Chaotic and fractal maps in higher-order derivative dynamical systems

Hamiltonian maps are considered a class of dynamical systems that hold meticulous properties used to model a large number of complex dynamical systems. When time flows in dynamical systems with two-dimensional degrees of freedom, the trajectories in phase space can be analyzed within bidimensional surfaces known as Poincaré sections. The Chirikov–Taylor standard map for two canonical dynamical variables (momentum and coordinate) is the most renewed map characterized by a family of area-preserving maps with a single parameter that controls the degree of chaos. In this study, a generalization of the standard map for two different problems is presented and discussed. The first problem deals with the higher-order derivative Hamiltonian system (up to the fourth order) since the fourth-order characteristic provides the possibility of chaotic behavior at all scales including nanoscales where high-order derivatives take place in nanosystems. The second problem concerns the time-dependent δ -kicked rotor in fractal dimensions characterized by a time-dependent potential due to its important implications in quantum chaos. This study shows that higher-order derivative maps and fractal dimensional δ -kicked rotor maps apparently exhibit a large number of chaotic orbits and fractal patterns, including the spiral fractal patterns comparable to the Julia set. Moreover, these problems are characterized by additional parameters which can be used to control chaos. Some of these parameters lead to chaos, and others lead to fractal patterns.