Singular Hopf Bifurcation in Systems with Two Slow Variables

Abstract

Hopf bifurcations have been studied intensively in two dimensional vector fields with one slow and one fast variable [É. Benoît et al., Collect. Math., 31 (1981), pp. 37–119; F. Dumortier and R. Roussarie, Mem. Amer. Math. Soc., 121 (577) (1996); W. Eckhaus, in Asymptotic Analysis II, Lecture Notes in Math. 985, Springer-Verlag, Berlin, 1983, pp. 449–494; M. Krupa and P. Szmolyan, SIAM J. Math. Anal., 33 (2001), pp. 286–314; J. Guckenheimer, in Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, NATO Sci. Ser. II Math. Phys. Chem. 137, Kluwer, Dordrecht, The Netherlands, 2004, pp. 295–316]. Canard explosions are associated with these singular Hopf bifurcations [S. M. Baer and T. Erneux, SIAM J. Appl. Math., 46 (1986), pp. 721–739; S. M. Baer and T. Erneux, SIAM J. Appl. Math., 52 (1992), pp. 1651–1664; B. Braaksma, J. Nonlinear Sci., 8 (1998), pp. 457–490; Y. Lijun and Z. Xianwu, J. Differential Equations, 206 (2004), pp. 30–54], manifested by a very rapid growth in the amplitude of periodic orbits. There has been less analysis of Hopf bifurcations in slow-fast systems with two slow variables where singular Hopf bifurcation occurs simultaneously with type II folded saddle-nodes [A. Milik and P. Szmolyan, in Multiple-Time-Scale Dynamical Systems, IMA Vol. Math. Appl. 122, Springer-Verlag, New York, 2001, pp. 117–140; M. Wechselberger, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 101–139]. This work contributes to our understanding of these Hopf bifurcations in five ways: (1) it computes the first Lyapunov coefficient of the bifurcation in terms of a normal form, (2) it describes global features of the flow that constrain the types of trajectories found in the system near the bifurcation, (3) it identifies codimension two bifurcations that occur as coefficients in the normal form vary, (4) it exhibits complex solutions that occur in the vicinity of the bifurcation for some values of the normal form coefficients, and (5) it identifies singular Hopf bifurcation as a mechanism for the creation of mixed-mode oscillations. A subtle aspect of the normal form is that terms of higher order contribute to the first Lyapunov coefficient of the bifurcation in an essential way.