Abstract forced symmetry breaking
- Peterhof, Daniela
- Recke, Lutz
2010 Mathematics Subject Classification
- 58E07 58E09 58F35
- Forced symmetry breaking, bifurcation from solution orbits, G-invariant implicit function theorem, locking cones, principle of reduced stability
We consider abstract forced symmetry breaking problems of the type F(x,λ) = y, x ≈ O(x0), λ ≈ λ0, y ≈ O. It is supposed that for all λ the maps F(·,λ) are equivariant with respect to representations of a given compact Lie group, that F(x0, λ0) = 0 and, hence, that F(x,λ0) = 0 for all elements x of the group orbit O(x0) of x0. We look for solutions x which bifurcate from the solution family O(x0) as λ and y move away from λ0 and zero, respectively. Especially, we describe the number of different solutions x (for fixed control parameters λ and y), their dynamic stability, their asymptotic behavior for y tending to zero and the structural stability of all these results. Further, generalizations are given to problems of the type F(x,λ) = y(x,λ), x ≈ O(x0), λ ≈ λ0, y(x,λ) ≈ 0. This work is a generalization of results of J. K. HALE, P. TÁBOAS , A. VANDERBAUWHEDE and E. DANCER to such extend that the conclusions are applicable to forced frequency locking problems for rotating and modulated wave solutions of certain S1-equivariant evolution equations which arise in laser modeling.
- J. Differential Equations, 144 (1998) pp. 233--262.