Ergodic and resonant torus doubling bifurcation in a three-dimensional quadratic map

We consider the rich dynamics and bifurcations exhibited by a three-dimensional quadratic map. Torus doubling bifurcations are central to bifurcation theory. Such bifurcations can only occur in a higher dimensional map of dimensions greater than or equal to three. After subsequent doublings, formation of Shilnikov attractors and hyperchaotic attractors are usually observed. It is shown that the map under consideration shows both resonant and ergodic torus doubling bifurcation. We show that the system exhibits resonant torus doubling bifurcation in which the doubled mode-locked periodic orbits lies on a Möbius strip. Additionally, we also illustrate the doubling of ergodic tori and analyze the bifurcation via the use of the second Poincaré section and the method of Lyapunov bundles. The analysis involves the techniques of construction of one-dimensional manifolds, a one-parameter continuation of saddle periodic orbits, and multi-dimensional Newton–Raphson method to locate the saddle periodic orbits.