Overview
Synchronization of 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 the 20-th 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. Original
notion and theory of synchronization implies periodicity of oscillators.
The discovery of deterministic chaos introduced a new kind of an oscillating
system, a chaotic generator. Chaotic oscillations are found in many dynamical
systems of various origins. The behavior of such systems is characterized by
instability and, as the result, limited predictability in time. 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.
Clear understanding of chaos synchronization phenomena and dynamical mechanisms
behind it opens new opportunities both for applications of chaotic signals in
engineering and for understanding functionality of neurobiological networks,
where irregular (chaotic) dynamics of neurons occurs naturally. This research
activity focuses on the developments of theoretical foundations for chaos
synchronization based on the experimental studies of synchronization phenomenon
in physical, neurobiological and other systems.
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Publications
N.F. Rulkov and V.S. Afraimovich Detectability of non-differentiable generalized synchrony.
submitted to PRE, 2002
V. Afraimovich, A. Cordonet and N. F. Rulkov. Generalized synchronization of chaos in
noninvertible maps. Phys. Rev. E. v.66, (2002) 016208.
N. F. Rulkov, V.S. Afraimovich, C.T. Lewis, J.-R. Chazottes, and A. Cordonet.
Multivalued mappings in generalized chaos synchronization.
Phys. Rev. E. v.64, (2001) 016217.
N. F. Rulkov and C.T. Lewis. Subharmonic destruction of generalized chaos
synchronization.
Phys. Rev. E. v.63, (2001) 065204.
N. F. Rulkov. Regularization of synchronized chaotic bursts.
Phys. Rev. Lett. v.86, (2001) 183-186.
V.S. Afraimovich, W.W. Lin, and N.F. Rulkov Fractal dimension for poincare
recurrences as an indicator of synchronized chaotic regimes. International
Journal of Bifurcation and Chaos , 10 (2000) 2323-2337.
N. F. Rulkov and L.S. Tsimring. Synchronization Methods for Communication
with Chaos over Band-Limited Channels.
Int. J. Circ. Theor. Appl v.27, (1999) 555-567.
R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I.
Rabinovich, and H. D. I. Abarbanel. Synchronous Behavior of Two Coupled Biological
neurons. Phys. Rev. Lett. v.81, (1998) 5692-95
M.M.Sushchik, N.F.Rulkov, and H.D.I.Abarbanel. Robustness and
Stability of Synchronized Chaos: An Illustrative Model. IEEE
Trans. Circuits Syst. 44 (1997) 867.
R.Brown and N.F.Rulkov.
Synchronization of chaotic systems: Transverse stability of
trajectories in invariant manifolds. CHAOS., 7
(1997) 395.
N.F.Rulkov and M.M.Sushchik.
Robustness of synchronized chaotic oscillations. International
Journal of Bifurcation and Chaos , 7
(1997) 625.
R.Brown and N.F.Rulkov. Designing coupling that guarantees
synchronization between identical
chaotic systems. Phys. Rev. Lett., 78 (1997) 4189.
N.F.Rulkov. Images of synchronized chaos: Experiments with
circuits.
CHAOS , 6 (1996) 262.
H.D.I.Abarbanel, R.Huerta, M.I.Rabinovich, N.F.Rulkov, P.F.Rowat
and A.I.Silverston. Synchronized Action of Syncptically Coupled Model
Neurons. Neural Computation , 8 (1996) 1567.
N.F.Rulkov and M.M.Sushchik. Experimental observation of synchronized
chaos with frequency ratio 1:2. Physics Letters A, 214
(1996) 145.
H.D.I.Abarbanel, N.F.Rulkov and M.M.Sushchik.
Generalized synchronization of chaos: The auxiliary system approach.
Phys. Rev. E, 53 (1996) 4528.
C. Letellier, G. Gouesbert and N.F.Rulkov.
Topological analysis of chaos in equivariant electronic circuit.
International Journal of Bifurcation and Chaos , 6
(1996) 2531.
H.D.I.Abarbanel, N.F.Rulkov and M.M.Sushchik.
Blending chaotic attractors using the synchronization of chaos.
Phys. Rev. E, 52 (1995) 214.
N.F.Rulkov, M.M.Sushchik, L.S.Tsimring and H.D.I.Abarbanel.
Generalized synchronization of chaos in directionally coupled
chaotic systems. Phys. Rev. E, 51 (1995) 980.
N.F.Rulkov, M.M.Sushchik, L.S.Tsimring, and H.D.I.Abarbanel.
Nontrivial cases of synchronized chaotic oscillators, in: Proceedings of
1995 Intl. Symp. on Nonlinear Theory and Appl., IEEE, 1995, v.2, 937-940.
M.M.Sushchik, N.F.Rulkov, L.S.Tsimring, and H.D.I.Abarbanel.
Generalized synchronization of chaos in directionally coupled chaotic
systems, in: Proceedings of 1995 Intl. Symp. on Nonlinear Theory and Appl.,
IEEE, 1995, v.2, 949-952.
N.F.Rulkov and A.R.Volkovskii. Experimenting with chaos in
electronic circuits. In: Nonlinear Dynamics In Circuits.
Editors T.Carroll and L.Pecora (World Scientific 1995) pp.139-173.
R.Brown, N.F.Rulkov and N.B.Tufillaro. The effect of additive noise
and drift in the dynamics of the driving on chaotic synchronization.
Physics Letters A, 196 (1994) 201.
R.Brown, N.F.Rulkov and N.B.Tufillaro. Synchronization of chaotic
systems: The effect of additive noise and drift in the dynamics of the
driving. Phys. Rev. E, 49 (1994) 3784.
R.Brown, N.F.Rulkov and E.R.Tracy. Modeling and synchronization
chaotic system from time-series data. Phys. Rev. E, 49
(1994) 3784.
R.Brown, N.F.Rulkov and E.R.Tracy. Modeling and synchronizing
chaotic system from experimental data. Physics Letters A, 194
(1994) 71.
E. Del Rio, M.G.Velarde, A.Rodriguez-Lozano, N.F.Rulkov and
A.R.Volkovskii. Experimental evidence for synchronous behavior of
chaotic nonlinear oscillators with unidirectional or mutual driving.
International Journal of Bifurcation and Chaos, 4 (1994) 1003.
N.F.Rulkov, A.R.Volkovskii, A.Rodriguez-Lozano, E. Del Rio and
M.G.Velarde. Synchronous Chaotic Behaviour of a Response Oscillator
with Chaotic Driving. Chaos, Solitons & Fractals, 4
(1994) 201.
N.F.Rulkov and A.R.Volkovskii. Threshold synchronization of chaotic
relaxation oscillations. Phys. Letters, 179A (1993)
332.
N.F.Rulkov, A.R.Volkovskii, A.Rodriguez-Lozano, E. Del Rio and
M.G.Velarde. Mutual synchronization of chaotic self-oscillators with
dissipative coupling. International Journal of Bifurcation and Chaos,
2 (1992) 669.
N.F.Rulkov and A.R.Volkovskii. Synchronized chaos in electronic
circuits. ( In: Proceedings of Exploiting Chaos and Nonlinearities,
Chaos in Communications, SPIE Annual Meeting, San Diego, California,
11-16 July 1993), SPIE Press, 2038, 132.
N.F.Rulkov and A.R.Volkovskii. Synchronization of pulse-coupled chaotic
oscillators. In: Proceedings of 2-nd Experimental Chaos Conference
Wahsington DC, 6-8 October, 1993.
A.R.Volkovskii and N.F.Rulkov. Synchronous chaotic response of a
nonlinear oscillator system as a priciple for the detection of the
information component of chaos. Tech. Phys. Lett., 19 (1993) 97.
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