Universal unfolding

The prediction of the period 2 saddle-node curve terminating at J remains until someone develops a universal unfolding for this co-dimension two bifurcation. 

The qualitative multiplicity features of a lumped-parameter system in which several reactions occur simultaneously can be determined in a systematic fashion by finding the organizing singularities of the steady-state equation and its universal unfolding. To illustrate the technique we determine the maximal number of solutions of a CSTR in which N parallel, first-order reactions with equal and high activation energies occur as well as the influence of changes in the residence time on the number and type of solutions. 

The qualitative features of the local bifurcation diagrams (0 vs. Da) of Eq. (13) in the neighborhoood of any singular point defined by (14) are same as those of its universal unfolding... 

The qualitative features of the steady-state Eq. (11) in the neighborhood of these singular points are the same as those of the universal unfolding... 

It is important to note that while the selection of the bifurcation variable does not affect the maximal number of steady-state solutions, it affects the number and type of bifurcation diagrams. For example, if we selected the coolant or feed temperature as the bifurcation variable then Eq. (18) would be the universal unfolding for both the adiabatic and the cooled case and no isolas would exist [1 2 ]. 

The death of the Universe unfolds in several acts. In our current era (106 to 1014 years after the Big Bang), the energy generated by stars drives astrophysical processes. Even though our Universe is 10 to 20 billion years old, the vast majority of stars have barely begun to twinkle (via fusion). Massive stars burn brightly but die fast. Stars as heavy as the sun live for about 10 billion years. Low-mass stars have not even begun to evolve. 

A function/of the essential variables, is said to be of finite co-dimension if a small perturbation of/gives rise to only a finite number of topological types. It is in this case that a map / of co-rank k can be embedded in a family of deformations (/, x R), parametrized by q variables which form the control space a R. By definition, q, the dimension of the control space, is the co-dimension of the singularity represented by /. The family of functions (/, / X R R) thus defined, describes the universal unfolding of /, in which the function/itself is the member associated with the origin of control space. In our applications the control space is a subset of the nuclear configuration space. 

Thom has classified these universal unfoldings according to their corank and to the dimension of the control space W called the codimension. Thom s classification is reported in Table 1. 

The problem of universal unfolding for the functions listed in Table 2.1 may be solved similarly to the problem of finding a form of the function g(x a, b) in (2.8). Universal unfoldings obtained in such a way are compiled in Table 2.2 (this is the first part of the Thom theorem, which pertains to potential functions in one state variable). 

Function Designation Point type Number of parameters in universal unfolding... 

In the formulation of the Thom theorem occur such important concepts of catastrophe theory as equivalence, determinacy, universal unfolding, codimension. Due to the vital role of these notions in catastrophe theory, we shall try to describe them in more detail. Let us add that the material presented in this Appendix is derived, to a large extent, from the papers of Mather. 

As follows from the above example the function x2y is not /e-determined. Hence, it is not structurally stable, since each perturbation containing terms missing from A(F) changes the nature of its degenerate critical point. This is a more serious defect of the potential function than, for example, that of the function from Example A.3, which is fc-determined and a universal unfolding containing a finite number of terms may be found for it. 

The above example shows why a codimension is equal to the number of parameters in a universal unfolding. It follows from the Thom theorem (see Table 2.5) that the universal unfolding of the function F(x, y) = x3 + y3 is, indeed, expressed by the above equation. 

Proceeding now as in Example A.5 we establish that in the set complementing A to M2 are the following monomials xly , where i,j = 1, 2, 3. Hence, a universal unfolding of this function must contain the term x3y3. 

Calculating the Jacobian ideal of the following functions in one and two variables having a degenerate critical point x3, + x4, x5, x6, x7, x2y — y3, x3 + y3, (x2y + y4), x2y + y5, (x3 + y4) according to the rules presented in Section A2.2 we conclude that the above functions are /c-deter-mined, because their codimension is finite. Furthermore, on the basis of the form of Jacobian ideal we establish that the respective universal unfoldings of these functions have a form consistent with the functions compiled in Tables 2.2, 2.5 (in Section A2.2 we computed the Jacobian ideal, among other functions, for the functions x, x2, x3, x4,..., and for x3 + y3, x2y + y4). 

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. 

A codimension of k of the vector field f(x) implies that at least k control parameters c, on which the family of vector fields f(x c) must depend, are required to describe all possible modifications of the phase portrait of (5.61) nearby its stationary state. Such the family of functions f(x c) is called, by analogy with elementary catastrophe theory, a universal unfolding of codimension k, of the field f(x). 

II) in the second step the form of a universal unfolding is found ... 

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