Arkady Pikovsky Universität Potsdam
Group "Statistical Physics / Theory of Chaos" Department of Physics

Research Topics

(current activities highlighted)


Synchronization of chaos
Phase synchronization
Synchronization in networks
Chimera states and other patterns of synchrony
Controlling synchrony

Data analysis

Inferring synchrony and coupling properties
Network reconstruction
Biomedical applications

Nonlinearity and disorder

Strongly nonlinear lattices: First and second sound
Destruction of Anderson localization in nonlinear lattices
Lyapunov exponents in disordered systems

Compactons and scattering in nonlinear lattices

Phase compactons
Compactons vs chaos in lattices
Chaotic scattering

Effects of noise

Coherence resonance
System-size resonance
Noise in systems with delay
Synchronization by common noise
Common noise and coupling

Space - Time Chaos

Lyapunov exponents and vectors
Globally coupled chaotic systems
Mixing flows

Dynamics between order and chaos

Strange nonchaotic attractors
Singular continuous spectra
Renormalization group
Production systems

Statistical theory of chaos

Simple systems with hyperbolic strange attractors
Attractors and repellers in reversible systems
Coupling sensitivity of chaos


First recognized in 1665 by Christiaan Huygens, synchronization phenomena are abundant in science, nature, engineering, and social life. System as diverse as clocks, singing crickets, cardiac pacemakers, firing neurons, and applauding audience exhibit a tendency to operate in synchrony. These phenomena are universal and can be understood within a common framework based on modern nonlinear dynamics.
Complete synchronization in chaos appears when two interacting chaotic system completely adjust their instant states. This happens if the coupling is strong enough to suppress chaotic instability. Complete synchronization can happen for space-time chaos as well (see figure). Synchronization by common noise happens without interaction, due to common driving of systems.
In the case of Phase synchronization of chaotic oscillators only the their phases are adjusted while amplitudes remain independent.
In ensembles of globally coupled systems the synchronization manifests itself via the appearance of a macroscopic mean field. Quite often clusters of synchronized elemnts are observed. Chimera states are patterns where synchronous and asynchronous domains coexist (figure)
Review and popular publications:
A. Pikovsky, M. Rosenblum
Dynamics of globally coupled oscillators: progress and perspectives
Chaos, 25, 097616 (2015)

M. Rosenblum and A. Pikovsky
Synchronization: from pendulum clocks to chaotic lasers and chemical oscillators
Contemporary Physics, 44, n.5, 4011-416 (2003)

U. Parlitz, A. Pikovsky, M. Rosenblum, J. Kurths
Schwingungen im Gleichtakt
Physik Journal, 5, Nr. 10, 33-40 (2006)

Some publications:
L. Smirnov, G. Osipov, and A. Pikovsky,
Chimera patterns in the Kuramoto-Battogtokh model
J. Phys. A: Math. Theor., 50, 08LT01 (2017)

M. Zaks and A. Pikovsky
Chimeras and complex cluster states in arrays of spin-torque oscillators
Scientific Reports, 7, 4648 (2017)

Michael Rosenblum and Arkady Pikovsky
Self-Organized Quasiperiodicity in Oscillator Ensembles with Global Nonlinear Coupling
Phys. Rev. Lett. 98, 064101 (2007)

V. Ahlers and A. Pikovsky
Critical properties of the synchronization transition in space-time chaos
Phys. Rev. Lett, 88, 254101 (2002)

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 cross-correlation analysis.
In physiology there one often accounts for interacting oscillatory systems. E.g., respiration and cardiac cycle interact. The properties of interaction, including the coupling function itself (figure) can be infered from the observations. Another example is synchrony between oscillations measured via multichannel magnetoencephalography, it is useful for a detection of a source of pathological rhythmic activity causing Parkinson tremor.
In many cases interacting oscillatory system build a network (figure). From observations of the dynamics, it is possible to extract information about the coupling links. This network reconstruction can be performed for different dynamics properties of the coupled units.
Some publications:
A. Pikovsky
Reconstruction of a random phase dynamics network from observations
Phys. Lett. A, 382, 147-152 (2018)

B. Kralemann, M. Frühwirth, A. Pikovsky, M. Rosenblum, T. Kenner, J. Schaefer, and M. Moser
In vivo cardiac phase response curve elucidates human respiratory heart rate variability
Nature Communications, 4, p. 2418 (2013)

B. Kralemann, L. Cimponeriu, M. Rosenblum, A. Pikovsky, and R. Mrowka
Phase dynamics of coupled oscillators reconstructed from data
Phys. Rev. E 77, 066205 (2008)

P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H.-J. Freund
Detection of n:m phase locking from noisy data: application to magnetoencephalography
Phys. Rev. Lett. 81 , n. 15, 3291-3294, 1998

Compactons and scattering 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.
So-called 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.
For nonlinear lattices one can formulate a scattering problem: an incoming wave is transferred to a reflected and a transmitted waves. Both can be chaotic, and the nonlinear scattering is generally non-reciprocal (figure).
Some publications:
S. Lepri and A. Pikovsky
Nonreciprocal wave scattering on nonlinear string-coupled oscillators
Chaos, 24, 043119 (2014)

K. Ahnert, A. Pikovsky
Traveling waves and Compactons in Phase Oscillator Lattices
CHAOS 18, n. 3, 037118 (2008)

A. Pikovsky and P. Rosenau
Phase Compactons
Physica D 218, 56-69 (2006)

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 one-dimensional 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.
Typically, in disordered strongly nonlinear lattices a turbulent state appears. Perturbations on top of this set can be interpreted as first and second sound modes, with certain dispersion characteristics (figure).
Some publications:
A. Pikovsky
First and second sound in disordered strongly nonlinear lattices: numerical study
J. Stat. Mech. P08007 (2015)

S. Roy, A. Pikovsky
Spreading of energy in the Ding-Dong Model
CHAOS, 22, n. 2, 026118 (2012)

S. Tietsche and A. Pikovsky
Chaotic destruction of Anderson localization in a nonlinear lattice
Europhysics Letters 84 n. 1, 10006 (2008)

A. Pikovsky and D. Shepelyansky
Destruction of Anderson Localization by a Weak Nonlinearity
Phys. Rev. Lett 100, 094101 (2008)

V. Ahlers, R. Zillmer, and A. Pikovsky
Lyapunov exponents in disordered systems: Avoided crossing and level statistics
Phys. Rev. E, 63, 036213 (2001)

Effects of noise

Effect of noise on a nonlinear dynamical system may be highly nontrivial. One often speaks on noise-induced phenomena, when some features are observed solely due to presence of noise. In many cases a dependence of some observed quantities on the noise amplitude is non-monotonic, these phenomena are called noise-induced resonances.
Coherence resonance is an effect of maximal order of noise-induced oscillations at a certain noise level. It is usually observed when the process has several characteristic time scales which differently depend on the noise level. Most spectacular is coherence resonance in an excitable system (in the figure we depict the experiment of Miyakawa and Isikawa with an excitable chemical reaction).
In ensembles of coupled noisy systems a phase transition to a collective dynamics of the mean field may occur. In finite ensembles this dynamics is effectively noisy due to finite-size effects. In such systems a resonance dependence on the noise level appears as a system size resonance, when the maximal coherence or the maximal response to an external force happens for a certain system size (figure). This effect occurs also for usual phase transitions, e.g. for the Ising model.
Many interesting effects appear when noise is acting on a system with a delayed feedback. Because the noise-induced dynamics has its own, noise-dependent time scale, nontrivial features can be observed when this time scale is close to the delay time. In this way effects of delayed feedback on the noise-induced oscillations in a bistable potential, in an excitable system (figure) has been considered.
Noise can lead to synchronization of the systems. If two identical nonlinear systems are driven with the same fluctuating force, their states can become identical. This effect is called synchronization by common noise. It occurs when the Lyapunov exponent becomes negative, due to noise. In neuroscience this phenomenon is known under name reliability of neuron spikes (figure shows experiments by Hunter et al).
As a week common noise always synchronizes oscillators, nontrivial effects of coexistence of phase locking and frequency anti-entrainment appear in ensembles of oscillators with repulsive coupling (figure).
Some publications:
A. V. Pimenova, D. S. Goldobin, M. Rosenblum, and A. Pikovsky
Interplay of coupling and common noise at the transition to synchrony in oscillator populations
Sci. Reports 6, 38158 (2016)

W. Braun, A. Pikovsky, M. A. Matias, P. Colet
Global dynamics of oscillator populations under common noise
EPL, 99, 20006 (2012)

Denis S. Goldobin and Arkady Pikovsky
Antireliability of noise-driven neurons
Phys. Rev. E 73, 061906 (2006)

A. Pikovsky and J. Kurths
Coherence resonance in a noise-driven excitable system
Phys. Rev. Lett. 78, 775-778, 1997

A. Pikovsky, A. Zaikin, and M. A. de la Casa
System size resonance in coupled noisy systems and in the Ising model
Phys. Rev. Lett, 88, 050601 (2002)

L. S. Tsimring and A. Pikovsky
Noise-induced dynamics in bistable systems with delay
Phys. Rev. Lett, 87, 250602 (2001)

Space - time chaos

Chaotic regimes in spatially distributed systems are called space-time chaos. Due to excitation of many degrees of freedom the field is irregular both in space and time. The challenging questions are statistical properties of dynamics for large systems, in particular in the thermodynamic limit when the system size tends to infinity.
Important objects in space-time chaos are Lyapunov exponents and vectors. We have developed a statistical theory of these objects based on the analogy to roughening interfaces and on the use of the Kardar-Parisi-Zhang scaling. Based on this one can characterize space-time chaos with local growth rates (figure).
In ensembles of coupled chaotic systems a transition to a collective mode can occur. We have developed a theory of this transition based on the linear response approach. For an ensemble of globally coupled Bernoulli maps it predicts a transition to an oscillating global mode (figure).
Interesting effects appear when temporal chaos is combined with spatial mixing. If the spatial mixing is strong enough, only spatially homogeneous state is stable, which varies chaotically in time. For weaker mixing a spatial structure of the field can appear (figure).
Some publications:
A. V. Straube, A. Pikovsky
Mixing-induced global modes in open active flow
Phys. Rev. Lett. 99, 184503 (2007)

A. Pikovsky and A. Politi
Dynamic localization of Lyapunov vectors in Hamiltonian lattices
Phys. Rev. E, 63, 036207 (2001)

D. Topaj, W.-H. Kye, and A. Pikovsky
Transition to coherence in populations of coupled chaotic oscillators: A linear response approach
Phys. Rev. Lett, 87, 074101 (2001)

A. Pikovsky and O. Popovych
Persistent patterns in deterministic mixing flows
Europhysics Letters, 61, n. 5, 625-631 (2003)

Dynamics between order and chaos

Ordered (e.g. periodic and quasiperiodic) and chaotic behavior are well-established 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 non-chaotic 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 two-frequency 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.
Some publications:
E. Neumann and A. Pikovsky
Quasiperiodically driven Josephson junctions: strange nonchaotic attractors, symmetries and transport
European Physical J. B 26, 219-228 (2002)

U.Feudel, A.S.Pikovsky, and M.A.Zaks
Correlation properties of quasiperiodically forced two-level system
Phys. Rev. E, 51, n.3, 1762-1769, 1995

S. P. Kuznetsov, E. Neumann, A. Pikovsky, and I. R. Sataev
Critical point of tori collision in quasiperiodically forced systems
Phys. Rev. E 62 , n. 2, 1995-2007, 2000

I. Katzorke and A. Pikovsky
Chaos and complexity in a simple model of production dynamics
Discrete Dynamics in Nature and Society, 5, 179-187, 2000

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 Fokker-Planck 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.
Coupled chaotic systems can synchronize. Quite nontrivial are properties of chaos synchronization at nonlinear coupling, where more than two systems interact (figure).
Hyperbolic chaos is the strongest, mathematically pure form of chaos. Remarkably, it can be observed in realistic physical systems (figure).
Some publications:
J. Petereit and A. Pikovsky,
Chaos synchronization by nonlinear coupling
Commun Nonlinear Sci Numer Simulat 44, p. 344 - 351 (2017)

P. V. Kuptsov, S. P. Kuznetsov, and A. Pikovsky
Hyperbolic chaos at blinking coupling of noisy oscillators
Phys. Rev. E 87, 032912 (2013)

R. Zillmer, V. Ahlers and A. Pikovsky
Coupling sensitivity of localization length in one-dimensional disordered systems
Europhysics Letters, 60, n. 6, 889-895 (2002)