Naimark–Sacker bifurcations in a delay quartic map

In this paper, we consider a two-dimensional map in which one of the fixed points is destabilized via a supercritical Naimark–Sacker bifurcation. We investigate, via numerical simulations, phenomena associated with the appearance, in the phase-space, of closed invariant curves involved in the Naimark–Sacker bifurcation. Lyapunov exponents, parameter-space and phase-space diagrams are used to show that the transition from quasiperiodic to chaotic states generally do not happen in this case. We determine numerically the location of the parameter sets where the Naimark–Sacker bifurcation occurs.