Analysis of fractal fractional Lorenz type and financial chaotic systems with exponential decay kernels

In this work, we formulate a fractal fractional chaotic system with cubic and quadratic nonlinearities. A fractal fractional chaotic Lorenz type and financial systems are studied using the Caputo Fabrizo (CF) fractal fractional derivative. This study focuses on the characterization of the chaotic nature, and the effects of the fractal fractional-order derivative in the CF sense on the evolution and behavior of each proposed systems. The stability of the equilibrium points for the both systems are investigated using the Routh-Hurwitz criterion. The numerical scheme, which includes the discretization of the CF fractal-fractional derivative, is used to depict the phase portraits of the fractal fractional chaotic Lorenz system and the fractal fractional-order financial system. The simulation results presented in both cases include the two- and three-dimensional phase portraits to evaluate the applications of the proposed operators.