A Novel Discrete Chaotic Dynamical System with Intriguing Periodicity and Bifurcation Structure

We introduce a novel chaotic dynamical system formulated as a second-order difference equation. The model includes one time-delay linear term and one nonlinear term composed of the product of two different time-delay terms. As a result, the nonlinear term gives rise to strange periods. The dynamical behaviors of the model are investigated by the Lyapunov spectrum and bifurcation diagram. We found that the model can have a positive Lyapunov exponent. We also found that while the bifurcation diagram has a typical Arnol’d tongue structure, there are unique shapes that show the progression of overlapping fish hooks in the synchronization region.