INTRODUCTION
Since, the ideal of synchronizing two identical chaotic systems from different
initial conditions was first introduced by Carroll and Pecora, Chaos synchronization
has gained a lot of attention among scientists from variety of research fields
over the last few years (Carroll and Pecora, 1990, 1991;
Chen and Dong, 1998). Chaos synchronization can be applied
in the vast areas of physics and engineering science, especially in secure communication
(Kocarev and Parlitz, 1995; Murali
and Lakshmanan, 1998). In order to achieve the synchronization, a nonlinear
controller that obtains signals from the master and slave systems and manipulates
the slave system should be designed. Recently, many control methods have been
developed to achieve Chaos synchronization between two identical chaotic systems
with different initial conditions (Yassen, 2003; Liao,
1998; Fang et al., 1999; Yau
et al., 2005, 2006; Yau,
2004). However, most of these methods are only applicable to the Chaos synchronization
of two systems that are identical in every aspect and which contain only low
dimensional attractors. This is in stark contrast to many realword applications
of the technology. In fact, in systems such as laser array, biological systems
and cognitive processes, it is hardly the case that every component can be assumed
to be identical. In the area of communications security for example, the adoption
of higher dimensional chaotic systems as well as systems with more than one
positive Lyapunov exponents has been proposed for use to generate more complex
dynamics. Methods are therefore needed to synchronize chaotic systems that are
both different and are of high dimensions. Moreover, when the controller is
realized in practical physical systems, due to physical limitations of actuators,
the nonlinearities in control input do exist. The presence of nonlinearities
in control input may cause serious influence of system performance and decrease
the system response. Besides, the nonlinearity in control input may cause the
chaotic system perturbed to unpredictable results because the chaotic system
is very sensitive to any system parameters. Therefore, its effect cannot be
ignored in analysis of control design and realization for Chaos synchronization.
Thus, the derivation of controller with input nonlinearity for Chaos synchronization
is an important problem.
In the study, Ho and Hung (2002), Yassen
(2005), Zhang et al. (2006) and Agiza
and Yassen (2001) used active control techniques to synchronize two different
chaotic systems are either only concerns some low dimension chaotic systems
or the input nonlinearity is not discussed. In this case of input nonlinearity,
the applications of above method are shown by Ho and Hung
(2002), Yassen (2005), Zhang
et al. (2006) and Agiza and Yassen (2001)
are hard to achieve.
In this study, the goal is to force the two different hyperchaotic Rössler system and hyperchaotic Chen system to be synchronized even if they are subjected to input nonlinearity. The method of active sliding mode control law is applied to control the Chaos synchronization system. The technique requires two stages. The first stage is to select stable sliding surfaces for the desired dynamics and the second stage is to design a switching control law to achieve the stable sliding surfaces. Finally, numerical simulation is carried to confirm the validity of the proposed theoretical approach.
SYSTEM DESCRIPTION AND PROBLEM FORMULATION
In this study, two different hyperchaotic systems included Rössler system and Chen system are described in the follows. In order to observe the synchronization behavior in these two systems, it is assumed that the hyperchaotic Rössler system drives the hyperchaotic Chen system. Therefore, the master and slave systems are shown in the follows:
Slave system:
where, φ_{1}(u_{1}), φ_{2}(u_{2}), φ_{3}(u_{3}), φ_{4}(u_{4}) are the nonlinear control inputs attached in the slave system. Let the synchronization error vector state is:
Substitution Eq. 1 and 2 into the error state,
the error dynamic equations can be obtained as follows:

Fig. 1: 
A scalar nonlinear function φ_{i}(u_{i}(t))
inside sector [ς_{i}, ρ_{i}], i = 1, 2, 3, 4 
The
is a continues nonlinear function with φ_{i}(0) = 0 and
is inside sector [ς_{i}, ρ_{i}] (i = 1, 2, 3, 4),
i.e.,
where, ς_{i} and ρ_{i} are nonzero positive constants. A nonlinear function φ_{i}(u_{i}(t)) is shown in Fig. 1.
Now, the sliding surfaces suitable for the application can be defined as:
where, S_{i}(t)∈0R and λ_{i} is the design parameters
which can be determined later. For the existence of the sliding mode (Slotine
and Li, 1991), it is necessary and sufficient that:
and
Therefore, the following sliding mode dynamics can be obtained as:
Obviously, if the design parameters λ_{i}>0, i = 1, 2, 3, 4,
the stability of Eq. 6 are surely guaranteed, that is .
Thus, the slave system will be derived to master system by designing the appropriate
signal control inputs u_{i}(t), i = 1, 2, 3, 4. Meanwhile, it is worthy
of that the values of parameters λ_{i}>0, i = 1, 2, 3, 4, are
also relative to the speed of system response.
SLIDING MODE CONTROL LAW WITH INPUT NONLINEARITY
We choose a control law of the form:
Where:
Based on the control law (Eq. 9), the reaching condition
is guaranteed in the following theorem, that is, the proposed Eq.
9 will derive the Eq. 3 with nonlinear inputs onto the
sliding mode s(t) = 0.
Theorem 1: Consider the error dynamics Eq. 3 with input
nonlinearities. The hitting condition of the sliding mode is satisfied, if the
control u_{i}(t) is given by Eq. 9 for i = 1, 2, 3,
4.
Proof: Letting the Lyapunov function of the system be:
Then its derivative with respect to time is:
Where:
Therefore, if:
then ,
confirming the presence of reaching condition. Thus the proof is achieved completely.
Remarks: The controller designed in this study is robust. Therefore,
we can increase the value of η_{i} to overcome the effect of disturbances
which are bounded. The performance of proposed algorithm is still kept under
the disturbance.
NUMERICAL SIMULATIONS
In this simulation, the 4th order RungeKutta algorithm was used to solve the sets of differential equations related to the master and slave systems with a time grid of 0.0001. We selected the parameters of the hyperchaotic Rössler system as a_{1} = 0.25, b_{1} = 3, c_{1} = 0.5, d_{1} = 0.05 and the parameters of the hyperchaotic Chen systems as a_{2} = 35, b_{2} = 3, c_{2} = 12, d_{2} = 7, κ = 0.5. The initial values of hyperchaotic Rössler and Chen systems are x(0) = [x_{1}(0) x_{2}(0) x_{3}(0) x_{4}(0)] = [15 10 20 15], y(0) = [y_{1}(0) y_{2}(0) y_{3}(0) y_{4}(0)] = [10 15 10 5]. In the synchronization example, we selected λ_{1} = λ_{2} = λ_{3} = λ_{4} = 2 to result in stable sliding modes and the nonlinear inputs are defined as:
Furthermore, it is assumed that the slope of nonlinear sectors in these three
synchronization examples are ς_{1} = ς_{2} = ς_{3}
= ς_{4} and ρ_{1} = ρ_{2} = ρ_{3}
= ρ_{4} = 0.9 and the parameters γ_{1} = γ_{2}
= γ_{3} = γ_{4} = 5 are selected to satisfy the Eq.
11. The time responses of the hyperchaotic Chen system controlled by the
hyperchaotic Rössler system is shown in Fig. 2ad.
It can be see that the slave system synchronizes with the master system in spite
of input nonlinearity. Obviously, the synchronization errors converge asymptotically
to zero after the control is active at time t = 10 sec in Fig.
3.


Fig. 2: 
The time history of controlled hyperchaotic Rössler (x_{1},
x_{2}, x_{3}, x_{4}) and Chen (y_{1}, y_{2},
y_{3}, y_{4}) chaotic systems: (a) x_{1}, y_{1}
versus time t, (b) x_{2}, y_{2} versus time t, (c) x_{3},
y_{3} versus time t and (d) x_{4}, y_{4} versus
time t 

Fig. 3: 
The synchronization time response of error dynamics of controlled
hyperchaotic Rössler and Chen systems 
CONCLUSION
In this study, we introduced a sliding mode control technique to synchronize the hyperchaotic Rössler system and hyperchaotic Chen system. Based on Lyapunov stability theorem, an effective control method for synchronizing different chaotic systems has been proposed using variable structure design. The proposed sliding mode control enables stabilization of synchronization error dynamics to zeros asymptotically in spit of input nonlinearity. Numerical simulation results are presented to verify the effectiveness of the proposed synchronization technique. The main feature of this approach is that it gives the flexibility to construct a control law so that the control strategy can be easily extended to any dimensional chaotic systems.
ACKNOWLEDGMENT
The financial support of this research by the National Science Council of the
Repulbic of China, under Grant No. NSC 972221E269 011 is greatly appreciated.