REVERSIBLE HOPF-BIFURCATION IN 4-DIMENSIONAL MAPS

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 epsilon crosses a threshold value. We construct a perturbation scheme to show that, depending on a control parameter 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 epsilon is varied, all these invariant curve 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.