Nonlinear analysis of a four-dimensional fractional hyper-chaotic system based on general Riemann–Liouville–Caputo fractal–fractional derivative

Abstract

In this study, a four-dimensional fractional hyperchaotic model is analyzed based on general Riemann–Liouville–Caputo (RLC) fractal–fractional derivative (FFD). A series of new operators are constructed using three different elements, namely, the general Mittag–Leffler function, exponential decay, and power law. The operators have two parameters: One is considered as fractional order and the other as fractal dimension. The Qi hyperchaotic fractional attractor is modeled by using these operators, and the models are solved numerically using a very efficient numerical scheme. Meanwhile, the existence and uniqueness of solutions have been investigated to justify the physical adequacy of the model and the numerical scheme proposed in the resolution. The numerical simulations for some specific fractional order and fractal dimension are presented. Furthermore, these results obtained via generalized Caputo–Fabrizio and fractal–fractional derivative show some crossover effects, which is due to non-index law property. Finally, these obtained from generalized fractal–fractional derivative show very strange and new attractors with self-similarities.