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In this paper, a novel three-dimensional fractional-order chaotic system without equilibrium, which can present symmetric hidden coexisting chaotic attractors, is proposed. Dynamical characteristics of the fractional-order system are analyzed fully through numerical simulations, mainly including finite-time local Lyapunov exponents, bifurcation diagram, and the basins of attraction. In particular, the system can generate diverse coexisting attractors varying with different orders, which presents ample and complex dynamic characteristics. And there is great potential for secure communication. Then electronic circuit of the fractional-order system is designed to help verify its effectiveness. What is more, taking the disturbances into account, a finite-time synchronization of the fractional-order chaotic system without equilibrium is achieved and the improved controller is proven strictly by applying finite-time stable theorem. Eventually, simulation results verify the validity and rapidness of the proposed method. Therefore, the fractional-order chaotic system with hidden attractors can present better performance for practical applications, such as secure communication and image encryption, which deserve further investigation.

Since Lorenz system [

In recent years, however, increasing attention has been paid to hidden attractors, which were discovered first in the Chua’circuit [

However, the existing research findings about hidden attractors have been mostly focused on integer-order dynamical systems and relatively few efforts have been done for fractional-order systems with hidden attractors [

Based on the above issues, this paper discusses the fractional-order form of a new 3D symmetric system without equilibrium and gives emphasis to the influence of the order on dynamic behavior of the system. The detailed dynamic analysis was carried out, mainly including existence of multiple attractors. And a finite-time synchronization of the fractional-order chaotic system with hidden attractors is realized. It has enormous value for the application of chaos technology in practical engineering field, especially in secure communication.

This paper is organized as follows. In Section

So far, a few different definitions [

In definition (

Sprott A system [

The chaotic attractor of the fractional-order system (

3D Poincaré map of the fractional-order system (

To further confirm the chaotic motion, by the Benettin–Wolf algorithm, the finite-time local Lyapunov exponents on the time interval

Dynamics of the finite-time local LEs of the fractional-order system (

In numerical simulations, the Lyapunov exponents may differ significantly from different trajectories and only a finite-time interval can be considered for their computation. Therefore, we follow the concept of the finite-time local Lyapunov exponents in the article.

For system (

While

Bifurcation diagram can help us observe the further dynamic behaviors of the fractional-order system (

Bifurcation diagram with the variable parameter

Largest finite-time local LEs on the time interval

Coexisting attractors with initial conditions

When the parameters and fractional-order are chosen as

Bifurcation diagram with the variable parameter

Dynamic map of Largest finite-time local LEs on the time interval

In Figure

Bifurcation diagram with the variable fractional-order

Cross-section for

Coexisting chaotic attractors of the fractional-order system (

Coexisting limit cycles of the fractional-order system (

Cross-section for

Nevertheless, there are other cases of coexisting attractors. With order chosen as

Coexistence of a chaotic attractor and a limit cycle of system (

Cross-section for

Coexistence of a chaotic attractor and a limit cycle of system (

Circuit implementation of the fractional-order chaotic system plays a crucial role in actual application, which can also verify whether the results obtained by theory analysis and numerical computation are consistent and correct or not. Based on PSpice software, the schematic of the fractional-order chaotic system with order

The electronic circuit schematic of the fractional-order system (

Therein, with the proposed transfer function approximations method [

With the Kirchhoff’s circuit laws, the mathematical model of system (

Hidden chaotic attractors of the fractional-order system (

There is a wide variety of literature [

For the fractional-order system in general, if it satisfies

Making allowances for the internal disturbances

The disturbances

Afterwards, we define the error system as

It can realize the synchronization of the master system (

According to (

By reference to Lemma

Therefore, from Theorem

The numerical simulations are carried out to verify the effectiveness of the proposed scheme. The disturbances

Synchronization errors between the master and slave systems.

Synchronization results of the master and slave systems: (a)_{1}; (b)_{1}; (c)_{1}.

In this paper, a novel three-dimensional symmetric fractional-order system without equilibrium, which can generate various hidden coexisting attractors, has been introduced. We have analyzed dynamical characteristics of the fractional-order system in detail by numerical simulations and found the system presents abundant complex dynamic characteristics. Especially, through bifurcation diagram, the basins of attraction, and phase diagram, the influence of the order on dynamic behavior of the fractional-order system has been explored in depth. With the approximations of fractional-order integrator

The data used to support the findings of this study are included within the article.

There are no conflicts of interest regarding the publication of this paper.

This project was supported by the National Natural Science Foundation of China (Grant no. 51877162).