Abstract
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.
Highlights
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
Using a symmetry-breaking bifurcation theorem [1, Theorem 3.1] and the numerical verification methods, we proved the existence of a Z2-symmetry-breaking bifurcation point for a nonlinear forced vibration system described by a wave equation in [2], and Nakao et al verified some symmetrybreaking bifurcation points for two-dimensional Rayleigh-Benard heat convection system in [3, 4]
For our purpose, which is to check that Theorem 1 is applicable to the numerical verification method
Summary
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. We have various excellent bifurcation theorems from the theoretical point of view It needs in general, 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. We establish codimension-m bifurcation theorems applicable to the numerical verification methods. We can apply Theorem 1 to Hopf bifurcation by setting m = 2 and choosing an appropriate space of periodic functions as X, the subspace of steady functions as X1, and the complementary subspace of X1 as X2 (see Section 4 below).
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