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The epitome of this paper centers on the application of the differential transformation method (DTM) the renowned Chen system which is described as a three-dimensional system of ODEs with quadratic nonlinearities. Numerical comparisons are made between the DTM and the classical fourth-order Runge–Kutta method (RK4). Our work showcases the precision of the DTM as the Chen system transforms from a non-chaotic system to a chaotic one. Since the Lyapunov exponent for this system is much higher compared to other chaotic systems, we shall highlight the difficulties of the simulations with respect to its accuracy. We wrap up our investigations to reveal that this direct symbolic–numeric scheme is effective and accurate.

This paper emphasizes the strength of variational iteration method (VIM) in numerically solving the chaotic Rossler system. The Rossler system is a three-dimensional system of ODEs with quadratic nonlinearities. An adaptation of the VIM was implemented and is called the multistage VIM (MVIM). Numerical comparisons are made between MVIM and the classical fourth-order Runge–Kutta (RK4) with results showing extremely good performance by MVIM, yielding great accuracy and efficiency.

Based on the nonlinear control and Lyapunov stability theory, the anti-synchronization between two identical and different chaotic systems is investigated, where the antisynchronization can be characterized by the vanishing of the sum of relevant variables in chaotic systems, such that the Lu system is controlled to be the Lorenz system. Theoretical analysis and numerical simulations are shown to verify the results.

In this paper, the exact solutions of Helmholtz equation are obtained by means of the homotopy analysis method (HAM). This analytical method is employed to give approximate analytical solutions of Helmholtz equation. The auxiliary parameter ℏ in the HAM solutions has provided a convenient way of controlling the convergence region of series solutions. It is also shown that the solutions which are obtained by the Adomian decomposition method (ADM) and variational iteration method (VIM) are special cases of the solution obtained by HAM.

This is a case study of solving the Genesio system by using the classical variational iteration method (VIM) and a newly modified version called the multistage VIM (MVIM). VIM is an analytical technique that grants us a continuous representation of the approximate solution, which allows better information of the solution over the time interval. Unlike its counterpart, numerical techniques, such as the Runge–Kutta method, provide solutions only at two ends of the time interval (with condition that the selected time interval is adequately small for convergence). Furthermore, it offers approximate solutions in a discretized form, making it complicated in achieving a continuous representation. The explicit solutions through VIM and MVIM are compared with the numerical analysis of the fourth-order Runge–Kutta method (RK4). VIM had been successfully applied to linear and nonlinear systems of non-chaotic in nature and this had been testified by numerous scientists lately. Our intention is to determine whether VIM is also a feasible method in solving a chaotic system like Genesio. At the same time, MVIM will be applied to gauge its accuracy compared to VIM and RK4. Since, for most situations, the validity domain of the solutions is often an issue, we will consider a reasonably large time frame in our work.

The aim of this paper is to investigate the accuracy of the Adomian decomposition method for solving the hyperchaotic Rossler system, which is a four-dimensional system of ODEs with quadratic nonlinearities. Comparisons between the decomposition solutions and the fourth-order Runge–Kutta (RK4) solutions are made. It is found that the comparable accuracy of the ADM with the RK4 for the chaotic Lorenz system does not carry to the hyperchaotic Rossler system.

In this paper, He’s variational iteration method (VIM) is applied to the generalized Burgers–Huxley equation. The VIM produces an approximate solution of the equation without any discretization. The VIM is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. Comparisons with the Adomian decomposition method (ADM) reveal that the VIM is very effective and convenient.

In this paper, the Adomian decomposition method (ADM) is applied to the Chen system which is a three-dimensional system of ODEs with quadratic nonlinearities. The ADM yields an analytical solution in terms of a rapidly convergent infinite power series with easily computable terms. Comparisons between the decomposition solutions and the classical fourth-order Runge–Kutta (RK4) numerical solutions are made. In particular we look at the accuracy of the ADM as the Chen system changes from a non-chaotic system to a chaotic one. To highlight some computational difficulties due to a high Lyapunov exponent, a comparison with the Lorenz system is given.

In this paper, the Adomian decomposition method (ADM) is applied to the famous Lorenz system. The ADM yields an analytical solution in terms of a rapidly convergent infinite power series with easily computable terms. Comparisons between the decomposition solutions and the fourth-order Runge–Kutta (RK4) numerical solutions are made for various time steps. In particular we look at the accuracy of the ADM as the Lorenz system changes from a non-chaotic system to a chaotic one.