By David H. Sattinger

A dialogue of advancements within the box of bifurcation idea, with emphasis on symmetry breaking and its interrelationship with singularity conception. The notions of common recommendations, symmetry breaking, and unfolding of singularities are mentioned intimately. The booklet not just experiences contemporary mathematical advancements but additionally offers a stimulus for additional study within the box.

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**Sample text**

We denote the reflection x-^ -x by P. % = %. A general point ^ in jVj can be represented by ^ = zltyl + z2^2. A set of coordinates on the real subspace of Ker / is thus given by (z l5 z2, z\, z2). We denote time translations as before by cre. The operations cre, Ty and P then have the following matrix representation relative to this basis: The invariants of

We investigate the qualitative structure of the solution set of F = 0 by embedding the reduced bifurcation equations Q = 0 in a universal unfolding of Q. Then F is ^-equivalent to that universal unfolding. Let F be an element of ^ and consider the orbit GF through F under 'S-contact equivalence: The "tangent space" to CF at F in the module # is formally obtained by differentiating along arbitrary one-parameter curves through F. Thus, let us EQUIVARIANT SINGULARITY THEORY 51 suppose we have where T(x, A, 0) = J, X(x, A, 0) = x and A(A, 0) = A.

A set of coordinates on the real subspace of Ker / is thus given by (z l5 z2, z\, z2). We denote time translations as before by cre. The operations cre, Ty and P then have the following matrix representation relative to this basis: The invariants of