Synchronization of periodic oscillations has been known to scientists since the historical observation of this phenomenon by Huygens in pendulum clocks. With the development of radio and electronics in this century, synchronization occupied a very special place in science and technology. As many phenomena studied by nonlinear dynamics, synchronization was observed and shown to play an important role in many problems of a most diverse nature (physical, ecological, physiological, meteorological, to name a few). There is hardly a single communication or data storage application that does not rely on synchronization.

The discovery of deterministic chaos introduced a new kind of an oscillating system, a chaotic generator. Intuitively it would seem that chaos and synchronization are two mutually exclusive terms. Yet it has been shown that synchronization can be observed even in chaotic systems. However, the special features of chaotic systems make it impossible to directly apply the methods developed for synchronization of periodic oscillations. Even defining the notion of synchronization for chaotic systems is difficult without running into a paradox or controversy.

It is the objective of this course to introduce students to the basic concepts and philosophy of nonlinear dynamics and to apply this philosophy to the analysis of synchronization problems in physical, electrical and biological systems. The first part of the course is dedicated to synchronization of periodic oscillations, including the classical example of synchronization in Van der Pol generator, synchronization in relaxation oscillators and phase lock loops. The second part of the course introduces the notion of chaos and discusses various cases of synchronized chaotic oscillation, such as identical synchronization, generalized synchronization and phase synchronization. The examples used here are based on electronic circuits and biological systems with an emphasis on applications, in particular on the use of chaos in secure communications. The course will emphasize the use of analytical and semi-analytical methods in studying synchronization.
 
 

1. Introduction. Synchronization phenomena in nature and technology. Elements of oscillation theory: phase space, classification of equilibrium states on a phase plane.

2. Introduction to bifurcations on a phase plane.

3. Parametric resonance. Discrete maps. Floquet theorem. Floquet multipliers and characteristic exponents.

4. Passive oscillator under the action of an external periodic force:linear and nonlinear resonance.

5-6. Self-sustained oscillations in autonomous dissipative systems. Limit cycles. Synchronization. Arnold tongues.

7. Systems with cylindrical phase space. Dynamics of phase lock loops.

8. Synchronization in large ensembles of periodic oscillations.

9-10. Relaxation oscillators. Synchronization in integrate-and-firesystems.
 
 

1-2. Introduction to chaos. Chaotic oscillations in nature and technology. Transition to chaos: period doubling, quasiperiodicity, intermittency.

3. Chaos in electronic circuits. Experimental methods in chaos studies.

4-6. Synchronization between identical chaotic systems. Synchronization manifold. Stability of synchronization manifolds. Conditional Lyapunov exponents. On-off intermittency. Riddled basins of attraction.

7. Synchronization between non-identical chaotic systems. Generalized synchronization.

8. Phase synchronization of chaotic oscillations.

9-10. Examples and applications of synchronized chaos. Chaos in communications.Synchronization in neurobiology.