^{1}

^{2}

^{2}

^{2}

^{1}

^{2}

A 3D fractional-order nonlinear system with coexisting chaotic attractors is proposed in this paper. The necessary condition of the existence chaos is

Chaotic behaviors in nonlinear is a very interesting phenomenon. The high irregularity, unpredictability, and complexity in chaotic systems [

On the other hand, the fractional-order differential equations [

Motivated by the above discussions, based on a 3D autonomous continuous chaotic system reported by Zhou and Ke [

The outline of this paper is organized as follows. In Section

In this paper, the Caputo definition of the fractional derivative will be used in next. The Caputo definition of the fractional derivative is described as

Next, based on the 3D autonomous continuous chaotic system reported by Zhou and Ke [

Now, we can obtain the eigenvalues of the five equilibrium points. The eigenvalues of equilibrium point

Tavazoei and Haeri [

Now, we can obtain the necessary condition of the existence chaos in fractional-order system (

In this paper, the improved version of Adams-Bashforth-Moulton [

The error of this IVABM numerical algorithm is

We can yield the largest Lyapunov exponent varying as

The largest Lyapunov exponent varying as fractional order

According to Figure

The chaotic attractor in fractional-order system (

Same as the results reported by Zhou and Ke [

(1) If the initial point (initial condition) is near the unstable

(2) If the initial point (initial condition) is near the unstable

Next, some numerical simulations are given for

For example, choose the initial conditions as (

The “positive attractor” and “negative attractor” in fractional-order system (

For example, choose the initial conditions as (

For example, choose the initial conditions as (

The “positive attractor” and “negative attractor” in fractional-order system (

For example, choose the initial conditions as (−4, −3, −2). Therefore,

According to Figures

There are overlaps between the coexisting chaotic attractors in [

First, a result on stability of fractional-order nonlinear system is recalled. Consider the fractional-order nonlinear system as follows:

Fractional-order nonlinear system (

where

Next, we discuss how to stable the unstable equilibrium point in fractional-order chaotic system (

Let real matrix be

Let

One can easily obtain that point

Now, controlled system (

First, it is easy to obtain that

Second, according to

Therefore, according to Lemma

In our control scheme, the linear controller is only determined by one single state variable, so our control scheme is different from many previous control schemes.

Next, in order to show the effectiveness of the proposed control approach, the numerical simulations are performed for

For the unstable equilibrium point

The time evolution of state variables

For the unstable equilibrium point

The time evolution of state variables

For the unstable equilibrium point

The time evolution of state variables

For the unstable equilibrium point

The time evolution of state variables

For the unstable equilibrium point

The time evolution of state variables

The simulative results in Figures

In this paper, a fractional-order chaotic system is proposed. The necessary condition of existence chaos in this fractional-order chaotic system is

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare no conflicts of interest.