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`= `G`_{T} = `G`_{TT} = 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` = `G`_{T} = `G`_{L} = 0.
(3)

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

G(T_{1}, L, p) | = |
G(T_{2}, L, p) | = | 0 |
T_{1} \neq T_{2} (4) |

G_{T}(T_{1}, L, p) | = |
G_{T}(T_{2}, L, p) | = | 0 | (5) |

At a double-limit point two limit points, at
`T`_{1} and
`T`_{2}, 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].

**References**

- 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. - V. Balakotaiah and D. Luss.
Analysis of the multiplicity patterns of a cstr.
`Chemical Engineering Communications`,**13**:111--132, 1981. - V. Balakotaiah and D. Luss. 1982a.
Analysis of the multiplicity patterns of a cstr.
`Chemical Engineering Communications`,**19**:185--189, 1982. - 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. - V. Balakotaiah and D. Luss . Global analysis of the multiplicity features of multi-reaction lumped-parameter systems.
- M. Golubitsky and D. Schaeffer.
A theory for imperfect bifurcation theory via singularity theory.
`Communications on Pure and Applied Mathematics`,**32**:21--98, 1979. - M. Golubitsky and D. Schaeffer. 1985a.
`Singularities and Groups in Bifurcation Theory`, volume 1. Springer, New York, first edition, 1985. - 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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Last Updated: 23rd July 2004.