M lefranc, S. Boccaletti, B. Gluckman, C. Grebogi, J. K¨ Urths, L. Pecora

Department of Physics, Anna University, Chennai - 600 025, India

Centre for Dynamics of Complex Systems, University of Potsdam, 14469 Potsdam, Germany

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6Department of Physics, Humboldt University, 12489 Berlin, Germany [e-mail]

We have investigated the synchronization transitions from anticipatory to lag synchronization via complete synchronization [Physical Review E 71 016211 (2005)] and their inverse counterpart [Chaos 19 023107 (2009)] in unidirectionally coupled time-delay systems with excitatory and inhibitory time-delay couplings, respectively. The transition between different types of synchronization can be realized, for a fixed set of parameters, as a function of the coupling delay  along with a suitable stability condition following the Krasovskii-Lyapunov theory. We demonstrate the experimental realization of the above synchronization transitions in coupled time-delay electronic circuits with a threshold nonlinearity

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How and why the analysis of a dynamics can depend on the choice of the observable

Christophe LETELLIER

Complexe de Recherche Interprofessionnal en A ´ erothermochimie (CORIA), Universit ´ e de Rouen, France

(with Luis A. Aguirre and Robert Gilmore) The Takens theorem ensures us that a phase portrait equivalent to the original one can be reconstructed pro-

vided a large enough embedding dimension. Unfortunately, a weakness in this beautiful theorem is related to the ”generic” measurement function. Indeed, how can we be sure that the measurement function we use is ”generic”? For instance, if you investigate the Lorenz dynamics, you may choose to measure variable z, but this is not a measurement function since the complete lack of symmetry is broken as soon you perturb z with a small amount of x or y. But there is a more serious problem. It may arise that some states which are different in the original phase space cannot be distinguished in the reconstructed space or, worse, that they cannot be observed at all. This is the so-called observability problem which has a similar but not identical counterpart in control theory. Surprisingly, this is very rarely mentioned while investigating dynamical systems. But many techniques like global modeling, dimension estimation, synchronization, recurrence plots and related estimators, provide results that are dependent on the choice of the observable. It will be showed that most of these discrepancies can be interpreted in terms of lack of observability.

Observability is related to the quality of the coordinate transformation between the original m-dimensional phase space and the corresponding m-dimensional differentiable embedding. Lack of observability is always related to singularities occurring in this coordinate transformation. This presentation will introduce the concept of observability coefficients, explain how they can be computed from the original equations and illustrate in different situations how they can be used to explain heterogeneities in the results obtained while using different variables from a given system.

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Acoustic target identification with chaos based waveforms

Frederic Rachford & Thomas Carroll

Naval Research Laboratory, Code 6362, Washington, DC 20375 [e-mail]

We propose a method of distinguishing two known targets using their their acoustic signatures in cross correlation with selected chaos based waveforms. Initially acoustic chirp waveforms were digitally generated, broadcast from a tweeter and scattered off several similarly sized objects for a number of object orientations. A microphone aligned with the tweeter received the scattered waveforms and the waveforms were digitized with an oscilloscope. The digitized waveforms received from two distinct objects were sorted into angular windows. A computer program generated a large number of test waveforms with the same band width (20%) and center frequency (3.3 or 5 KHz) as the original chirp. Two methods both derived from chaotic time series were employed to generate the test waveforms. In one case constant amplitude waveforms were assembled from concatenated sinusoids whose periods were specified by the time series. In the other case the time series its self was run through a band pass filter. The time series were generated by taking the modulus of a six parameter chaotic map. The shift register parameters were randomly varied and the generated test waveforms were selected to maximize the averaged cross correlation of the return from one target, while minimizing the averaged cross correlation of the other and vis versa. Contrast ratios, ratios of the cross correlations, were then calculated for each target for return waveforms within each angular window. Waveforms that maximized the difference in contrast between the two targets were retained and optimized via a standard downhill simplex routine. Using these optimized waveforms we can distinguish between targets for orientations within our orientation windows.

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Using experimental data to estimate the states of models of neural circuits

Henry D.I. ABARBANEL

Department of Physics and Marine Physical Laboratory (Scripps Institution of Oceanography), University of California, San Diego, La Jolla, CA 92037 (USA)

The talk at this Experimental Chaos Conference will address how one can bring information from laboratory measurements and field observations into nonlinear models of those systems. This entails a path integral formulation of the problem that then connects it to the extensive body of work in statistical physics and opens new ways to think about this critical aspect of our scientific inquiry. The application of the methods to neurobiology and numerical weather prediction will be discussed.

Quinn, J. C., P. H. Bryant, D. R. Creveling, S. R. Klein, and H. D. I. Abarbanel, “State and Parameter and State Estimation of Experimental Chaotic Systems Using Synchronization,” Physical Review E, 80 016201 (2009).

Gibb, L. T. Q. Gentner, and H. D. I. Abarbanel, “Inhibition and Recurrent Excitation in a Computational Model of Sparse Bursting in Song Nucleus HVC,” Journal of Neurophysiology (2009) Jun 10. [Epub ahead of print]

Gibb, L. T. Q. Gentner, and H. D. I. Abarbanel, “Brainstem Feedback in a Computational Model of Birdsong Sequencing,” Journal of Neurophysiology (2009) Jun 24. [Epub ahead of print]

Abarbanel, H. D. I., “Effective actions for statistical data assimilation,” Physics Letters A, 373, 4044-4048 (2009). doi:10.1016/j.physleta.2009.08.072

Abarbanel, H. D. I., M. Kostuk, and W. Whartenby, “Data Assimilation with Regularized Nonlinear Instabilities,” accepted in Quarterly Journal of the Royal Meteorological Society, February, 2010.

Quinn, J. and H. D. I. Abarbanel, “State and Parameter Estimation using Monte Carlo Evaluation of Path Integrals,” submitted to Quarterly Journal of the Royal Meteorological Society, December, 2009.

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Non-linear Kalman filtering techniques for estimation and prediction of rat sleep dynamics

Madineh Sedigh-Sarvestani1, Steven L. Weinstein3, Steven J. Schiff1;2, & Bruce J. Gluckman1;

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3Engineering Science and Mechanics, Pennsylvania State University, University Park, PA,

Department of Neurosurgery, Pennsylvania State University, University Park, PA.

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0Pediatric Epilepsy, Weill Cornell Medical College, New York City, NY [e-mail]

Our laboratory has ongoing efforts to utilize model based controller-predictor systems to better understand the non-linear dynamics of the brain, with particular attention to sleep and seizure. Towards this end, we have implemented several published and novel computational models of the brain to investigate its behavior in a variety of different states (i.e. sleep vs. wake). These models have been modified so that they simulate the sleep dynamics of our experimental rodents within small sampling times. We have implemented these models in an Unscented Kalman Filter (UKF) framework to serve as duplicate source and tracker models and show that the UKF-based data assimilation algorithm we have developed is extremely robust and can reconstruct hidden dynamics even when the tracker model is intentionally made inadequate. In parallel with these computational efforts, we have obtained a feature set of experimental data from our continuously cabled rodents and have used these features to classify state of sleep and to develop a seizure prediction algorithm. Several of these discrete and continuous features are then used as the noisy observables in the implemented UKF framework to recursively reconstruct all of the inaccessible variables of the dynamic sleep model. Results from this reconstruction are promising and allow us access to hidden variables, such as sleep driven changes in neurotransmitter concentrations that would be hard or impossible to measure directly from our rodents. Furthermore, we have augmented our algorithm to make shorttime predictions of sleep state. We then use these predictions of sleep-state transitions to improve the performance of our seizure prediction algorithm by reducing the confounding effect of sleep state on seizure prediction.

Recent published literature has begun to illuminate the intimate link between seizure and sleep, a relationship with a long clinical history in human patients. It is becoming increasingly clear that in order to predict and control seizure dynamics, we must first be able to grasp the non-linear dynamics of sleep and sleep-state transitions. Thus, our work bridges experimental and control theory techniques to investigate a crucial missing link which will give us insight into the dynamics of seizures and may drastically improve seizure prediction

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Forecasting and pattern control in Rayleigh-Benard convection

Adam PERKINS

Center for Nonlinear Dynamics and School of Physics Georgia Institute of Technology Atlanta, Georgia, USA

Predictive power in spatiotemporally complex systems is limited by several factors. Foremost among them is inherent system instability that can cause small initial uncertainty to grow rapidly. Often, the dynamically important modes of instability are unknown or characterized insufficiently. We are addressing these issues in a RayleighBenard convection experiment, in which a novel technique of pattern control provides a tool for the repeatable imposition of a given convection pattern.

We apply selected perturbations to a given pattern to create an ensemble with nearby initial conditions, close to a particular instability. An Arnoldi-inspired analysis of the ensemble reveals directly the physical structure of the dominant modes of that instability as well as the corresponding growth rates. The extracted modal information may be used for pattern control; moreover, our general methodology may be applied to a large number of patternforming systems, so long as an acceptable method of pattern actuation can be realized.

n addition, we employ an efficient forecasting algorithm, the Local Ensemble Transform Kalman Filter (LETKF), to produce system state and parameter estimates of convection patterns observed experimentally. State estimation refers to the synchronization of a numerical system state with (noisy and incomplete) experimental measurements, prior to time evolution in a forecasting process. This estimation procedure is motivated by and directly applicable to other spatiotemporally complex systems, such as the weather or cardiac dynamics. Our experimental pattern control gives us a way of testing systematically the effects of small changes to initial conditions on this crucial step of the forecasting process.

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Adaptive synchronization of a network of chaotic oscillators

Bhargava Ravoori, Adam Cohen, Francesco Sorrentino, Thomas Murphy, Edward Ott, & Rajarshi Roy

Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA [e-mail]

Synchronization among networks of coupled chaotic systems is an interesting phenomenon with potential applications in sensor and communication networks. In order for a network of chaotic oscillators to admit a synchronous solution, each node must receive the same cumulative coupling from its peers. This constraint implies that the coupling matrix describing the network has a uniform row-sum. This condition is difficult to achieve in practice. Moreover, even if synchrony is attained, environmental drifts and other network perturbations can cause the coupling strengths to change, making it impossible to maintain synchrony over time.

We present here an adaptive control system [1] that overcomes these limitations. We experimentally show that the system can both acquire and maintain a state of global synchronization in a network of chaotic oscillators even when the coupling matrix is unknown and time-varying [2]. Each node in the network uses locally measured signals to construct a real-time estimate of its total input coupling strength. A suitable multiplicative scaling is then applied to the coupling signal to ensure that all nodes in the network receive the same cumulative coupling, thus making synchronization feasible.

The network is comprised of three optoelectronic nonlinear time-delayed feedback loops which exhibit highdimensional chaotic dynamics [2, 3]. Each node is coupled to every other through a bidirectional fiber-optic link, and the coupling strengths are controlled using variable optical attenuators. Using the adaptive algorithm we successfully synchronize the network under time-varying coupling conditions. Furthermore, we show that from the computed scale factors obtained at each node, we can deduce the coupling matrix, thereby enabling us to both track and localize disturbances and perturbations in the network.

References: [1] F. Sorrentino and E. Ott, Phys. Rev. Lett. 100, 114101 (2008); Phys. Rev. E 79, 016201 (2009). [2] B. Ravoori et al., Phys. Rev. E 80, 056205 (2009). [3] A. B. Cohen et al., Phys. Rev. Lett. 101, 154102 (2008). This work was supported by DOD MURI grant (ONR N000140710734).

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VKS experiment: a chaotic turbulent dynamo?

Franc¸ois DAVIAUD

Service de Physique de l’ ´ Etat Condens ´ e, Commissariat ` a l’´ Energie Atomique, Saclay, France

The VKS experiment studies dynamo action in the flow generated inside a cylinder filled with liquid sodium by the rotation of coaxial impellers. We first report observations related to the self-generation of a stationary dynamo when the flow forcing is Rπ symmetric, i.e., when the impellers rotate in opposite directions at equal angular velocities. The bifurcation is found to be supercritical with a neutral mode whose geometry is predominantly axisymmetric. We discuss the role of turbulence in the dynamo mechanism. We then report the different dynamical dynamo regimes observed when the flow forcing is not symmetric: stationary dynamos, transitions to relaxation cycles or to intermittent bursts, and random field reversals. We show that these dynamics result from the interactions of a few modes and display characteristic features of low dimensional dynamical systems despite the high degree of turbulence in the flow. (VKS collaboration: CEA - CNRS - ENS Paris - ENS Lyon).

References R. Monchaux et al., Phys. Rev. Lett. 98, 044502 (2007) M. Berhanu et al., Europhys. Lett. 77, 59007 (2007) F. Ravelet et al., Phys.Rev. Lett. 101 074502 (2008) R. Monchaux et al. Phys Fluids 21, 025104 (2009)

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Kinematic dynamo threshold in time dependent velocity fields

Miguel L ´ opez & Javier Burguete

C/Irunlarrea S.N., Dep. of Physics and Appl. Mathematics Edificio Los Casta ˜ nos. , Pamplona, Navarra, Spain, 31008 [e-mail]

Conducting neutral fluid flows can be dramatically different from the non-conducting case because of their interaction with magnetic fields, either internal (self-sustained) or external (forcing). In this work we present an experimental analysis of a von K ´ arm ´ an swirling flow and the influence of this hydrodynamics in the generation of a magnetic filed.

6The objective is to determine the effect of time dependent flows in the threshold of the dynamo action. To achieve this goal, we have characterized the flow before this instability in a model experiment (using water). This velocity field, determined only by the hydrodynamics, has been used to find out the MHD effects. The fluid has been stirred in a cylindrical cavity up to a Reynolds number of 10. We show that the average velocity field of the turbulent flow bifurcates subcritically breaking some symmetries of the problem and becomes time-dependent because of equatorial vortices moving with a precession movement. This subcriticality produces a bistable regime, with a hysteresis region for an extremely small range of parameters. Three different time-scales are relevant to the dynamics, two of them very slow compared to the impeller frequency.

We have studied the different time scales of the system, changing a enclosure volume (neutrally buoyant spheres) assuming that the density of the sphere is homogeneous. We follow this volume in a period of time and we compare the results in different spatial scales.

The effect of these different time-scales and symmetry-breaking’s has been tested in a kinematic dynamo code. The threshold strongly depends on the existence of these features.

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Large scale fluctuations and dynamics of the Bullard - von Karman dynamo

Nicolas Plihon, Gautier Verhille, Mickael Bourgoin, Romain Volk, & Jean-Franc¸ois Pinton

Laboratoire de Physique ENS Lyon - CNRS UMR 5672 [e-mail]

The importance of turbulent induction processes in dynamo action has been recognized for most natural dynamos. More recently, the von-K ´ arm ´ an Sodium dynamo showed the importance of turbulent fluctuations in the generation and dynamics of the magnetic field. We will present and analyze the features of an experimental synthetic fluid dynamo built in the spirit of the Bullard dynamo. It is a two-step dynamo in which one process stems from the fluid turbulence, while the other part is achieved by a linear amplification of currents in external coils, as in the Bullard device. The fluid turbulent process is based on a von-K ´ arm ´ an gallium flow; hence the designation ”Bullard-von-K ´ arm ´ an dynamo”.

The Bullard-von-K ´ arm ´ an dynamo allows to investigate the influence of the statistical properties of the turbulent induction process on the dynamics of the dynamo. Modifications in the flow forcing are introduced in order to change the dynamics of the flow, and hence of the turbulent induction.

On-off intermittency at onset of dynamo action has been characterized. The on-off intermittent feature appears to be very robust at onset but its range of existence strongly depends on the low frequency spectrum of the turbulent induction process. For some conditions, magnetic field reversals have been observed. The waiting-time distribution between reversals has been found to evolve from power-law to Poisson-like depending on the distance from onset. The large scales fluctuations also have a significant impact on these reversals.

Most of these experimental results can be understood as emerging form a supercritical system subject to multiplicative noise. Some other features (such as reversals) requires the presence of additive noise and their precise understanding remains a challenge. The links and differences with the dynamics of the von-K ´ arm ´ an Sodium dynamo will also be discussed.

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How does El-Nino influence the dynamics of climate network in their basin, and around the globe?

Shlomo HAVLIN

Department of Physics, Bar Ilan University, Israel

(with A. Gozolchiani) The temporal correlations between records of temperature and between records of height can be regarded as a

dynamical climate network. The network’s response in different worldwide regions to El-Nino Southern Oscillation (ENSO) is shown to be much stronger compared to the response of the classical measures such as mean and variance of temperature or height level [1]. The network dynamical response to El-Nino is found to be in the form of links that become unstable appear and disappear during El-Nino periods [2]. We find that the responding links tend to be the same during all the strong events. When studying the behavior of weighted nodes we find that during El-Nino events many nodes inside the El-Nino basin lose their strong dependence on their surrounding nodes, but still keep influencing them.

References [1] K. Yamasaki, A. Gozolchiani, S. Havlin, Phys. Rev. Lett. 100, 228501 (2008). [2] A. Gozolchiani, K. Yamasaki, O. Gazit, S. Havlin, EPL 83, 28005 (2008).

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Synchronization of time-delayed diffusively coupled systems: an experimental case study with Hindmarsh-Rose oscillators

Erik Steur, Patrick Neefs, & Henk Nijmeijer

Eindhoven University of Technology, Dept. Mechanical Engineering, Dynamics and Control Group, P.O.Box 513, 5600 MB Eindhoven, The Netherlands

[e-mail]

We discuss synchronization in networks of systems that are interconnected via diffusive coupling. We present theoretical results for a general class of nonlinear systems that are interacting with or without time-delay. These theoretical results are supported by experiments with a setup consisting of sixteen electronic Hindmarsh-Rose neurons. The experiments are performed for the non-delayed case as well as the situation where interaction delay is explicitly taken into account. We will focus in particular on the influence of the network topology on the synchronization in case of delayed interactions.

References [1] Erik Steur and Henk Nijmeijer, Synchronization in networks of linearly time-delay coupled systems: a

passivity based approach, (submitted for publication) 2009 [2] P.J. Neefs, E. Steur and H. Nijmeijer, Network complexity and synchronous behavior: an experimental

approach, accepted for publication in Int. J. Neuro. Syst., 2010

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Network dynamics in collective motion

Tam ` as VICSEK

Department of Biological Physics, E ¨ otv ¨ os Lor ´ and University, Hungary

Collective motion patterns are perhaps the most widespread and spectacular manifestations of collective behaviour. The ultimate goal we face is to find unifying principles describing the essential aspects of flocking. On the way in this direction it is a natural approach to investigate the delicate dynamics of the inetractions between the co-moving individual units. After an introduction to the topic, a recent model and two new experiments will be discussed. The model has been designed to capture the basic aspects of network dynamics in a very simple system of collectively moving particles. The experimental observations involve a system of self-propelled toy boats and a study of the hierarchical network dynamics in pigeon flocks.

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Dynamics and augmentation patterns in adaptive networks

Casey Schneider-Mizell1, Jack Parent2, Eshel Ben-Jacob3, Leonard Sander1, & Michal Zochowski