Computation of Takens–Bogdanov Type Bifurcations with Arbitrary Codimension

Abstract

Let $\dot u = f(u,\alpha )$, $u(0) = u^0$ describe a dynamical system, where $u \in R^n $ is the state variable, $\alpha \in R^k $ is a parameter variable, and $f(u,\alpha )$ takes values in $R^n$. Here, n can be large, but k is usually small (say,$ \leqslant 10$). A solution $(u,\alpha )$ to the equilibrium equations $f(u,\alpha ) = 0$ is a Takens–Bogdanov type bifurcation of codimension m if the Jacobian $f_u (u,\alpha )$ contains an m-dimensional Jordan block corresponding to a zero eigenvalue and has no other eigenvalues equal to zero. This includes the cases of turning point bifurcation $(m = 1)$, Takens-Bogdanov bifurcation $(m = 2)$, and triple point bifurcation $(m = 3)$. A unified direct method is proposed, based on ideas of A. Griewank and G. W. Reddien in the case of Takens–Bogdanov points, for both the detection and computation of parameter values a at which Thkens–Bogdanov type bifurcations of any given codimension occur. As an application we compute Tkens–Bogdanov type bifurcations with codimensions 1, 2, and 3 in a trimolecular Brusselator reaction scheme. The method extends to any parametrized family of matrices in which matrices with the above-mentioned Jordan structures are sought.