Codimension-m Bifurcation Theorems Applicable to the Numerical Verification Methods

We establish codimension-m bifurcation theorems applicable to the numerical verification methods. They are generalization of codimension-1 bifurcation theorems established by (Kawanago, 2004). As a numerical example, we treat Hopf bifurcation, which is codimension-2 bifurcation. 1. Introduction By the recent growth of the computer power, we can observe numerically bifurcation phenomena of solutions without difficulty for a lot of differential equations and systems. It is in general difficult, however, to analyze rigorously such phenomena by the use of pure analytical methods. Actually, it seems impossible at least at present to analyze by the use of pure analytical methods the Hopf bifurcation phenomena in the Brusselator model treated in Section 4. We need some computer-assisted analysis to treat it. We now have various excellent bifurcation theorems from the theoretical point of view. It needs in general, however, some particular devices to apply them to a given concrete dynamical system since we are usually not able to check some conditions in such theorems directly by numerical methods. Another important approach to computer-assisted analysis for bifurcation problems is to establish new bifurcation theorems applicable directly to numerical verification methods. This approach is our theme in this paper. It is useful from the practical and applied mathematical point of view. In [1], the author established some codimension-1 bifurcation theorems applicable directly to numerical verification methods. Using a symmetry-breaking bifurcation theorem [1, Theorem 3.1] and the numerical verification methods, we proved the existence of a -symmetry-breaking bifurcation point for a nonlinear forced vibration system described by a wave equation in [2], and Nakao et al. verified some symmetry-breaking bifurcation points for two-dimensional Rayleigh-Bénard heat convection system in [3, 4]. In this paper, we establish codimension- bifurcation theorems applicable to the numerical verification methods. They are generalization of codimension-1 bifurcation theorems mentioned above. In Section 4, we apply our new theorem to Hopf bifurcation, which is codimension-2 bifurcation. Here, we present our main theorem. Let and be real Banach spaces. Let and be closed subspaces of and let and be closed subspaces of . We assume that and . Here, means the direct sum. Let and have the following properties: We denote by the first row vector of the identity matrix of order . We define by Here, , and we assume that for any . We define projections and by In what follows, we always set