Abstract
In the field of complex systems, there is a need for better methods of knowledge discovery due to their nonlinear dynamics. The numerical simulation of chaotic or hyperchaotic system is mainly performed by the fourth-order Runge–Kutta method, and other methods are rarely reported in previous work. A new method, which divides the entire intervals into N equal subintervals based on a meshless collocation method, has been constructed in this paper. Some new complex dynamical behaviors are shown by using this new approach, and the results are in good agreement with those obtained by the fourth-order Runge–Kutta method.
1. Introduction
This paper shows some novel complex dynamical behaviors of a class of four-dimensional chaotic or hyperchaotic systems, and a four-dimensional system (1) is adopted as an example to elucidate the solution process.
We consider the following four-dimensional system:with the following initial conditions:where are the state variables and are the positive parameters of the system. We assume that the solution of systems (1) and (2) exists and it is unique.
In recent years, some new four-dimensional chaotic or hyperchaotic systems [1–7] are presented. In [1], the authors presented a four-dimensional hyperchaotic system and investigated and analyzed some complex dynamical behaviors such as ultimate boundedness, chaos, and hyperchaos. In [2], the authors reported a four-dimensional dissipative chaotic system, and the coexistence of rich chaotic dynamics in the system was investigated through the Lyapunov spectrum, bifurcation diagram, Poincarė map, frequency spectrum, and attractor plot. In [3], the authors discussed the synchronization between two chaotic dynamical systems of different order using an adaptive control scheme. In [4], a four-dimensional autonomous system with complex hyperchaotic dynamics was presented, and the complex dynamical behaviors were investigated by dynamical analysis approaches, such as time series, Lyapunov exponent spectra, bifurcation diagram, and phase portraits. In [1–8], the authors used the fourth-order Runge–Kutta method to simulate chaotic or hyperchaotic systems. In this paper, we mainly introduce an improved meshless collocation method to simulate hyperchaotic systems with long-time dynamic behavior.
2. The Meshless Collocation Method
A hyperchaotic system has two or more positive Lyapunov exponents. At least four dimensions for the integer order continuous autonomous system are needed in order to generate a hyperchaotic system. Chaotic sequences of such system are more dependent on the parameters and initial conditions, and their dynamic behaviors are more difficult to predict. Its attractors are more complex than general attractor. Diffusion and confusion can be carried out simultaneously in several dimensional spaces. Therefore, the hyperchaotic system has a distinct advantage over low dimensional chaos, and its high-precision numerical solution is very important.
In this section, we introduce an improved meshless collocation method to solve the four-dimensional chaotic or hyperchaotic system, and the main points are as follows.
We transform system (1) into equation (5) using the meshless collocation method [9]. We construct the following linear iterative format of system (1):
Using the meshless collocation method, can be written as [9, 10]where the barycentric interpolation primary function and is the center of gravity interpolation weight.
Inserting (4) into (3) and then letting , the format (3) can be transformed into the following linear algebraic equations [9, 10]:where is the M-order matrix, I is the M-order unit matrix, and the vector
Schneiaer and Werner [11] introduced the meshless collocation method by using higher-order rational interpolation functions. The barycentric representation of the rational interpolation function possesses various advantages in comparison with other representations such as continued fractions. Baltensperger and Berrut [12] introduced the meshless collocation method for solving general hyperbolic problems and proved its stability and its convergence in weighted norms. Li and Wang [13, 14] presented the algorithm and program of the meshless collocation method and had indicated the meshless collocation method has a high accuracy, good stability, and convergence [15–19]. However, the method cannot apply directly the solving of hyperchaotic system due to great errors. Therefore, an improved mesheless collocation method, which divides the entire intervals into N equal subintervals, is constructed in the following section.
We obtain the solution of equation (5) on every intervals .
Dividing the interval into N equal subintervals , letting with and . On , selecting the second kind Chebyshev nodes , and using (4) and the initial conditions (2), we can get
Replacing the first line of equation (5) with the initial conditions (6) and solving equation (5) that has been replaced by the initial conditions (6), we can get a numerical solution of systems (1) and (2) on .
On , select the second kind Chebyshev nodes because have been obtained on . Using (4), we can get
Replacing the first line of equation (5) with the initial conditions (7) and solving equation (5) that has been replaced by the initial conditions (7), we can get a numerical solution of systems (1) and (2) on .
Similarly, on , we can get the numerical solution of systems (1) and (2) on . After obtaining the numerical solution for all subintervals, these solutions are combined to obtain a numerical solution to equation (1) over the entire interval .
It is easy to see that the present method provides a globally smooth numerical solution due to the continuity of the numerical solution obtained and its first-order derivative at the common end point of two adjacent intervals.
Based on above formulas, a computer code has been programmed. The existence, uniqueness, and convergence of the approximate solution are ensured by using the computer code.
From above the meshless collocation method, we can get the following theorem.
Theorem 1. If is the countable dense point in , the linear iterative format (3) is the convergence, is the solution of equation (1), is the solution of equation (3), and is the solution of equation (5). Then,In numerical experiments, we select the initial iteration value . The accuracy of iteration control is , and let .
3. Numerical Experiment
In this section, some numerical experiments are studied to demonstrate the accuracy of the present method. The experiments are computed using Matlab R2017a.
Experiment 1. We consider the following four-dimensional system [1]:where are the positive parameters of the system, which satisfy the following initial conditions:Lyapunov exponents of Experiment 1 are shown in Figure 1. The complex dynamic behaviors of Experiment 1 are shown in Figures 2–9.
Experiment 2. We consider the following four-dimensional system:where a is the real parameter, which satisfies the following initial conditions (Figure 10):Lyapunov exponents of Experiment 2 are shown in Figure 1. The complex dynamic behaviors of Experiment 2 are shown in Figures 11–13.
Experiment 3. We consider the following four-dimensional system [7]:where a is the parameter of the system, which satisfies the following initial conditions:Lyapunov exponents of Experiment 3 are shown in Figure 1. The complex dynamic behaviors of Experiment 3 are shown in Figures 14–19.
4. Conclusions and Remarks
In this paper, an improved meshless collocation method, which divides the entire intervals into N equal subintervals, has been constructed for a class of four-dimensional chaotic or hyperchaotic systems. The numerical results demonstrate that this approach is more effective and accurate than other mesheless collocation methods. Some new complex dynamical behaviors are shown by using the improved method. It is worth noting that the improved method can also be used to solve other similar problems [19, 20]. In the further work, we will be devoted to studying some fractional-order chaotic systems.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This study was supported by the Natural Science Foundation of Inner Mongolia (2017MS0103), Inner Mongolia Maker Collaborative Innovation Center of Jining Normal University, and National Natural Science Foundation of China (11361037).