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

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 ( ̇ 푥 = 푎푥 + 푑푧 − 푦푧, ̇ 푦 = 푥푧 − 푏푦, 0 ≤ 푡 ≤ 푇, ̇푧 = 푐(푥 − 푧) + 푥푦, ̇푤 = 푐(푦 − 푤) + 푥푧,) as examples. Numerical simulations are used to verify the effectiveness of the present method.

We consider the following 4D butterfly hyperchaotic system with butterfly phenomenon [24]: where 푥, 푦, 푧, 푤 are the state variables and 푎, 푏, 푐, 푑 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 푥 0 (푡), 푦 0 (푡) and construct the following linear iterative format of system (1): Next, we use the barycentric Lagrange interpolation collocation method to solve (3).
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 1 of (6) are replaced separately by the equation of initial conditions (8) in turn.

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 휀 = 10 −10 , and the initial iteration value 푥 0 = 푦 0 = 푧 0 = 0; 푥 1 = 푦 1 = 푧 1 = 푇.Parameters of the numerical Experiments 1-5 are listed in Table 1.
Experiment 1.We consider the following hyperchaotic system [25]: where 푥, 푦, 푧, 푤 are the state variables and 푎, 푏, 푐, 푑 are the positive parameters of the system, which satisfy the following initial conditions: We choose Chebyshev nodes; the number of nodes 푀 = 40.Numerical results of Experiment 1 are given in Figures 1  and 2.
Experiment 5. We consider the following hyperchaotic system [28]: where 푥, 푦, 푧, 푤 are the state variables and 푎, 푏, 푐, 푑, 푘 are the positive constant parameters of the system, which satisfy the following initial conditions: We choose Chebyshev nodes; the number of nodes 푀 = 40.Numerical results of Experiment 5 are given in Figures 11  and 12.

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.

Complexity
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.