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We investigate the bifurcations and basins of attraction in the Bogdanov map,a planar quadratic map which is conjugate to the H\'enon area-preserving map inits conservative limit. It undergoes a Hopf bifurcation as dissipation isadded, and exhibits the panoply of mode locking, Arnold tongues, and chaos asan invariant circle grows out, finally to be destroyed in the homoclinictangency of the manifolds of a remote saddle point. The Bogdanov map is theEuler map of a two-dimensional system of ordinary differential equations firstconsidered by Bogdanov and Arnold in their study of the versal unfolding of thedouble-zero-eigenvalue singularity, and equivalently of a vector fieldinvariant under rotation of the plane by an angle $2\pi$. It is a useful systemin which to observe the effect of dissipative perturbations on Hamiltonianstructure. In addition, we argue that the Bogdanov map provides a goodapproximation to the dynamics of the Poincar\'e maps of periodically forcedoscillators.Comment: Replacement PDF version converted from old 1990s file

THE BOGDANOV MAP: BIFURCATIONS, MODE LOCKING, AND CHAOS IN A DISSIPATIVE SYSTEM