Perturbation of periodic boundary conditions. II. Bifurcation and symmetry

AbstractWe continue our study (SIAM J. Math. Anal., 15, No. 4 (1984)) of perturbations of the problem (∗) − x″+ bx = λax, x(0) − x(1) = 0, x′(0) − x′(1) = 0 both by changes of boundary conditions and by addition of nonlinear terms. We assume that for λ = λ0 there are two linearly independent solutions of (∗). The method of Liapunov and Schmidt is used to analyze the full nonlinear problem. For three examples we do the thorough local bifurcation analysis of determining the bifurcation set and bifurcation diagrams. These examples are endowed with varying amounts of the symmetry of (∗), and we see that the examples are amenable to varying methods: (1) When the perturbed problem loses all symmetry, one can use the generic methods of Chow, Hale, and Mallet-Paret, (2) when the perturbed problem retains some of the symmetry and is not too degenerate, one can use results of Shearer and basic group-theoretical techniques, (3) when the perturbed problem retains much symmetry but is degenerate, one can use some of the same methods as for (2), Vanderbauwhede's generalization of the simple eigenvalue idea, and some ad hoc methods