BIFURCATION ANALYSIS OF DOUBLE TAKENS–BOGDANOV POINTS OF NONLINEAR EQUATIONS WITH A Z2-SYMMETRY

Abstract

We consider the nonlinear equation f (x, Λ) = 0, f: X × ℝ3 → X, where X is a Banach space and f satisfies a Z2-symmetry relation. We are interested in a singular point (x0, Λ0) called a double Takens–Bogdanov point, where fx has a zero eigenvalue of geometric multiplicity 2 and algebraic multiplicity 3. We show that only three parameters are needed to recognise and compute this singular point when the Z2-symmetry is present. Paths of the following three kinds of singular points bifurcating at this double Takens–Bogdanov point are studied: double s-breaking fold points (where fx has a zero eigenvalue of algebraic and geometric multiplicity 2), Takens–Bogdanov points (a zero eigenvalue of geometric multiplicity 1 and algebraic multiplicity 2) and s-breaking G points (a simple zero eigenvalue and a pair of simple purely imaginary eigenvalues). The theory is developed in such a way as to produce nondegeneracy conditions and tests which may be utilised in numerical calculations. Numerical results are presented for the one-dimensional Brusselator model, described by a system of four reaction–diffusion equations.