Singularity theory with a distinguished parameter

Many models of physical systems can be reduced to a scalar equation of the form,

G(T,L,p)=0. (1)

The scalar equation contains a state variable (T), a distinguished parameter, (L), sometimes called the primary bifurcation parameter, and several secondary bifurcation parameters (p). The graph of T versus L for fixed p is called a steady-state diagram or a response curve. This graph contains bifurcation points at which the number of solutions to the equation G=0 changes. We call a figure showing how the value of L at which a particular type of bifurcation occurs varies as one of the secondary parameters is changed an unfolding diagram.

The parameter space p consists of regions with different kinds of steady-state diagrams. The fundamental task in the study of equation (1) is to identify the types of steady-state diagrams that occur and their location in parameter space. We refer to a figure showing where the different types of steady-state diagrams occur in the parameter space p as a bifurcation diagram. There is no consistency of notation in the literature. What we call a steady-state diagram has been referred to as a bifurcation diagram by some authors. Similarly what we call an unfolding diagram has also been called a bifurcation diagram.

Singularity theory with a distinguished parameter enables degenerate points (singular points) to be located. At these points the boundaries of some of the various regions coalesce, so that several types of steady-state diagrams exist in a neighbourhood of the singularity. The singular points are characterised by the vanishing of several partial derivatives of G with respect to T and L. The real strength of singularity theory is that it is able to predict all the steady-state diagrams existing locally to a singular point [Golubitsky and Schaeffer, 1985a].

However, in practice it is important to determine the global existence of the differing types of steady-state diagrams. This is required because parameter values are unlikely to correspond to those in the vicinity of a singular point. In a landmark paper Golubitsky and Schaeffer proved that that a qualitative change in a steady-state diagram occurs if and only if the bifurcation parameters cross the boundaries of one of three typyes of curves: the cusp, isola, and double limit point curves [Golubitsky and Schaeffer, 1979]. Thus the bifurcation diagam is constructed by determining the locus of these three curves in physical parameter space. This method divides parameter space into regions, each corresponding to a different steady-state diagram of the problem G=0. This methodology was first systematically applied to investigate multiplicity features of open chemically reacting systems by Balakotaiah and Luss [1981; 1982a; 1982b; 1984]. It is now a standard approach in studying such problems.

An attractive feature of this method is that the location of the boundaries is determined directly in the physical parameter space, whereas in singularity theory proper, the boundaries are defined in terms of the unfolding parameters appearing in the normal form of the singularity; in practice it is very difficult to relate the unfolding parameters to the physical ones.

The cusp variety is the set of p satisfying the equations

G= GT = GTT = 0. (2)

(A set of non-degeneracy conditions must also be satisfied. These are given in [ Golubitsky & Schaeffer, 1985b].) Typically when the cusp curve is crossed a hysteresis loop appears or disappears in the steady-state diagram as two limit points appear or disappear.

The isola variety is the set of p satisfying the equations

G = GT = GL = 0. (3)

When the isola variety is crossed two limit points appear or disappear. Two types of behaviour may occur. In the first, the steady-state diagrams separate locally into two isolated curves (transcritical singularity). In the second, an isolated branch of connected solutions appears or disappears (isola singularity).

The double-limit variety is the set of p satisfying the four equations

G(T1, L, p) = G(T2, L, p) = 0 T1 \neq T2 (4)
GT(T1, L, p) = GT(T2, L, p) = 0 (5)

At a double-limit point two limit points, at T1 and T2, occur at the same value of the distinguished parameter. As the the double limit point variety is crossed the the relative position of these limit points changes.

A heuristic description of this theory with a focus on applications to chemical systems has been written by Balakotaiah [1986].


  1. V. Balakotaiah. Steady-state multiplicity features of open chemically reacting systems. In G.S.S. Ludford, editor, Reacting Flows: Combustion and Chemical Reactors Part II, volume 24 of Lectures in Applied Mathematics, pages 129--161. American Mathematical Society, 1986.
  2. V. Balakotaiah and D. Luss. Analysis of the multiplicity patterns of a cstr. Chemical Engineering Communications, 13:111--132, 1981.
  3. V. Balakotaiah and D. Luss. 1982a. Analysis of the multiplicity patterns of a cstr. Chemical Engineering Communications, 19:185--189, 1982.
  4. V. Balakotaiah and D. Luss. 1982b. Structure of the steady-state solutions of lumped-parameter chemically reacting systems. Chemical Engineering Science, 37(11):1611--1623, 1982.
  5. V. Balakotaiah and D. Luss
  6. . Global analysis of the multiplicity features of multi-reaction lumped-parameter systems. Chemical Engineering Science, 39(5):865--881, 1984.
  7. M. Golubitsky and D. Schaeffer. A theory for imperfect bifurcation theory via singularity theory. Communications on Pure and Applied Mathematics, 32:21--98, 1979.
  8. M. Golubitsky and D. Schaeffer. 1985a. Singularities and Groups in Bifurcation Theory, volume 1. Springer, New York, first edition, 1985.
  9. M. Golubitsky and D. Schaeffer. 1985b. The classification theorem. In Singularities and Groups in Bifurcation Theory, volume 1, chapter IV.2, pages 196--202. Springer-Verlag, first edition, 1985.

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