Chaotic Dynamics of a Three-Dimensional Endomorphism in the Vicinity of the Arnold Tongue

The objective of this work is to investigate the transition between periodicity and chaos in the vicinity of Arnold tongues, within the framework of a three-dimensional, nonlinear, and noninvertible discrete dynamical system, which is a specific instance of an endomorphism. Arnold tongues, which represent families of resonance regions in the parameter space, are typically associated with the emergence of periodic orbits. As parameters vary and the system exits these regions, chaotic dynamics can emerge, notably through Neimark–Sacker bifurcations or period-doubling cascades. The analysis reveals alternating regimes of stability and chaos, highlighting how complex dynamics can arise at the boundaries of resonance zones in nonlinear systems.