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# Numerical Simulation of the Fractional-Order Lorenz Chaotic Systems with Caputo Fractional Derivative

1 School of Physics and Electronic Information Engineering, Jining Normal University, Jining, 012000, China

2 Department of Mathematics, Inner Mongolia University of Technology, Hohhot, 010051, China

3 Institute of Economics and Management, Jining Normal University, Jining, 012000, China

* Corresponding Author: Yulan Wang. Email:

(This article belongs to this Special Issue: Fractal-Fractional Models for Engineering & Sciences)

*Computer Modeling in Engineering & Sciences* **2023**, *135*(2), 1371-1392. https://doi.org/10.32604/cmes.2022.022323

**Received** 04 March 2022; **Accepted** 09 June 2022; **Issue published** 27 October 2022

## Abstract

Although some numerical methods of the fractional-order chaotic systems have been announced, high-precision numerical methods have always been the direction that researchers strive to pursue. Based on this problem, this paper introduces a high-precision numerical approach. Some complex dynamic behavior of fractional-order Lorenz chaotic systems are shown by using the present method. We observe some novel dynamic behavior in numerical experiments which are unlike any that have been previously discovered in numerical experiments or theoretical studies. We investigate the influence of , , on the numerical solution of fractional-order Lorenz chaotic systems. The simulation results of integer order are in good agreement with those of other methods. The simulation results of numerical experiments demonstrate the effectiveness of the present method.## Keywords

In 1963, Edward Lorenz discovered a mathematical model for atmospheric convection. This model is also known as Lorenz chaotic system. The Lorenz system is widely used in electric circuits, forward osmosis and chemical reactions. In recent years, people have studied the chaotic behavior in the fractional dynamic system and found that the fractional dynamic system has unique properties that the integer dynamic system does not have. Therefore, the numerical simulation of fractional chaotic system is very important. In this paper, we simulate the fractional-order Lorenz chaotic dynamical systems [1–6] is as

with the initial conditions

The solutions of the fractional-order Lorenz chaotic dynamical systems are very hard to obtain analytically, and researchers, therefore, rely on numerical methods to provide an approximate solution that could be used for prediction. In the last decades, several numerical methods have been proposed. In [1], based on the qualitative theory, the authors investigated the existence and uniqueness of solutions for a class of fractional-order Lorenz chaotic systems (1). In [2], compared the dynamical regimes of fractional-order systems with dynamical regimes of the corresponding standard systems. In [3], Complex dynamics with interesting characteristics were presented by means of phase portraits, the largest Lyapunov exponent and bifurcation diagrams. In [4–6], the authors gave a dynamic analysis of a fractional-order Lorenz chaotic system. Although some numerical and analytical methods of the FDEs have been announced, such as spectral method [7–11], reproducing kernel method [12–19], homotopy perturbation method [20–23], high-precision numerical approach [24–27], and so on numerical and analytical methods [28–36]. These researchers all say their own approach can accurately simulate chaotic systems. In fact, since chaotic systems have no exact solution, researchers do not know which method is more accurate. For the numerical simulation of chaotic systems, it is necessary to use numerical methods to study the long time properties of solutions of the fractional order chaotic systems. This paper introduces a high-precision numerical method [24–27] for solving system (1). Some complex dynamic behavior of the fractional-order Lorenz chaotic systems are discovered by using the present numerical approach. We observe some novel dynamic behavior in numerical simulations which are unlike any that have been previously discovered in numerical simulations or theoretical studies. The simulation results of numerical experiments demonstrate the effectiveness of the present method.

Fractal and fractional calculus [37–48] have been widely concerned. In the last three decades, there have existed many inequivalent definitions [49–51] of fractional derivatives. The most famous of these definitions that have been widely popularized in the world of fractional calculus is Riemann-Liouville fractional definition, Gr

Definition 1.1. Riemann-Liouville fractional derivative of order

Definition 1.2. Caputo fractional derivative of order

Definition 1.3. The Gr

where

Let

Therefore, the

Using Newton’s binomial theorem, we know

We can proof the following form:

Theorem 1.4. If

and, the Gr

Proof 1. For the proof, please refer to [49].

From Theorem (1.4), it follows that, if

In [24–27], in order to obtain higher precision, author given the construction method of generating function for arbitrary p, and then give a recursive method of fractional derivative and integral based on the generating function.

Definition 2.1. A

Theorem 2.2. The p order polynomial function

where

Proof 2. It can be seen from Eqs. (13) and (14)

Substituting

Multiply both ends of Eq. (16) by z and find the first derivative of z, then

Then substitute

Multiply both ends of Eq. (16) by z, and then find the first derivative of

Substituting

Repeating the above process, the following equation can be established:

The matrix form of the equation is formula (15), which is proved by the theorem.

Definition 2.3. Ihe

Theorem 2.4. If the generating function

where the subsequent coefficient

Proof 3. Let

where, if

Both sides of Eq. (24) are multiplied by

Substituting Eq. (23) into Eq. (25), we can get

Using the

then its right end can be written

By comparing the same square coefficient of z in Eq. (27), we can get

Move Eq. (29) back one step, let

If

where, when

Corollary 2.5. The

where

The

Applying (34), an approximate computation scheme of the

and

So, a high-precision numerical approach of the fractional-order Lorenz chaotic systems (1) is given by

In this section, some numerical examples are studied. Some novel chaotic behaviors are shown. We consider the systems (1) with the initial conditions

In this paper, some complex dynamic behavior of fractional-order Lorenz chaotic systems are shown by using the present method. We observe many novel dynamic behaviors in numerical experiments which are unlike any that have been previously discovered in numerical experiments or theoretical studies. We investigate the influence of

All computations are performed by the MatlabR2017b software.

Acknowledgement: The authors would like to express their thanks to the unknown referees for their careful reading and helpful comments.

Funding Statement: This paper is supported by the Natural Science Foundation of Inner Mongolia [2021MS01009] and Jining Normal University [JSJY2021040, Jsbsjj1704, jsky202145].

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

Dai, D., Li, X., Li, Z., Zhang, W., Wang, Y. (2023). Numerical Simulation of the Fractional-Order Lorenz Chaotic Systems with Caputo Fractional Derivative.*CMES-Computer Modeling in Engineering & Sciences, 135(2)*, 1371–1392.