Observer based synchronization of chaotic systems

Abstract

We show that the synchronization of chaotic systems can be achieved by using the observer design techniques which are widely used in the control of dynamical systems. We show that local synchronization is possible under relatively mild conditions and global synchronization is possible if the chaotic system can be transformed into a special form. We also give some examples including the Lorenz, the Rössler systems, and Chua's oscillator which are known to exhibit chaotic behavior, and show that in these systems synchronization by using observers is possible.

An observer looks at synchronization

“…Expressing (39) in terms of the equivalent coordinates we obtain (40) The subsystem formed by is linear, time-invariant and can be rendered stable by selecting appropriately (e.g., and ). It is therefore suitable to construct a reduced observer.…”

“…In [34], the following hysteretic circuit is described: (39) The parameters and are all assumed to be positive. The output (described in [8] as the drive signal) is given by…”

On the Stabilization of Periodic Orbits for Discrete Time Chaotic Systems by Using Scalar Feedback

“…We note that the form given above is also called Brunowsky canonical form and has an important application in chaos synchronization, see e.g. [Morgül & Solak, 1996]. Let p s (λ) be an arbitrary but stable polynomial given as…”

Robust Control Approaches for Synchronization of Biochemical Oscillators

“…In particular, from control theory perspective, there are basically two ways that are used for synchronization of nonlinear systems. The first is related with observer based synchronization which is applied for coupling identical systems (i.e., same structure and order) and different initial conditions (Alvarez-Ramirez et al, 2002;Nijmeijer & Mareels, 1997;Morgul & Solak, 1996). In these cases, identical synchronization is reached which implies the coincidence of the states of the coupled systems.…”

“…Different approaches have been used to synchronize individual biochemical oscillators (Afraimovich et al, 1997;Boccaletti et al, 2002;Canavier et al, 1999;Collins & Stewart, 1994;Goldbeter, 1996;Mirollo & Strogatz, 1990;Morgul & Solak, 1996;Nijmeijer & Mareels, 1997;Pikovsky et al, 1996;Zhou et al, 2008). Classical synchronization approaches includes different coupling approaches and the periodic modulation of an external forcing (periodical or noisy).…”