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This paper centres on the application of the new piecewise successive linearization method (PSLM) in solving the chaotic and nonchaotic Chen system. Numerical simulations are presented graphically and comparison is made between the PSLM and Runge-Kutta-based methods. The work shows that the proposed method provides good accuracy and can be easily extended to other dynamical systems including those that are chaotic in nature.

In this paper, we consider the Chen dynamical system [

The solution of nonchaotic, chaotic and hyperchaotic Chen system equations has been obtained by different analytical and numerical methods such as Runge-Kutta-based numerical schemes. In recent years, several analytical and seminumerical methods have been utilized in solving the Chen system equations [

The aim of this paper is to obtain the solution of the Chen system using the successive linearization method (SLM) that is implemented in a sequence of intervals. This new algorithm is called the piecewise successive linearization method (PSLM). The standard SLM was developed primarily for solving nonlinear boundary value problems (see, e.g., [

The remainder of the paper is organized as follows. In Section

In this section, we give a brief description of the basic idea behind the successive linearization method (SLM) [

The SLM approach assumes that the solution of (

It was observed through numerical experimentation that, in its standard form, the SLM will not give accurate solutions for initial value problems of the Chen system type. Thus, it can be remarked that the SLM is ideally suited for boundary value problems whose solutions do not rapidly change in behaviour or oscillate over small regions of the domain of the governing problem. The SLM solution can thus be considered to be local in nature and may not be suitable for initial value problems at very large values of the independent variable

Let

In this section, numerical simulations are given to validate the proposed piecewise successive linearization method. The PSLM results were obtained using

The results of the PSLM simulation of the Chen system for the nonchaotic case are shown in Figures

Time series PSLM (solid line) and

Phase portraits of the Chen system using the PSLM for the nonchaotic case.

The results of the PSLM simulation of the Chen system for the chaotic case are shown in Figures

Time series PSLM (solid line) and

Phase portraits of the Chen system using the PSLM for the chaotic case.

In this paper, we carefully develop a novel piecewise successive linearization method (PSLM) which seeks to overcome the difficulties arising when the successive linearization method is extended to initial value problems with complex behaviour. The proposed method of solution is implemented on the Chen system and results are generated using known parameters that give both chaotic and nonchaotic outcomes. The validity of PSLM is verified by comparing its results with the results of built in Matlab-based numerical Runge-Kutta method,