Dynamics of fractional chaotic systems with Chebyshev spectral approximation method

Abstract

The dynamical behavior of chaotic processes with a noninteger-order operator is considered in this work. A lot of scientific reports have justified that modeling of physical scenarios via non-integer order derivatives is more reliable and accurate than integer-order cases. Motivated by this fact, the standard time derivatives in the model equations are formulated with the novel Caputo fractional-order operator. The choice of using the Caputo derivative among several existing fractional derivatives has to do with the fact that it gives way for both the initial conditions and boundary conditions to be incorporated in the development of the chaotic model. Numerical approximation of fractional derivatives has been the major challenge of many scholars in different areas of engineering and applied sciences. Hence, we developed a numerical approximation technique, which is based on the Chebyshev spectral method for solving the integer-order and non-integer-order chaotic systems which are largely found in physics, finance, biology, engineering, and other areas of applied sciences. The proposed numerical method used here is easy to implement on a digital computer, and capable of solving higher-order problems without reduction to the system of lower-order ordinary differential equations with limited computational costs. Experimental results are presented for different instances of fractional-order parameters.