The topics of memristive system and synchronization are two hot fields of research in nonlinear dynamics. In this paper, we introduce a memristor-based chaotic system with no equilibrium. It is found that the memristor-based system under investigation exhibits fruitful dynamic behaviors such as coexisting bifurcation, multistability, transient chaos, and transient quasiperiod. Thus, it is difficult to reproduce the accurate dynamics of the system, which is highly advantageous in encryption and communication. Then, a simple intermittent control scheme with adaptive mechanism is developed to achieve complete synchronization for the introduced system. Because the output signal is transmitted intermittently to the receiver system, more channel capacity can be saved and the security performance can be improved naturally in practical communication.

As the fourth basic circuital element along with resistor, inductor, and capacitor, the memristor was postulated by Chua in 1971 [

The memristor is currently used to design flash memory, improve neural networks, and construct chaotic circuits, for the intrinsic characteristics of memory, nanoscale device, and inherent nonlinearity. Itoh and Chua constructed the memristive chaotic oscillator in 2008, by replacing Chua’s diodes in Chua’s circuit with the piecewise linear memristor [

Meanwhile, close attention was paid to chaotic system without equilibrium [

Because of its application in secure communication, digital signal, neural network, and other fields, the synchronization of chaotic system is a fashionable subject in nonlinear science. Since the first scheme was carried out by Pecora and Carroll for the synchronization of two identical chaotic systems [

In this paper, we introduce a memristor-based chaotic system with no equilibrium. The dynamical evolution of the memristive system is studied by using phase diagram, time-domain trajectory, bifurcation diagram, and Lyapunov exponent. It is found that by changing system parameters or initial condition, the reported system exhibits different topological structures of coexisting bifurcation, multistability, transient dynamics. The coexisting hidden attractors signify that the system has fruitful and complex dynamic behaviors, which is highly advantageous in encryption and communication for the difficulty of reproducing the accurate dynamics of the system. Then, a simple intermittent control scheme with adaptive mechanism is developed to achieve complete synchronization for the introduced memristive system. Since the output signal is transmitted intermittently to the receiver system, more channel capacity can be saved and the security performance of the communication system can be improved naturally in practical communication. Theoretical analysis and illustrative examples are executed to verify the effectiveness of the proposed synchronization scheme.

Based on Sprott A system, the constructed memristive chaotic system can be described by the following differential equations:

The dissipativity is decided by

When choosing the parameters

Phase portrait projected onto the plane of (a)

It is found that the memristive system under consideration can experience rich bifurcation structures when continuously monitoring the bifurcation parameter. Also, the memristive system has completely different bifurcation behaviors when the initial conditions are set to different values.

We assign the parameters

(a) Bifurcation diagrams and (b) spectrums of maximal Lyapunov exponent versus parameter

Enlargement of the bifurcation diagrams of Figure

Coexistence of different attractors with different parameter

We also study the dynamics evolution of system (

(a) Bifurcation diagram; (b) Lyapunov exponent spectrum versus

Dynamic distributions of system (

Symmetric coexisting attractors with respect to

It is surprising to see in Figure

Firstly, the case of system parameters

(a) Time-domain waveform of

Then we take the selection of system parameters

(a) Time-domain waveform of

We consider the master-slave synchronization scheme for the introduced chaotic system, and the corresponding master-slave systems are described by the compact form:

In which,

To realize the synchronization of systems (

The synchronization error is defined by

When

In fact, to reduce the control consumption, we can optimize the controller as

The practical significance of the optimized control scheme is that one imposes the controller

We impose the controller

We first choose the initial condition of system (

(a) Phase diagram and (b) state trajectories with

(a) Time response of the states and (b) synchronization error with

Then, we also set

(a) Phase diagram and (b) state trajectories with

(a) Time response of the states and (b) synchronization error with

We know that no matter what the dynamic state of the memristive system is, the synchronization control of the memory system can be easily realized by adopting the designed method.

In this paper, we introduce a memristor-based chaotic system with no equilibrium. Various tools including phase diagram, time-domain trajectory, bifurcation diagram, and Lyapunov exponent are exploited to establish the connection between the system parameters and dynamical behaviors. It is found that the reported system exhibits complex dynamics such as coexisting bifurcation, multistability, symmetric coexisting attractors, and transient dynamics, which is helpful for the security improvement of encryption and communication due to the difficulty of reproducing the accurate dynamics. Then, a simple control scheme with single linear couple is developed to achieve complete synchronization for the memristive system. Since the output signal is transmitted intermittently to the receiver system with the adaptive mechanism, the communication channel capacity will be reserved for more message transmission. Also, the security of chaotic communication system will be improved for the reduction of redundancy of synchronization information in the channel.

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

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the Hunan Provincial Natural Science Foundation of China (no. 2019JJ40109); Research Foundation of Education Bureau of Hunan Province of China (no. 18A314); and Science and Technology Program of Hunan Province (no. 2016TP1021).

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