Reversible Hopf bifurcation in four-dimensional maps

Abstract

Following a recent work [Sevryuk and Lahiri, Phys. Lett. A 154, 104 (1991)], we study the bifurcation of four-dimensional reversible maps in which the eigenvalues of the Jacobian of the map at a symmetric fixed point move off the unit circle along a pair of conjugate rays as some parameter \ensuremath{\epsilon} crosses a threshold value. We construct a perturbation scheme to show that, depending on a control parameter \ensuremath{\gamma}, the bifurcation can be either ``normal'' or ``inverted'' in nature. In the former case, two one-parameter families of elliptic invariant curves passing arbitrarily close to the fixed point (which coexist with Kolmogorov-Arnold-Moser tori) merge together and move away from the fixed point. In the latter case, the families of elliptic invariant curves meet a family of hyperbolic invariant curves. As \ensuremath{\epsilon} is varied, all these invariant curves shrink to the fixed point and are annihilated. The problem of determining whether an invariant curve is elliptic or hyperbolic is related to a tight-binding model on a linear quasiperiodic chain familiar in solid-state theory. Numerical evidence confirming these results is presented. A few areas for further study are indicated.