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Ahmed E. Matouk (Mansoura University, Egypt; Hail University, Saudi Arabia)

Source Title: Chaos Synchronization and Cryptography for Secure Communications: Applications for Encryption

Copyright: © 2011
|Pages: 25
DOI: 10.4018/978-1-61520-737-4.ch007

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TopLet us consider the system

By dividing system (1) into two parts arbitrarily as

WhereIf we make a copy of the response subsystem called which driven by the variables *x _{D}* (from the original system), it follows that; the system

Following the suggestion of Pecora and Carroll, as the variation of the time the variables will converge asymptotically to the *x _{R}* variables and remain with it in step instantaneously. Our goal is to make the response system a stable subsystem in order to obtain synchronization. Suppose we have trajectory with initial condition then if a trajectory started at a nearby point at t = 0, then the conditions needed for synchronization implies that

This leads to

In Ref. (10), Pecora and Carroll used a matrix Z in place of in Eqn. (6), such that Z(0) is equal to identity matrix. Thus

The solution Z(t) is called the *transfer function* or the *principal matrix solution* (Pecora & Carroll, 1991). Z(t) will determine whether perturbations will grow or shrink in any particular directions. Using the principal matrix solution Z(t) as we can determine the Lyapunov exponents for subsystem for a particular drive trajectory *x _{D}*, which help us to estimate the average convergence rates and their associated directions.

Now, calculating Z(t) for large t from the variational equation (7), the Lyapunov exponents are given by:

**Definition 1.**The Lyapunov exponents for subsystem (see equation (4)) are called sub Lyapunov exponents or conditional Lyapunov exponents (CLEs).

Now, we have the following theorem:

**Theorem 1.****(Pecora and Carroll)**According to Pecora and Carroll, the coupled chaotic systems can be regarded as drive and response systems which will perfectly synchronize only if CLEs are all negative (Carroll & Pecora, 1991; Pecora & Carroll, 1990).

The above theorem is necessary, but not sufficient condition for synchronization. It says nothing about the set of initial conditions between the drive and response systems.

However, synchronization can be achieved even with positive CLEs (Güémez et al, 1997). **Intermittent synchronization** can occur when CLEs are very small positive or negative values close to zero, while **permanent synchronization** occurs when CLEs take sufficiently large negative values.

Now, suppose that and *x _{R}* are subsystems under perfect synchronization. Hence, if we define then .

Assume that the subsystem is linear, then

In the following we are going to apply Pecora and Carroll method to a modified autonomous Duffing-Van der Pol system (Matouk & Agiza, 2008; Matouk, 2005), and known here as MADVP. This system is described by the following equations:

The parameters are all positive real parameters and . System (10) exhibits chaotic behaviors at the parameter values and (see Figure 1).

The x-drive system is given by equations (10) will be used to drive the following response system:

The difference system for and in matrix form is

The real parts of the eigenvalues of the matrix J are the conditional Lyapunov exponents (CLEs). Then the response system synchronizes if all the real parts of the eigenvalues of J are negative.

The characteristic equation of (12) is

then the roots are given as followsIt is clear that the term is always less than then we have two cases;

Case (i) ifThis means that the two roots will be real and negative and the system with x-drive configuration does synchronize.

Case (ii) ifThis is the case used in observing the chaotic motion of this system. The eigenvalues will be

Hence all eigenvalues have negative real parts, which imply that all solutions tend to zero as t tends to ∞. The conditional Lyapunov exponents (CLEs) are and as expected, and then the response system with x-drive configuration does synchronize.

By solving the drive and response systems (10) and (11) numerically and using the above-mentioned parameter values, we find that in spite of the differences in initial conditions of and , the response system synchronizes so that for and (see Figure 2).

Now, we will calculate the conditional Lyapunov exponents (CLEs) numerically. We used a MATLAB code based on the Eckmann and Ruelle QR decomposition technique to calculate the exponents (Eckmann & Ruelle, 1985). Table (1) shows a calculation of the CLEs for various subsystems. From the table we see that, the subsystem (y, z) driven by x is stable subsystem as we expected and the numerical values of its CLEs are close to the analytical values shown above . We also deduce that synchronization can not occur for z-drive configuration because the largest CLE of the (x, y) subsystem is sufficiently positive (see Table 1).

Drive | Response | CLEs |

x | (y, z) | -0.7999, -0.8004 |

y | (x, z) | 0.0021, -12.053 |

z | (x, y) | +6.0177, -19.4448 |

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