Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 7693547, 14 pages

http://dx.doi.org/10.1155/2016/7693547

## Finite-Time Synchronization of Chaotic Systems with Different Dimension and Secure Communication

Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, China

Received 8 May 2016; Accepted 12 July 2016

Academic Editor: Ricardo Aguilar-López

Copyright © 2016 Shouquan Pang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Finite-time synchronization of chaotic systems with different dimension and secure communication is investigated. It is rigorously proven that global finite-time synchronization can be achieved between three-dimension Lorenz chaotic system and four-dimension Lorenz hyperchaotic system which have certain parameters or uncertain parameters. The electronic circuits of finite-time synchronization using Multisim 12 are designed to verify our conclusion. And the application to the secure communications is also analyzed and discussed.

#### 1. Introduction

Synchronization of chaotic systems recently has become one of the most interesting topics because of the seminal work of Pecora and Carroll [1]. These phenomena have arisen in various fields of science, like engineering technology [2], ecological system [3], biological neurons [4], and others [5–8] various types of synchronization. They have been deeply investigated in the previous literature. A large number of results have been reported in the past two decades [9].

From a practical point of view, it will be more reasonable to realize synchronization in a given time [10–13]. This means optimality in convergence time [14]. This problem is important in all fields where synchronization finds or will find practical interest [15–17]. If we consider, for example, the application of synchronization in secure communications, the range of time during which the chaotic oscillators are not synchronized corresponds to the range of time during which the encoded message can unfortunately not be recovered or sent. More than a grave and irreversible loss of information, this is a catastrophe in digital telecommunications, since the first bits of standardized bit strings always contain signalization data, that is, the “identity card” of the message. Hence, it clearly appears that the synchronization time has to be minimized, so that the chaotic oscillators synchronize as fast as possible. In this context, it is fair to say that there is a need to study finite-time chaos synchronization and optimization problems of nonlinear systems with uncertainties.

This paper mainly addresses the problem of finite-time master-slave synchronization of the three-dimensional (3D) Lorenz system and four-dimensional (4D) hyperchaotic Lorenz system. A family of feedbacks has been proposed to achieve the finite-time synchronization of the two systems with different dimension. It is rigorously proven that global finite-time synchronization can be achieved between three-dimension Lorenz chaotic system and four-dimension Lorenz hyperchaotic system which have certain parameters or uncertain parameters. Finally, the experimental setup for the coupling process has also been designed via Multisim 12 simulations. The numerical simulation and circuit experiment show that the proposed control technique is feasible and reliable.

#### 2. Preliminary Definitions and Lemma

Finite-time synchronization means that the state of the slave system can track the state of the master system after the finite time. The definition of finite-time synchronization is given as follows:where and are two* N*-dimensional state vectors. , , and are differentiable functions in . If there exists a constant , such that , one says that when , synchronization of the and is achieved in a finite time.

Lemma 1. *Let the nonlinear systemwhere , . For any initial value , system (2) is stable in finite time .*

#### 3. Finite-Time Synchronization of Different Dimension Systems

Lorenz system is considered a paradigm, since it captures many of the feathers of chaotic dynamics. The Lorenz system is described by the following nonlinear equations:where , and are the constants. With , , and , the Lorenz system shows chaotic behavior.

By introducing a nonlinear feedback controller to the second equation of the system (3), the follow hyperchaotic system is obtained [18]:When , , , and , the four Lyapunov exponents are , , , and . The Lyapunov dimension is , and numerical simulations have verified that system (4) indeed has a hyperchaotic attractor.

##### 3.1. Finite-Time Synchronization of Systems with the Certain Parameters

This subsection deals with finite-time synchronization of Lorenz system and hyperchaotic Lorenz system. Considering any initial conditions, system (3) is regarded as the master and system (4) as the slaver.

The master system could be described bywhere , , , and , , and are the state constants and , and are state parameters of system.

Hyperchaotic Lorenz system (4) is regarded as the slaver system, and is controllers, which is determined for achieving synchronization between master system (3) and slaver system (4). Described by where is the state vector of the responder system, is a continuous smooth function and is a control function. To make the same structures between (6) and (7), a new artificial variable is introduced into system (6) and set . Subtract (6) from (7) and define the state errors asThen the error system between the Lorenz system and the controlled hyperchaotic Lorenz system could be written asThe above four equations merge into a unified form:where

Our aim is to design a suitable controller, in which we can achieve the finite-time synchronization of Lorenz system (3) and hyperchaotic Lorenz system (4). This problem can be converted to design a controller to attain finite-time stability of error system (10).

For Lorenz system (3) and hyperchaotic Lorenz system (4) define the control functions , , , and as follows:where . Substituting (12) into (10) yields

From Lemma 1, system (13) is stable in the finite time. That meansThe synchronization of Lorenz system (3) and hyperchaotic Lorenz system (4) is achieved in a finite time.

Theorem 2. *Suppose that . Let controller be defined as (12). Then error system (10) can be stabilized in finite time by the controller . That is to say, slave system (4) will synchronize with master system (3) in finite time.*

##### 3.2. Finite-Time Synchronization of Systems with Uncertain Parameters

We consider the more realistic and practical case, where some of the parameters of the master system and slave system are unknown. It is valuable because practical chaos systems are often disturbed by different factors. It is that the parameters of the systems have been disturbed.

Consider the master systemand the slave systemwhere are the increments of the parameter, which is caused by the interruption, and , , , and are the control functions. Analogously, as in Section 3.1, a new artificial variable is given and , by defining the synchronization error in the following way:and then the error system can be written asIn order to solve the finite-time master-slave synchronization problem between system equations (15) and (16), we select the Lyapunov function:Differentiation of this function with respect to time yieldsCombining (18) and (20), we haveWe defined the active control functions , , , and as follows:One obtainsFrom Lemma 1, it follows that (18) is finite-time stabilized. Thus, uncertain slave system (16) can synchronize uncertain master system (15) in finite time.

Theorem 3. *Suppose that , . Let controller be defined as (22). Then uncertain slave system (16) synchronizes with uncertain master system (15) in finite time.*

#### 4. Numerical Simulations

##### 4.1. Case of the Certain Parameters

To verify the effectiveness of the proposed finite-time synchronization method, the 4th-order Runge-Kutta algorithm is used to solve the sets of differential equations in connection with the master and slave systems. The initial values of the two systems are taken as , , and the simulation parameters are , , and , , , and . The error variables of system (4) and system (3) are shown in Figure 1. The error vector is achieved as zero, which implies that system (3) and system (4) have achieved finite-time synchronization. It is clear that synchronization time of the state variables is S, S, S, S, and S, respectively. These results are in line with . Figure 2 shows the synchronization of the Lorenz systems with the certain parameters.