Complexity
Volume 2019 (2019), Article ID 1739785, 13 pages
https://doi.org/10.1155/2019/1739785

Research Article

Numerical Simulation of a Class of Hyperchaotic System Using Barycentric Lagrange Interpolation Collocation Method

Xiaofei Zhou,¹ Junmei Li,² Yulan Wang,² and Wei Zhang¹

¹Institute of Economics and Management, Jining Normal University, Jining 012000, Inner Mongolia, China
²Department of Mathematics, Inner Mongolia University of Technology, Hohhot 010051, China

Correspondence should be addressed to Yulan Wang; wylnei@163.com and Wei Zhang; jnsfxyzw@163.com

Received 12 December 2018; Accepted 3 February 2019; Published 13 February 2019

Academic Editor: Diyi Chen

Copyright © 2019 Xiaofei Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Hyperchaotic system, as an important topic, has become an active research subject in nonlinear science. Over the past two decades, hyperchaotic system between nonlinear systems has been extensively studied. Although many kinds of numerical methods of the system have been announced, simple and efficient methods have always been the direction that scholars strive to pursue. Based on this problem, this paper introduces another novel numerical method to solve a class of hyperchaotic system. Barycentric Lagrange interpolation collocation method is given and illustrated with hyperchaotic system () as examples. Numerical simulations are used to verify the effectiveness of the present method.

1. Introduction

Many chaotic systems have been developed such as Lorenz system [1], Rossler system [2], and Chen system [3]. As chaos theory progresses, many new chaotic systems [4–8] have been proposed, specially hyperchaotic systems [9–15]. A hyperchaotic system is usually characterized as a chaotic system with more than one positive Lyapunov exponent, implying that the dynamics expand in more than one direction, giving rise to more complex chaotic dynamics. Barycentric interpolation collocation method [16, 17] is a high precision method. Some authors have used barycentric interpolation collocation method to solve various kinds of problems [16–23]. This paper suggests the barycentric interpolation collocation method to solve a class of hyperchaotic system, and a hyperchaotic system (1) is adopted as an example to elucidate the solution process.

We consider the following 4D butterfly hyperchaotic system with butterfly phenomenon [24]:

where are the state variables and are the positive constant parameters of the system which satisfy the following initial conditions:

2. The Numerical Solution of System (1)

First of all, we give initial function and construct the following linear iterative format of system (1):

Next, we use the barycentric Lagrange interpolation collocation method to solve (3).

In the interval takes different nodes, . The barycentric interpolation of can be written as [16, 17]

is, respectively, barycentric Lagrange interpolation primary function and is center of gravity interpolation weight.

Use formula (4), the functions can be expressed as

So, linear iterative format (3) can be written in following partitioned matrix form:

The matrix is order matrix. is order unit matrix, diagonal matrix , and diagonal matrix . The vector

At last, we use initial conditions (2).

Take formula (4) into initial conditions (2); we can get the following discrete equations of initial conditions:

In this paper, we use displacement method to impose the initial conditions. The detailed procedure is as follows.

The first of (6) are replaced separately by the equation of initial conditions (8) in turn.

So, we can get that are approximate solution of (1) and (2).

3. Numerical Experiment

In this section, six numerical experiments are studied to demonstrate the effectiveness of the present method. All experiments are computed using MatlabR2017a. In Experiments 1–6, we choose Chebyshev nodes, the accuracy of iteration control is , and the initial iteration value ; . Parameters of the numerical Experiments 1–5 are listed in Table 1.

Table 1: Parameters used in Experiments 1–5.

Figures
a
b
c
d
k

Figure 1
1.378
0.5
0.6
0.097

Figure 2
1.378
0.5
0.6
0.097

Figure 3
0.2
0.5
0.8
0.063

Figure 4
0.3
0.5
0.8
0.063

Figure 5
0.6
0.5
0.8
0.063

Figure 7
1
0.5
2
1

Figure 8
1
0.5
2
1

Figure 9
8
3
4
-2
0.2

Figure 10
8
3
4
-2
0.2

Figure 11
15
2.5
0.75
2
0.2

Figure 12
15
2.5
0.75
2
0.2

Experiment 1. We consider the following hyperchaotic system [25]:

where are the state variables and are the positive parameters of the system, which satisfy the following initial conditions:

We choose Chebyshev nodes; the number of nodes . Numerical results of Experiment 1 are given in Figures 1 and 2.

(a)

(b)

Figure 1: States of the hyperchaotic system for Experiment 1 with , , , and : (a) states; (b) states.

Figure 2: Hyperchaotic attractors of the system for Experiment 1 with (c) on plane, (d) on plane, (e) on plane, and (f) in space.

Figure 1 is states of the hyperchaotic system for Experiment 1 with , which is obtained by using the current method, and is the states of and and is the states of and . Figure 2 is hyperchaotic attractors of the system for Experiment 1 with , which is obtained by using the current method. Among them, is the graph projected on -plane; is the graph projected on -plane; is the graph projected on -plane; is the graph in three-dimensional -space.

Experiment 2. We consider the following hyperchaotic system [26]:

where are the state variables and are the positive parameters of the system, which satisfy the following initial conditions:

We choose Chebyshev nodes, the number of nodes , and the parameters . Numerical results of Experiment 2 are given in Figures 3–6.

Figure 3: Phase portraits of a new hyperchaotic system for Experiment 2 with on plane, on plane, on plane, on plane, on plane, and in space.

Figure 4: Phase portraits of a new hyperchaotic system for Experiment 2 with on plane, on plane, on plane, on plane, on plane, and in space.

Figure 5: Phase portraits of a new hyperchaotic system for Experiment 2 with on plane, on plane, on plane, on plane, on plane, and in space.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6: The time series plots of a new hyperchaotic system for Experiment 2. and represent time series when ; and represent time series when ; and represent time series when .

Figure 3 is phase portraits of a new hyperchaotic system for Experiment 2 with by using the current method. is the graph projected on -plane; is the graph projected on -plane; is the graph projected on -plane; is the graph projected on -plane; is the graph projected on -plane; is the three-dimensional space graph. Figures 4 and 5 are phase portraits of a new hyperchaotic system for Experiment 2 obtained by using the current method with and , respectively. Figure 6 is time series plots of a new hyperchaotic system for Experiment 2 with different parameter value . and represent time series when ; and represent time series when ; and represent time series when .

Experiment 3. We consider the following butterfly hyperchaotic system [24]:

where are the state variables and are the positive constant parameters of the system, which satisfy the following initial conditions:

We choose Chebyshev nodes; the number of nodes . Numerical results of Experiment 3 are given in Figures 7 and 8.

(a)

(b)

Figure 7: States of a novel butterfly hyperchaotic system for Experiment 3 with : states; states.

Figure 8: Phase portraits of a novel butterfly hyperchaotic system for Experiment 3 with projected on the -plane, projected on the -plane, projected on the -plane, projected on the -plane, and in the three-dimensional space.

Figure 7 is states of a novel butterfly hyperchaotic system for Experiment 3 with , which is obtained by using the current method, and is the states of and and is the states of and . Figure 8 is phase portraits of a novel butterfly hyperchaotic system for Experiment 3 with , which is obtained by using the current method. Among them, is the graph projected on -plane; is the graph projected on -plane; is the graph projected on -plane; is the graph projected on -plane; is the graph in three-dimensional -space.

Experiment 4. We consider the following hyperchaotic Chen system [27]:

where are the state variables and are the positive constant parameters of the system, which satisfy the following initial conditions:

We choose Chebyshev nodes; the number of nodes . Numerical results of Experiment 4 are given in Figures 9 and 10.

(a)

(b)

Figure 9: Time response of the hyperchaotic Chen system’s variable states for Experiment 4 with : states; states.

Figure 10: The hyperchaotic Chen system for Experiment 4 with . Phase plane . Phase plane . Phase plane . Phase plane . in the three-dimensional space. in the three-dimensional space.

Figure 9 is time response of the hyperchaotic Chen system’s variable states for Experiment 4 with , which is obtained by using the current method, and is the states of and and is the states of and . Figure 10 is phase portraits of the hyperchaotic Chen system for Experiment 4 with , which is obtained by using the current method. Among them, is the graph projected on -plane; is the graph projected on -plane; is the graph projected on -plane; is the graph projected on -plane; is the graph in three-dimensional -space; is the graph in three-dimensional -space.

Experiment 5. We consider the following hyperchaotic system [28]:

where are the state variables and are the positive constant parameters of the system, which satisfy the following initial conditions:

We choose Chebyshev nodes; the number of nodes . Numerical results of Experiment 5 are given in Figures 11 and 12.

(a)

(b)

Figure 11: The time series plots of a new hyperchaotic system for Experiment 5 with : states; states.

Figure 12: Hyperchaotic attractors of a new hyperchaotic system for Experiment 5 with . plane; plane; plane; plane; plane; plane.

Figure 11 is the time series plots of a new hyperchaotic system for Experiment 5 with , which is obtained by using the current method, and is the states of and and is the states of and . Figure 12 is hyperchaotic attractors of a new hyperchaotic system for Experiment 5 with , which is obtained by using the current method. Among them, is the graph projected on -plane; is the graph projected on -plane; is the graph projected on -plane; is the graph projected on -plane; is the graph projected on -plane; is the graph projected on -plane.

Experiment 6. We consider the following 4D hyperchaotic system [29]:

where are the state variables and are the positive constant parameters of the system, which satisfy the following initial conditions:

We choose Chebyshev nodes; the number of nodes . Numerical results of Experiment 6 are given in Figures 13–15.

Figure 13: Phase portraits of a 4D hyperchaotic system for Experiment 6 with , projected on the -plane, projected on the -plane, projected on the -plane, projected on the -plane, and projected on the -plane.

Figure 14: Phase portraits of a 4D hyperchaotic system for Experiment 6 with projected on the -plane, projected on the -plane, projected on the -plane, projected on the -plane, and projected on the -plane.

Figure 15: The time series plots of a 4D hyperchaotic system for Experiment 6: and ; and .

Figure 13 is phase portraits of a 4D hyperchaotic system for Experiment 6 with , , , , , and , which is obtained by using the current method. Among them, is the graph projected on -plane; is the graph projected on -plane; is the graph projected on -plane; is the graph projected on -plane; is the graph projected on -plane. Figure 14 is phase portraits of a 4D hyperchaotic system for Experiment 6 with , , , , , and , which is obtained by using the current method. Figure 15 is the time series plots of a 4D hyperchaotic system for Experiment 6. and are obtained by using the current method with , , , , , and . and are obtained by using the current method with , , , , , and .

4. Conclusions and Remarks

In this paper, a class of hyperchaotic system has been solved by using barycentric Lagrange interpolation collocation method. The numerical simulation results are in accord with the theoretical analyses and circuit implementation. Numerical simulations are provided to verify the effectiveness and feasibility of the proposed numerical results, which are in agreement with theoretical analysis. In the further work, we will be devoted to studying fractional-order hyperchaotic system.

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 paper is supported by the Natural Science Foundation of Inner Mongolia 2017MS0103, Inner Mongolia maker Collaborative Innovation Center of Jining Normal University, and the National Natural Science Foundation of China 11361037.

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