Numerical Simulation of the Lorenz-Type Chaotic System Using Barycentric Lagrange Interpolation Collocation Method

Although some numerical methods of the Lorenz 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 a novel numerical method to solve the Lorenz-type chaotic system which is based on barycentric Lagrange interpolation collocation method (BLICM). The system (1) is adopted as an example to elucidate the solution process. Numerical simulations are used to verify the effectiveness of the present method.


Introduction
In 1963, Edward Lorenz developed a simplified mathematical model for atmospheric convection.The model is a system of three ordinary differential equations now known as the Lorenz equations (see [1][2][3] with the initial conditions  (0) =  1 , where () is proportional to the rate of convection, () to the horizontal temperature variation, and () to the vertical temperature variation.The constants , ,  are system parameters proportional to the Prandtl number, Rayleigh number, and certain physical dimensions of the layer itself.
As chaos theory progresses, many new Lorenz-type systems [4][5][6] have been proposed, specially Lorenz hyperchaotic systems [7][8][9][10].The Lorenz system is widely used in electric circuits, chemical reactions, and forward osmosis.Although some numerical methods of the Lorenz system have been announced, simple and efficient methods have always been the direction that scholars strive to pursue.

The Numerical Solution of the System (1)
First of all, we give initial function  0 (),  0 (),  0 () and construct following linear iterative format of system (1) Next, we use BLICM to solve (3).

Numerical Experiment
In this section, some numerical examples are studied to find some new chaotic behaviors and verify the existing chaotic dynamic behaviors.In Experiments 1-5, the accuracy of iteration control is  = 10 −10 , the initial iteration value  0 =  0 =  0 = 0;  1 =  1 =  1 = , and for parameters , , , and  see Table 1.
We choose Chebyshev nodes, and the number of nodes  = 40.
We choose Chebyshev nodes, the number of nodes  = 40, and the parameters  = 7,  = 5 and the initial conditions (0) = 0, (0) = 1, (0) = 0.   Figure 7 is time series plots of chaotic system for Experiment 2 at different parameter value .We can see that the frequency of fluctuations of , , and  accelerates obviously with the increase of .When  = 3, the fluctuations of , , and  change obviously, and their fluctuations become smaller and smaller and finally stop at a certain value.Figures 5 and 6 are strange attractors of chaotic system for Experiment 2 by using the current method at  = 2 and  = 3 respectively.

Advances in Mathematical Physics
Experiment .We consider the 3D autonomous chaotic Lorenz-type system [7] We choose Chebyshev nodes and the number of nodes  = 60 and the parameters  = 10,  = 15 and the initial conditions (0) = 10, (0) = −0.2,(0) = 0.75.projected on (, )-plane; (c) is the graph projected on (, )plane.Figure 11 is time series plots of the 3D chaotic Lorenz type system for Experiment 3 at different parameter value .We can see that the fluctuation range of  changes obviously with the increase of .When  = 1, the fluctuation range of  is −5 to 15, and when  = 3, the fluctuation range of  is −15 to 5. Figures 9 and 10 are phase portraits of the 3D chaotic Lorenz type system for Experiment 3 by using the current method at  = 2 and  = 3, respectively.
Experiment .We consider the Lorenz system where , , and  are real parameters, which satisfy the following initial conditions: We choose Chebyshev nodes and the number of nodes  = 40.Experiment .We consider the new chaotic system [8] where , , and  are state variables and , , , and  are real parameters, which satisfy the following initial conditions: We choose Chebyshev nodes and the number of nodes  = 40.Figure 13 is obtained by using the current method with the parameters  = 0.5,  = −0.1, = 1.5, and  = 0.12.In Figure 13, (a

Conclusions and Remarks
In this paper, the Lorenz System has solved by using BLICM.These numerical experiments illustrate that the numerical results of the present method are the same as the experimental results.
All computations are performed by the MatlabR2017b software packages.

Figure 1
is obtained by using the current method with  = 1.Among them, (a) is the time series plot; (b) is the phase diagram of ; (c) is the three-dimensional space graph; (d) is the graph projected on (, )-plane; (e) is the graph projected on (, )-plane.Figures2 and 3are obtained by using the current method at  = 10 and  = 100, respectively.We can see that the fluctuation amplitude of  and  increases, while the fluctuation amplitude of  decreases with the increase of .The corresponding graphs , , , and  also have obvious changes.Experiment .We consider the Lorenz-type system[6]   =  ( − ) + ,   =  − ,   = − +
+  −  ( + ) , Figure 12 is obtained by using the current method with the parameters  = 10,  = 8/3, and  = 28.In Figure 12, (a) is the time series plot of ; (b) is the time series plot of ; (c) is the three-dimensional space graph; (d) is the graph projected on (, )-plane; (e) is the graph projected on (, )plane; (f) is the graph projected on (, )-plane.
) is the time series plot; (b) is the threedimensional space graph; (c) is the graph projected on (, )plane; (d) is the graph projected on (, )-plane; (e) is the graph projected on (, )-plane.