Computation of Hopf bifurcation with bordered matrices

Hopf bifurcation for dynamical systems x = f(x, lambda) is characterized by rank deficiency two of A(2) + omega(2)I, where A = f(x)(x, lambda) is an element of R(nxn) is the Jacobian at the Hopf point and +/-i omega are the Hopf eigenvalues. Replacing w(2) by upsilon and allowing upsilon less than or equal to 0 we introduce real and imaginary Hopf matrices A with Hopf numbers upsilon, defining a codimension-2 manifold M(H,upsilon) subset of R(nxn) x R, which can be described by two scalar equations alpha(A, upsilon) = 0, beta(A, upsilon) = 0 given by the solution of certain linear bordered systems. These two scalar equations form a suitable augmentation of f(x, lambda) = 0 for the numerical computation of branches of Hopf points in two-parameter systems. In contrast to [Chu, Govaerts, and Spence, SIAM J. Numer. Anal., 31 (1994), pp. 524-539] we suggest an A-dependent bordering having certain advantages. We construct a scalar test function for Hopf points (in the sense of [Werner, in Bifurcation and Symmetry, Internat. Ser. Numer. Math., 104 (1992), pp. 317-327]) by nonlinear elimination of upsilon from the two scalar equations alpha = 0 and beta = 0. A strict sign change of this test function during numerical path following of equilibria is shown to be equivalent to eigenvalue crossing conditions in (real or imaginary [Werner and Janovsky, in Bifurcation and Chaos, Internat. Ser. Numer. Math., 97 (1991), pp. 377-388]) Hopf bifurcation points and can hence be used for the detection of Hopf points. An extension to Hopf points for mappings (Naimark-Sacker bifurcation) is given. Finally we illustrate our method by several examples, particularly for the computation of Hopf curves in two-parameter families of vector fields.