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

Abstract

<abstract><p>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.</p></abstract>

Fractional order Jacobi wavelet-based numerical analysis of fractal-fractional multi-pantograph delay differential equation with variable coefficients

“…The function u(t) considered as in subsection 2.1 is called CFFFD with EDK, if the following relation holds, see in [4,17,46]…”

“…Similarly, as in subsection 2.1, the CFFFI (or anti-derivative of CFFFD) with EDK is defined for a continuous function u(t) such as, see in [4,19,46],…”

Investigation of fractal fractional nonlinear Korteweg-de-Vries-Schrödinger system with power law kernel

“…In addition to using fractal dimensions for statistical self-similarity, one may also use mathematical models that might approximate the production of actual fractal objects based on direct observation [15]. The FF operators have several applications in applied sciences such as mathematical physics [16,17], chaotic systems [18], mathematical biology [19] and many more [20,21].…”