Defining Functions for Multiple Hopf Bifurcations

Let $A(u,\alpha)=F_u(u,\alpha)$ ($u \in R^n, \alpha \in R^k$) be a family of real $n \times n$ matrices arising as the Jacobian matrices of equilibrium solutions to the dynamical system $ \stackrel{.}{u}=F(u,\alpha)$. An equilibrium point is called a Hopf point if A has a conjugate pair of pure imaginary eigenvalues $\pm i\omega$, $\omega> 0$. It is called a double Hopf point if there are two such pairs $\pm i\omega_1, \pm i\omega_2$ and a 1:1 resonant double Hopf point if, in addition, $\omega_1=\omega_2$. Defining functions are obtained for the numerical detection, computation, and continuation of Hopf, double Hopf, and 1:1 resonant double Hopf points. They are based on a combination of matrix biproduct and bordered matrix methods. Example computations are done in a fairly realistic and complicated neural model problem with $n=13$ and $k=29$. However, to make the methods applicable to large-scale problems (e.g., discretized boundary value problems) we reduce the state space to a subspace that essentially contains the generalized eigenspaces of the eigenvalues with largest real part.