Numerical Simulation for a High-Dimensional Chaotic Lorenz System Based on Gegenbauer Wavelet Polynomials

Abstract

We provide an effective simulation to investigate the solution behavior of nine-dimensional chaos for the fractional (Caputo-sense) Lorenz system using a new approximate technique of the spectral collocation method (SCM) depending on the properties of Gegenbauer wavelet polynomials (GWPs). This technique reduces the given problem to a non-linear system of algebraic equations. We satisfy the accuracy and efficiency of the proposed method by computing the residual error function. The numerical solutions obtained are compared with the results obtained by implementing the Runge–Kutta method of order four. The results show that the given procedure is an easily applied and efficient tool to simulate this model.

High-Dimensional Chaotic Lorenz System: Numerical Treatment Using Changhee Polynomials of the Appell Type

“…In Figure 3a-i, we give a comparison between our numerical results and those results obtained by the RK4 method at α = 1 with m = 6 and r = 14.1. The REF [22] of the solution is shown in Figure 4a-i In addition, to strongly prove and confirm the effectiveness of the given method, we present a comparison with a previously published work solving the same model by using the SCM with the help of other orthogonal polynomials, named Gegenbauer wavelet polynomials [23]. This comparison is presented in Tables 1 and 2 with different values of the parameter r = 14.1 (for the chaotic case) and r = 55 (for the hyperchaotic case), with α = 0.95 and m = 10 in each method, but with the same initial conditions and the same parameters as in Figure 2.…”

Jacobi spectral method for the fractional reaction–diffusion equation arising in ecology

“…This method has many applications to solve problems in science and engineering [21][22][23][24][25][26][27][28][29][30][31][32][33][34]. More details about these methods and their applications can be found in other studies [35][36][37][38][39][40][41][42][43][44][45].…”

“…This method has many applications to solve problems in science and engineering [21–34]. More details about these methods and their applications can be found in other studies [35–45].…”

The effect of migration on the transmission of HIV/AIDS using a fractional model: Local and global dynamics and numerical simulations

“…Many authors have also used the SIR epidemic model to study the complex dynamics of epidemic systems (see, e.g., earlier studies [18][19][20]). Recently, fractional calculations have often been used in many disciplines, including epidemiology, science, engineering, economics, and many others [21][22][23][24][25][26][27]. When representing values in these fields, whole integers are often insufficient.…”