The Anti-synchronization and Circuit Simulation of Hyperchaotic Systems with Uncertain Parameters

Shu-hua WU*, Zhen-yong LIU, Zhi-chun MA and Wei-de MENG

Department of Physics and Electrical Information Engineering, Shijiazhuang University, Shijiazhuang, Hebei, China, PRC, 050035

*Corresponding author

Keywords: Hyperchaotic system, Anti-synchronization, Parameter identification, Circuit simulation.

Abstract. In this paper, the hyperchaotic system in the same structure with unknown parameters are anti-synchronized using the active control technique. The designed controller ensures the full anti-synchronization between the drive and the response systems in state variables and the update rule of the unknown parameters ensures the identification of parameters. In addition, the drive and the response system as well as the identification circuit for unknown parameters are established and a series of circuit simulations are completed. The highly consistency between the numerical simulation and the circuit simulation has proved the validity of the scheme.

Introduction

Chaos synchronization research has caught great attention since 1990 due to its potential applications in many fields, such as chemical reaction [1], biological systems, communications and neural network [2]. Tracing the history thoroughly, Pecora introduced the notion of chaos synchronization control for the first time; in the following years, a variety of chaos synchronization control methods were found, for example, the generalized projective synchronization, phase synchronization, anti-synchronization, generalized synchronization and lag synchronization. Anti-synchronization is very interesting. When two anti-chaotic systems achieve the anti-synchronization state, the absolute values of the two chaotic state variables are equal, but their signs are opposite, i.e., x(t)=-y(t). Up to now, a great many chaos synchronization control methods have been proposed, for example, the linear feed-back method [3], the adaptive synchronization [4-5], the active synchronization [7], the back-stepping nonlinear synchronization method [8], the projective synchronization [9] and the modified projective synchronization [10].

The previous literature mainly focused on the synchronization of the low-dimensional chaotic system with known parameters, but less on the anti-synchronization of the high-dimensional ones with uncertain parameters and few people designed the anti-synchronous circuit simulation and the parameter-tracking circuit simulation. In fact, some or all of the parameters in many actual systems are uncertain, and thus the chaos synchronization researches are becoming more and more difficult because of some uncertain parameters [11, 12]. The hyperchaotic system has two or more positive Lyapunov exponents. Its dynamic behaviors is more complicated and even more unpredictable compared to the common chaotic systems. It has a higher practical value in the field of information security and communication encryption. Therefore, how to achieve the synchronization of hyperchaotic system is a challenging problem [13,14].

In this chapter, on the basis of Lyapunov stability theory, a synchronization controller is designed using the active control method to ensure the anti-synchronization of the response and the drive systems without knowing the parameters; the track rule for unknown parameters is designed to ensure the identification of unknown parameters; The response and drive systems as well as the unknown parameter identification circuit are established and a series of circuit simulations are completed. The numerical simulation and the circuit simulation match and consistent with each completely, which has demonstrated the effectiveness of the method.

International Conference on Modeling, Analysis, Simulation Technologies and Applications (MASTA 2019)

Copyright © 2019, the Authors. Published by Atlantis Press. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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Systems

Here, we take a nonlinear hyperchaotic system as the drive system to

),,( txfx (1) establish a response system as follows:

),,,(),( yxtutygy (2)

Where x, y∈Rn means the state vector of system, f, g∈Rn means the vector field functions, u(t, x, y)means the designed active controller. Given that e=y+x is the anti-synchronous error for achieving the anti-synchronization between x and y, the anti-synchronous error between the designed system (2) and the drive system (1) is e, that allows the response system (2) with y0 as its initial value condition and the drive system (1) with x0 as its initial value condition to approach asymptotically, namely anti-synchronization, in other words,

0 0lim lim ( , ) ( , ) 0t

e y t y x t x

, where

is the Euclidean norm.

This hyperchaotic system is derived from the Chen system [15]. Described by the following 4D autonomous system [13]:

x = a(y - x)

y = (c - a)x - xz+cy

z = -bz+ xy - yz+ xz - w

w= -dw+ yz - xz

(3)

where d is a new system parameter. When a=37, b=3, c=26 and d=38, System (3) has two positive Lyapunov exponents, indicating

that it is a hyperchaotic system with strange attractors showing complicated dynamic behaviors. The simulated results are shown in Fig. 1.

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Figure 1. The hyperchaotic attractors of System (3) when a=37, b=3, c=26 and d=38. (a) x-y-z; (b) x-y-w; (c) x-z-w; and (d) y-z-w.

The next three sections, we will study the anti-synchronization of the hyperchaotic system with unknown parameters using the active control method.

Active Control for Anti-synchronization

To make a study on the anti-synchronization about a hyperchaotic system without known parameters using active control, we take System (3) as a drive system meeting the following conditions:

1 1

1

1

1 1 1 1

1 1 1 1 1 1 1 1 1

1 1 1 1 1 1

x = a(y - x )

y = (c - a)x - x z +cy

z = -bz + x y - y z + x z - w

w = -dw + y z - x z

(4)

where a, b, c and d are parameters; a=37, b=3, c=26 and d=38. And the nonlinear autonomous

(b) (a) (c) (d)

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system in the same structure is used as the response system, where, a', b', c' and d represent the estimates of the unknown parameters of the system a, b, c and d. The system model is as follows:

2 2 2 1

2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 3

2 2 2 2 2 2 4

x = a (y - x )+u

y = (c - a )x - x z +c y +u

z = -b z + x y - y z + x z - w +u

w = -d w + y z - x z +u

(5)

where u=[u1(t), u2(t), u3(t), u4(t)]T is the active control function. Here our purpose is to design an active controller u to allow System (5) anti-synchronize with system (4). Accordingly, we get the error dynamics between System (5) and (4) as follows:

422222111144

23222222111111433

22222211212

122211

uwdzxzyzxzydee

zbuzxzyyxzxzyyxebee

uxayxczxzxceeace

uxyaaeaee

)()(

)(

(6)

where e1=x2+x1, e2=y2+y1, e3=z2+z1, e4=w2+w1. To study how to make active control on hyperchaotic anti-synchronization, the following theorem

is deduced:

Theorem

Considering the following nonlinear controller:

222211114

22222211111143

2211122

1

1

0

zxzyzxzyu

zxzyyxzxzyyxeu

zxzxceecu

u

)( (7)

and estimating the tracking rule for parameter a', b', c' and d:

2 2 2 2 1

2 3

2 2 2

2 4

( )

( )

a x e y x e

b z e

c x y e

d w e

(8)

and then, the drive system (4) and the response system (5) will be asymptotically anti-synchronized gradually in state variables.

Proof

Choosing the positive definite Lyapunov function

)~~~~( 22222

423

22

212

1dcbeeeeV (9)

Where , , ,a a a b b b c c c and d d d . a', b', c' and d' are the estimated values of the uncertain parameters a, b, c and d respectively, then, we get the following equation by computing the derivative for the Lyapunov function:

ddccbbeeeeeeeeV ~~~~~~~~ 44332211 (10)

Substituting Equation (7) and (8) into Equation (10), following equation is achieved:

024

23

22

21 debeeaeV (11)

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On the basis of the Lyapunov stability principle, we get

,,,, 04321 eeee 0dcba~

,~,~

,~ (12)

when t , the two hyperchaotic systems in the same structure (4) and (5) will achieve the anti-synchronization progressively, although the parameters of System (5) are unknown. Consequently, the asymptotic anti-synchronization of the two hyperchaotic systems is proved.

Numerical Simulation

In the numerical simulation, we verfy the feasibility of the designed controller u and the parameter updating laws. We set a=37, b=3, c=26 and d=38 in System (4) to ensure the system is a hyperchaotic one. Employ the Runge-Kutta integration method to make the computation with the time step h=0.01. The initial conditions for the drive system are (x1(0), y1(0), z1(0) and w1(0))=(-6, -7, -8, -9); the initial conditions for the response system are (x2(0), y2(0), z2(0), w2(0))=(6, 7, 8, 9); the estimating values of the parameters are (a'(0), b'(0), c'(0), d'(0))=(1, 2, -3, -4) initially; and activat the controllers at the time when t=0. Fig. 2 (a)-(b) are the phase diagrams of the response and the drive systems when they are completely anti-synchronized; (c)-(d) are the time-state variables of the drive and the response systems when they achieve the anti-synchronization. The symmetrical patterns indicate that the full anti-synchronization between the two hyperchaotic systems has been achieved with the designed controller. Fig. 2(e) shows the process that e1, e2, e3 and e4 evolve to 0 gradually after the controller is switched on and the hyperchaotic system achieves the anti-synchronized globally when t>30s. Fig. 2(f) shows the identification process of the unknown parameters. Obviously, with the evolution of time t, the 4 estimates are approaching a'=a=37, b'=b=3, c'=c=26 and d'=d=38 gradually, while the response system is getting to be completely anti-synchronized with the drive system.

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Figure 2. The phase digrams and state variables of the drive and the response systems when they are at anti-synchronization state. (a) z1-w1 and z2-w2; (b) x1-z1 and x2-z2; (c) x1, x2-t; (d) z1, z2-t; (e) the evolutions of the error

systems e1, e2, e3 and e4; and (f) the identification of unknown parameters.

Circuit Experiment

Circuit Design

The actual circuit simulation of the hyperchaotic system is carried out using the Multisim11.0 analog electronic circuit which comprises the multiplier electronic devices AD633, the operational amplifiers LF353, several linear resistances and capacitances; and the four state variables X1, Y1, Z1 and W1 are constructed into a new hyperchaotic system circuit. The power supply of the operational amplifier is the ±12V dual supply. To avoid the distortion of the circuit simulation, the output voltage of the system is adjusted to one tenth (1/10), so the multiplier gain is set to 0.1V/V.

The idea for designing the anti-synchronization response circuit for the hyperchaotic system is the same as the one for its parameter identification circuit and drive circuit. It is also to turn the output voltages of signals, x2, y2, z2 and w2 into the 1/10 of variables in Equation (4). The multipliers are all AD633 and the gain is set to 0.1V/V in the circuits, namely. The designed circuit diagrams are shown in Fig. 3.

(b) (a) (c) (d) (e) (f)

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(a) The drive system (4); (b) The response system (5); (c) The circuit diagrams for the estimate parameters a', b', c'

and d'

Figure 3. Circuit diagrams

Circuit Simulation

The circuit in Fig. 3 is simulated using the Multisim 13.0 software and the chaotic attractors and the oscillation curves for the two systems in Fig. 4(a) and 4(b) are achieved. Obviously, being coincide with the numerical simulation diagrams in Fig. 2(a-d), the phase diagrams and the sequence diagrams are symmetrical. And the anti-synchronization curves for the circuit simulation (Fig. 5) have further confirmed that the drive and the response systems have reached anti-synchronization.

Figure 4. The simulated chaotic attractors and state variable curves for the two hyperchaotic systems when they are anti-synchronized. (a) z1-w1 and z2-w2 (2v/div,1v/div); (b) x1-z1 and x2-z2 (1v/div,5v/div); (c) x1, x2-t (0.5ms/div,2v/div);

and (d) z1, z2-t (0.5ms/div,5v/div)

Figure 5. The anti-synchronization curves of the hyperchaotic system (4) and (5). (a) x1-x2 (2v/div); (b) y1-y2 (2v/div); (c) w1-w2 (5v/div); (d) z1-z2 (5v/div).

Figure 6. The identification of the estimated parameters a', b', c' and d'. (a) The estimated parameter a' (2ms/div, 2v/div); (b) The estimated parameter b' (5ms/div, 200mv/div); (c) The estimated parameter c' (10ms/div, 2v/div); and (d) The

estimated parameter d' (10ms/div, 2v/div).

Finally, the transient response of the uncertain parameters in the circuit is simulated and the

identification process of these parameters are shown in Fig. 6: the steady-state values of the estimated value a', b', c' and d' are 3.7V, 300mV, 2.6V and 3.8V respectively. The simulation data

(a) (b) (c) (d)

(a) (b) (c) (d)

(a) (b) (c) (d)

(a)

(b)

(c)

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coincide with the theoretical values, which further proves that the designed parameter updating rule is correct.

Conclusion

Under the condition without knowing the system parameters or with unknown parameters, active controllers are designed based on the Lyapunov stability principle so that two hyperchaotic systems accomplish the anti-synchronization, while the identification of the unknown parameters are completed. The method is simple and practical and its control effect is very satisfactory. The response and drive systems as well as the unknown parameter identification circuits are designed and a series of circuit simulations are carried out using the Multisim software. The simulation results are ideal. The circuit simulated and the numerical simulated are identical in phase diagrams, time-sequence diagrams and the unknown parameter identifications. The ant-synchronization curves prove that the two hyperchaotic systems have fully reached anti-synchronization state. The numerical and the circuit simulation results verify the effectiveness of the proposed method jointly.

Acknowledgement

This research was supported by the Hebei Provincial Science and Technology Plan Self-funded Project (No. 18210912, 18214320).

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