Catastrophe manifold

To conclude our considerations of functions of one variable, we shall examine the properties of several potential functions given in Table 2.2. In analysing the properties of these functions and the catastrophes described by them, the catastrophe manifold M defined by equation (2.1a) will be helpful. In the case of a function of one state variable it takes the form... 

It is useful to distinguish in the catastrophe manifold M a subset I on which the function V has degenerate critical points since, as we shall see later, at these points a catastrophe takes place. The set S is thus given by the following equation ... 

We shall begin the investigation of Thom potential functions with the family of functions V(x a) = x3 + ax corresponding to the fold catastrophe or A2, see Table 2.2. The catastrophe manifold corresponding to this potential function is obtained from equation (2.13), c = (a),... 

We shall now proceed to the examination of catastrophes modelled by Thom potential functions of two state variables. This will be done using the notions of the catastrophe manifold M, singularity set I and bifurcation set... 

B, employed in Section 2.2 to investigate potential functions of one variable. In the case of functions of two variables the catastrophe manifold M is given by the equation (cf. equation (2.1a))... 

Classification of catastrophes will be preceded by the centre manifold theorem which is a counterpart to the splitting lemma in elementary catastrophe theory. It will turn out that in the catastrophe theory of dynamical systems such notions of elementary catastrophe theory as the catastrophe manifold, bifurcation set, sensitive state, splitting lemma, codimension, universal unfolding and structural stability are retained. 

Let us recall that in elementary catastrophe theory critical points of potential functions are examined. A potential function can have noncritical points, nondegenerate critical points and degenerate critical points. To degenerate critical points correspond sensitive states lying in the state variable and control parameter space in the catastrophe manifold M their... 

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...

The Tikhonov theorem has an important generalization, called the centre manifold theorem, which will be discussed in Sections 5.4.5-5.4.7. In classification of catastrophes occurring in dynamical systems and represented by systems of autonomous equations, the centre manifold theorem plays the role of the splitting lemma (see Section 2.3.4). 

When conditions (1)—(3) are not fulfilled, the Grobman-Hartman theorem is not valid. As will be shown later, then we have to deal with the sensitive state of a dynamical system (this corresponds to a degenerate critical point in elementary catastrophe theory). A generalization of the Grobman-Hartman theorem, the centre manifold theorem which may be regarded as a counterpart to the splitting lemma of elementary catastrophe theory, has been found to be very convenient in that case. 

A counterpart to the splitting lemma of elementary catastrophe theory is the centre manifold theorem (also called a neutral manifold), generalizing the Grobman-Hartman theorem to the case of occurrence of sensitive states (5.28). The centre manifold theorem allows us to establish an equivalence (nonequivalence) of two autonomous systems. In this sense it is also a generalization to the case of autonomous systems of the equivalence relationship introduced in Chapter 2 for potential functions. 

The roots Ali2(1,2) in this model are identical with the eigenvalues, for a = D = 1, in the Fisher-Kolmogorov model. As the second pair of the roots does not lead to a generation of the sensitive states, the nature of a catastrophe for the Oregonator with diffusion is the same as for the Fisher-Kolmogorov model. In other words, the waves with the velocities v centre manifold theorem the sensitive state is associated only with the zlt z2 variables. 

However, in multicell pile configuration reserve batteries, there is a considerable chance for intercell short-circuiting in the common electrolyte manifold. This intercell shortcircuiting not only dissipates the capacity of the cells, but it can also allow for dendritic growth, which can lead to electronic short-circuiting of the cells with catastrophic results. Experience has shown that such dendritic growth can be minimized or eliminated if all interior metallic surfaces of the battery have a nonconductive (usually Teflon-based) coating. 

These examples have identified two types of catastrophe points, a distinction that arises as a corollary of a theorem on structural stability. This theorem, when used to describe structural changes in a molecular system, states that the structure associated with a particular geometry X in nuclear configuration space is structurally stable if p r X) has a finite number of cps such that (i) each cp is nondegenerate (ii) the stable and unstable manifolds of any pair of cps intersect transver-sally. The immediate consequence of this theorem is that a structural instability can be established solely through either of two mechanisms in the bifurcation mechanism the charge distribution exhibits a degenerate cp, while the conflict mechanism is characterized by the nontransversal intersection of the stable and unstable manifolds of cps in p(r X). 

A totally diflFerent situation becomes possible in the case where the system does not have a global cross-section, and when is not a manifold. In this case (Sec. 12.4), the disappearance of the saddle-node periodic orbit may, under some additional conditions, give birth to another (unique and stable) periodic orbit. When this periodic orbit approaches the stability boundary, both its length and period increases to infinity. This phenomenon is called a hlue-sky catastrophe. Since no physical model is presently known for which this bifurcation occurs, we illustrate it by a number of natural examples. 

Fig. 12.4.1. (a) Illustrates the mechanism of a blueunstable manifold returns to the saddle-node from the node region so that the circles of its intersection with the cross-section S tighten with each subsequent iterate, (b) The return map along... 

Fig. 12.4.3. A phemenological scenario of development of the blue sky catastrophe when the saddle-node equilibrium O disappears, the unstable manifold of the saddle-node periodic orbit L has the desired configuration, as the one shown in Fig. 12.4.1(a).

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