
Research Topics
(current activities highlighted)

Synchronization of chaos
Phase synchronization
Synchronization in networks
Controlling synchrony



Inferring synchrony and coupling
Biomedical applications 


Destruction of Anderson localization in nonlinear lattices
Lyapunov exponents in disordered systems



Phase compactons
Compactons vs chaos in lattices



Coherence resonance
Systemsize resonance
Noise in systems with delay
Synchronization by common noise



Lyapunov exponents and vectors
Globally coupled chaotic systems
Mixing flows 


Strange nonchaotic attractors
Singular continuous spectra
Renormalization group
Production systems



Simple systems with hyperbolic strange attractors
Coupling sensitivity of chaos


Data analysis

Synchronization generally appears due to coupling, and can be used for
detecting an interaction in oscillating systems. As the phases are mostly sensitive to interaction, one usually neglects the information contained in the amplitudes.


Hilbert transform is an effective tool to determine the phase from the
observed signal (figure). Based on phase relations one can judge on the presence of synchrony,
and in this way to prove the interaction between oscillating systems. Moreover, as the phases are
most sensitive variables, the analysis based on the phases is in some cases superior to the usual linear
crosscorrelation analysis.


In physiology there one often accounts for interacting oscillatory systems.
E.g., respiration and cardiac cycle can be synchronous. Another example is synchrony between
oscillations measured via multichannel
magnetoencephalography (see figure), it is useful for a detection of a source of pathological
rhythmic activity causing Parkinson tremor. Using phases one can also determine
direction of coupling.




Compactons in nonlinear lattices

Theory of solitons (exponentially localized waves in integrable nonlinear partial differential equations) is one of masterpieces of nonlinear science. remarkably, in strongly nonlinear systems one can find much stronger localized objects, called compactons.


Phase compactons appear in simple lattices of coupled oscillators. They can be nicely described as waves with a compact support when partial differential equations are used; on a lattice they are extended but with superexponentially decaying tails. Compactons naturally energe from an initial pulse (figure) but the collisions are not elastic. On a long time scale chaos appears.


Socalled newton craddle (figure) is an example of a lattice with strongly nonlinear interactions: the elastic force between balls follows the Hertz's law and is proportional to the displacement in power 3/2. The waves in such lattices are compactons.




Nonlinearity and disorder

Both nonlinearity and disorder may lead to highly nontrivial phenomena, and an interplay of these two effects substitutes a challenging problem, theoretical and numerical.


Anderson localization means that all eigenstates in a linear onedimensional disprdered lattice are exponentially localized, and no transport occurs. We have demonstrated that in presence of nonlinearity this is no more true: due to nonlinear interaction an initially localized packet spreads subdiffusively (figure). We have also approached the problem of transmission through a disordered layere as a statistical bifurcation one.


If disordered chaotic systems are weakly coupled (figure), their Lyapunov exponents are
random. One can pose a question of their distribution. Similarly to eigenvalues of
random matrices the Lyapunov exponents experience effective repulsion. The latter leads to depletion
in the distribution of exponent spacings.




Dynamics between order and chaos

Ordered (e.g. periodic and quasiperiodic) and chaotic behavior are wellestablished
objects of nonlinear dynamics. However, there is a large field of complex dynamics between order and
chaos. Probably, the best example of this is quantum chaos. However, also in classical systems
one accounts for complex dynamical states.


Strange nonchchaotic attractors can be observed in quasiperiodically forced systems.
They are nonchaotic because the Lyapunov exponent is negative, howevere they are fractals (figure).
Strange nonchaotic attractors appear, e.g., when a Josephson junction is driven with a twofrequency
force.


A power spectrum is a standard tool to study stationary processes. Usually one distinguishes
discrete and continuous spectra, these two types correspond to ordered and chaotic dynamics. However,
in between of these two major classes there are also singular continuous spectra. These fractal
spectra (see figure) have been studied, e.g., for strange nonchaotic attractors.




Renormalization group is a powerful method to study scaling behavior in dynamical systems.
It can be applied not only to transitions bewteen order and chaos, but also to the transitions between
strange nonchaotic, regular, and chaotic attractors (figure).


One of interesting applications of the behavior between chaos and order has appeared
in the modelling of production dynamics. Some models of deterministic queueing theory
give nonchaotic behavior with high complexity (figure). Production systems are nontrivial, because
they incorporate both discrete (symbolic) and continuous variables.




Statistical theory of chaos

Chaotic motion is disordered, and thus it appears natural to describe it statistically. In this way
one can describe many properties of chaos using models where deterministic motion is
replaced with a corresponding random process.


Coupling sensitivity of chaos is the effect of singular dependence of the Lyapunov exponent
in coupled systems on the coupling parameter. A statistical theory of this effect based on the FokkerPlanck
equation gives a good quantitative description. Qualitatively the phenomenon can be explained
in terms of coupled random walks (figure). It is also relevant for Anderson localization
and for description of the Ising model in a disordered magnetic field.



