Dynamical analysis of a complex logistic-type map

A complex logistic-type map is considered in the present work. The dynamic behavior of the underlined map is discussed in two different cases: the first case is when the parameter of the map being real, and the second case is when the parameter being complex. Existence and stability of fixed points are derived. The conditions for existence of a pitchfork bifurcation, flip bifurcation and Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. Numerical simulations including Lyapunov exponent, phase plane, bifurcation diagrams is carried out using matlab to ensure theoretical results and to reveal more complex dynamics of the map. The results show that expressing the logistic map in terms of complex variables leads to more distinguished behaviors, which could not be achieved in the logistic map with real variables. In addition, considering the control parameter as a complex parameter shows more interesting dynamics if compared to the case when considering it as a real parameter. Existence of a snapback repeller is proved in the sense of Marotto. Finally, chaos is controlled using the OGY feedback method.