1.1 Pecora and Carroll Method
Let us consider the system
(1)By dividing system (1) into two parts arbitrarily as
Where
xD are the drive subsystem variables,
, and
xR are the response subsystem variables,
. Now, equation (1) is written as
(2) where
and
If we make a copy of the response subsystem called 
which driven by the variables xD (from the original system), it follows that; the system
(3) drives the response system
(4)Following the suggestion of Pecora and Carroll, as the variation of the time the 
variables will converge asymptotically to the xR 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
(5) where
is the Jacobian of the response vector field with respect to the response variables and
represents the higher order terms. For small
equation (5) becomes:
(6)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
(7)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 xD, 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:
(8) where
are the eigenvalues of Z(t), each associated with an eigenvector
(Pecora & Carroll, 1991).
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 xR are subsystems under perfect synchronization. Hence, if we define 
then 
.
Assume that the subsystem is linear, then
(9) where J is a (m×m) constant matrix. The real parts of the eigenvalues of the matrix J defined in equation (9), are the conditional Lyapunov exponents (CLEs) we seek.
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:
(10)The parameters 
are all positive real parameters and 
. System (10) exhibits chaotic behaviors at the parameter values 
and 
(see Figure 1).
Figure 1. Phase portrait of system (10), showing the double-band chaos, using the parameter values 
and 
.
1.1.1 Pecora and Carroll Method for MADVP System
The x-drive system is given by equations (10) will be used to drive the following response system:
(11)The difference system for 
and 
in matrix form is
(12) Or in vector notation
where
and
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 follows
It is clear that the term 
is always less than 
then we have two cases;
Case (i) if
This means that the two roots will be real and negative and the system with x-drive configuration does synchronize.
Case (ii) if
This 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.
1.1.2 Numerical Results
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).
Figure 2. (a-b) shows that the synchronization error tends to zero using Pecora and Carroll method
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).
Table 1. Conditional Lyapunov exponents for various drive-response configurations for the MADVP system with the parameter set
,
.
| Drive | Response | CLEs |
| x | (y, z) | -0.7999, -0.8004 |
| y | (x, z) | 0.0021, -12.053 |
| z | (x, y) | +6.0177, -19.4448 |