This paper proposed a novel switching scroll hyperchaotic system based on a memristor device and explored its application to secure communication. The new system could be switched between the double-scroll chaotic system and multiscroll one by switch
Chaotic systems based on memristor have widely attracted attention recently. It has many applications such as in secure communication [
With respect to the chaotic system based on memrister, its application to secure communication has been studied in many works, such as in [
The rest of the paper was organized as follows. Section
The Chua’s circuit is composed of a linear resistor, a linear inductor, two linear capacitors, and a nonlinear Chua’s diode as shown in Figure
Chua’s circuit diagram.
Based on Figure
A projection of the double-scroll attractor onto the
In Figure
The new hyperchaotic memristive circuit.
The flux-controlled memristor could be expressed as
And basic model of the memristor can be found in [
According to the definition of memristor, we obtain a novel expression as
Projections of the double-scroll hyperchaotic attractors onto the
There are two forms of step function, when
When
N-scroll could be obtained by a reasonable set of step function parameters. For example, when
The 2 × 3 multiscroll hyperchaotic system based on a memristor device is obtained as
When system parameters are
Projections of the 2 × 3 multiscroll hyperchaotic attractors onto the
Similarly, 2 × 5 and 2 × 6 multiscroll hyperchaotic system could be also obtained by taking
When selecting system parameters
Projections of the 2 × 5 multiscroll hyperchaotic attractors onto the
Projections of the 2 × 6 multiscroll hyperchaotic attractors onto the
Lyapunov exponents diagram.
From Figure
Lyapunov exponents and bifurcation diagram with respect to parameter
The equilibrium points of the system could be calculated by making (
The Jacobian matrix at point
Eigenvalues at point
The Equilibria
The dynamic analysis of 2 × 3 multiscroll hyperchaotic system is introduced briefly as follows. Let
From (
Equilibrium points and attractors of the 2 × 3 multiscroll hyperchaotic system. (a) Equilibrium points. (b) 2 × 3 multiscroll attractors.
Circuit diagram of the double-scroll hyperchaotic system is displayed in Figure
The memristor and its internal circuit. (a) The memristor. (b) Its internal circuit.
Circuit implementation of the double-scroll hyperchaotic system. (a)
Symbolic function circuit is designed by the saturated output voltage of the operational amplifier to achieve step function. The saturated voltage of the operational amplifier in Figure
Step function circuit.
The step function circuit designed by the method is easy to expand and achieve, that is, more complex function could be realized by increasing the number of corresponding comparison circuit. In the following part, we choose the 2 × 3 scroll system as an example to discuss the multiscroll system. Based on Figure
Circuit diagram of the multiscroll hyperchaotic system.
Circuit implementation of the multiscroll hyperchaotic system. (a)
The synchronization of the switching systems is briefly stated as follows. The drive system from (
And the response system is
The state error is defined as
The synchronization controller of the response system is designed as
According to the systems (
Let
Next, we illustrate the validity of the proposed controller by MATLAB 2010a. Selecting initial values
The state trajectories of the error system. (a) State trajectories of
Drive-response synchronous circuit of the multiscroll hyperchaotic system is demonstrated in Figure
Drive-response synchronous circuit of the multiscroll hyperchaotic system.
The
The
The
The drive-response synchronous circuit of the double-scroll hyperchaotic system is similar to that of the multiscroll one and will be omitted here.
In the circuit implementation, the double-scroll and multiscroll circuits are encapsulated, respectively, so as to simplify the circuit structure.
In this subsection, the synchronization circuit above is applied to secure communications. A voice signal
The chaotic synchronous circuit given above could be applied to extract the message at the receiver. Some strategies could be used to make the actual transmitted signal
The switching encryption circuit is shown in Figure
The switching encryption circuit.
Encryption and decryption flow chart.
Results of encryption and decryption are displayed in Figure
Encrypted and decrypted waveforms. (a) Original voice signal. (b) Encrypted voice signal. (c) Decrypted voice signal.
Moreover, the decrypted voice signal and the original voice signal are identical, indicating that the original signal could be recovered well by the chaotic signal generated by the response system. This proves the effectiveness and reliability of the encryption method.
Wrong decryption result.
Comparison of encryption speed between different algorithms.
Algorithms | Encryption speed (MB/s) |
---|---|
Algorithm of [ |
4.80 |
Algorithm of [ |
6.21 |
The proposed algorithm | 7.91 |
In this paper, a new switching scroll hyperchaotic system was proposed, which consisted of the double-scroll and multiscroll hyperchaotic system. Based on a memristor device, the double-scroll hyperchaotic system was taken as an example to analyse the dynamic properties, such as Lyapunov exponents, bifurcation diagram, dissipation and equilibrium points, and stability. Furthermore, circuit implementation of the system was introduced in detail and the results are consistent with those of numerical simulations. At last, the switching scroll hyperchaotic system was successfully applied to voice encryption and showed its effectiveness.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
The work was supported by the National Natural Science Foundation of China (no. 31700644), Postdoctoral Science Foundation of China (nos. 2015M582122 and 2016T90644), Key Research and Development Project of Shandong Province (nos. 2016ZDJS02A07 and 2017GNC12105), the Agricultural Machinery Research and Development Project of Shandong Province (2018YF004).