Uniqueness and stability of unfoldings of equivariant bifurcation problems with two groups of state variables;

Infinitesimally Stable Unfolding of Equivariant Bifurcation Problems;

Graded Stability of Unfoldings of Smooth Map Germs;

With the present method, the non-linear one-degree-of-freedom oscillators′ steady periodic solutions can be obtained easily, and the transition set and bifurcation diagram can be achieved by the universal unfolding theory.

By the theory of center manifold,universal unfolding of the system with a single zero eigenvalue is discussed and the set of bifurcation are given.

In this paper transversality idea from singularity theory is applied to set up the universal unfolding theorem of bifurcation problems with multiparameter.

In [6], the universal unfolding of C∞ map germs under a subgroup of A was discussed.

Universal unfolding theorem of bifurcation problems with one parameterwere given in [1,2].

Based on the left-right equivalent relation of smooth map-germs in slgularity theory,the neessary and sufficient condition for unfolding of equivariant bifurcation problem with two types of state variables under a subgroup A1(Γ) of left-right equivalent group to be versal unfolding are studied.

The main results:(1) the equivalence classes of versal unfoldings of bifurcation problems are one-to-one corresponding to the orbits of the bifurcation problem; (2) the necessary and sufficient condition of stability of bifurcation problems.

On this base,some properties of ΓΔ-equivariant bifurcation problem under contact equivalence are obtained:the finite modules over the rings of ΓΔ-invariant functons and the property of its tangent space are obtained and an sufficient and necessary conditon that an unfolding of ΓΔ-bifurcation problem is a versal unfolding is given.