High order approach for solving chaotic and hyperchaotic problems

Abstract

In this work, the method of Taylor's decomposition on two points is suggested in order to find approximate solutions of chaotic and hyperchaotic initial value problems and to analyze the behaviors of these solutions. Unlike to the classical Taylor's method, the proposed numerical scheme is based on the application of the Taylor's decomposition on two points to the system of nonlinear initial value problems, and as a result an implicit method is obtained. Stability and error analysis of the method are presented, and its high-order accuracy and A-stability are proven. One of the advantages of the proposed method is that it is a stable and very efficient method for chaotic problems as it is an implicit one-step method. The most important advantage of the Taylor's decomposition method is that it has high order accuracy for large step sizes with a simple algorithm compared to other methods. The applicability of the proposed method has been examined in some famous chaotic systems; the Lorenz and Chen systems, and hyperchaotic systems; the Chua and Rabinovich-Fabrikant systems, to emphasize both its accuracy and effectiveness. The accuracy of the proposed method is checked by comparing the calculated results with semi-explicit Adams-Bashforth-Moulton method and ninth order Runge-Kutta method. The calculated results are also compared with multi-stage spectral relaxation method and multi-domain compact finite difference relaxation method. Comparisons have shown that the method is more accurate and efficient than the other mentioned methods for large step sizes. The obtained results are also compared with the theoretical findings and it is shown that the theoretical and numerical results are consistent.