out that differential equations of the gradient type are relatively easy to deal with /Thom, 1975/- Eurthermore - and this may prove more relevant - it has been proposed sometimes that gradient systems are only worth studying in thermodynamics. /Gyarmati, 1961a, b Edelen, 1973/. Easy treatment and physical relevance has also come up in connection with equations with other kinds of symmetries too in connection with Hamiltonian systems and systems having similar but different specialities. 

Catastrophic corrosion damage to buried structures may occasionally occur as a result of earth leakage faults on d.c. equipment or on traction systems using metallic posts to support overhead catenary wires. Care should be taken when making earth potential measurements because of the high potential gradients that may be present. 

A Semi-quantitative Approach Erosion and Deposition. Over the centuries the primary impact of human activity has been to deforest the surrounding countryside and increase the rate of erosion and deposition into rivers. This results primarily from the destruction of vegetation cover which stabilizes soil systems on gradient. The ecological impact of erosion has at present reached catastrophic proportions. The magnitude of continental erosion into rivers is illustrated in Figure 3. 

The Federal safety standards included in 49 C.F.R. 193 (1990) define four classes of impounding systems ranging from dikes constructed within 24 inches of the component served to remote impounding spaces (see 49 CFR 193.2153). The structural requirements specify performance reliability and integrity as a result of imposed loading caused by a full liquid head of spilled material, erosive spill action, thermal gradients, fire exposure, and catastrophic rupture of storage or transport vessels into or near the system (see 49 C.F.R. 193.2155). 

The gradient dynamical system and the catastrophe theories are two very useful and complementary mathematical tools for the study of the energetic and mechanisms of chemical reactions. We propose a classification of the potential functions and of the control space parameters. It emerges that the structural stability is a central concept for the understanding of chemical reactions and of chemical reactivity. 

In this chapter, we have tried to convince the reader of the usefulness of the dynamical system theory for chemical reactivity studies. Indeed, it is possible to predict which changes may be achieved when internal, external, or methodological parameters are varied from the shape of energy surface or from the topologies of local functions. The structural stability of the gradient vector fields of global and local functions describing chemical systems appears to be an important concept which has to be considered to understand the reactivity. Moreover, the application of the catastrophe theory to chemical reactions enables the description of the mechanisms [27-34,49-52],... 

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