This work investigates chaos synchronization between two different fractional order chaotic systems of Lorenz family. The fractional order Lü system is controlled to be the fractional order Chen system, and the fractional order Chen system is controlled to be the fractional order Lorenz-like system. The analytical conditions for the synchronization of these pairs of different fractional order chaotic systems are derived by utilizing Laplace transform. Numerical simulations are used to verify the theoretical analysis using different values of the fractional order parameter.
Fractional calculus has been known since the early 17th century [
The fractional
order derivatives have many definitions; one of them is the Riemann-Liouville
definition [
However, the
most common definition is the Caputo definition [
On the other hand, chaos has been studied and developed
with much interest by scientists since the birth of Lorenz chaotic attractor in
1963 [
The generalized synchronization between
two different fractional order systems is investigated in [
The fractional order Chen system isgiven as follows:
Chaotic attractor of the fractional order Chen
system (
The fractional order Lü system is
given as follows
Chaotic attractor of the fractional order Lü
system (
The fractional
order Lorenz-like system [
Chaotic attractor of the fractional order
Lorenz-like system (
It should be
also noted that, the systems (
Consider the master-slave (or drive-response) synchronization scheme of two autonomous different fractional order chaotic systems:
In this subsection, the goal is to achieve chaos synchronization between the fractional order Chen system and the fractional order Lü system by using the fractional order Chen system to drive the fractional order Lü system. The drive and response systems are given as follows:
By subtracting (
Now, by letting
By taking the Laplace transform in both sides of (
If
Rewrite (
Using the final value theorem of
the Laplace transform, it follows that
Since
Consequently, the synchronization
between the drive and response systems (
An efficient
method for solving fractional order differential equations is the
predictor-correctors scheme or more precisely, PECE (Predict, Evaluate, Correct,
Evaluate) technique which has been investigated in [
Based on the
above mentioned discretization scheme, the drive and response systems (
Synchronization errors of
the drive system (
In this case it is assumed that, the fractional order Lorenz-like system drives the fractional order Chen system. The drive and response systems are defined as follows:
By subtracting (
Now, by choosing
Take Laplace transform in both sides of (
If we assume
that
Thus, the states of the drive
system (
Numerical
simulations are carried out to integrate the drive and response systems (
Synchronization errors
of the drive system (
Chaos synchronization between two different fractional order chaotic systems has been studied using linear control technique. Fractional order Chen system has been used to drive fractional order Lü system, and fractional order Lorenz-like system has been used to drive fractional order Chen system. Conditions for chaos synchronization have been investigated theoretically by using Laplace transform. Numerical simulations have been carried out using different fractional order values to show the effectiveness of the proposed synchronization techniques.
The author wishes to thank the reviewers and the associate editor for their careful reading and efforts and for providing some helpful suggestions. Also I wish to thank Professor E. Ahmed and Dr. Faycal Ben Adda for discussion and help.