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

Embeddings with symmetry [13]

Daniel Cross & Robert Gilmore

Department of Physics, 3141 Chestnut St, Philadelphia, PA 19104 [e-mail]

A dynamical system may be reconstructed from scalar data taken along some trajectory of the system. A reconstruction is considered successful if it produces a system diffeomorphic to the original. However, if the original dynamical system is symmetric, it is natural to search for reconstructions that preserve this symmetry. These generally do not exist. It is possible to show that a differential reconstruction of any nonlinear dynamical system preserves at most a two-fold symmetry and that this is always a parity symmetry. Implications for embeddings of the Lorenz system will be discussed in detail.

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7

Probing nonlinearity through a measure inspired in the autocorrelation function [14]

David Carlo Almeida Barbato

R. Dr. Bento Teobaldo Ferraz 271 - Bl. II - Barra Funda 01140-070 - S ˜ ao Paulo, SP - Brasil [e-mail]

In this work, we propose a kind of generalization of the Autocorrelation function (ACF). This new measure was designed aiming the search for dependencies between points of the series other than the linear ones.

The main idea concerned in the development of this tool is to avoid on the measurement process the use of the series average value. Instead of multiply the demeaned values of the series, the ratios between first differences of the values are taken.

Even though this seens to be similar to the standard ACF of the first differenced series, some deviations from that occur. We believe them can help in discriminating some nonlinear features of series.

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8

Application of delay-observer design to forecast of irregular time-series [15]

Simon Pinsky, Jeffry Mortensen, & Mark Pinsky

Department of Mathematics and Statistics, University of Nevada.Reno, Reno NV 89557, USA [e-mail]

Various observers have been used for estimating directly immeasurable states of dynamical systems as well as estimating the derivatives of a time-series when the underlying model is unknown. This paper utilizes the concept of observer design for forecasting on relatively short time intervals future values and derivatives of irregular time series. This task is attained via feedback control of a polynomial model where observations are recorded with a certain delay. This allows us to express the forecast accuracy in terms of variability of a time-series and forecast horizon. Examples illustrating forecast of a financial time-series are presented.

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9

Non-linearity detection by the noise titration technique : another tool dependent on the choice of the observable [16]

Elise Roulin, Ubiratan Santos Freitas, & Christophe Letellier

CNRS UMR 6614 - CORIA, Universit ´ e de Rouen, Site Universitaire du Madrillet, BP 12, 76801 SAINT ETIENNE DU ROUVRAY CEDEX

[e-mail]

Identifying chaotic dynamics from biological data is very challenging, mainly because it requires a conclusive proof for an underlying determinism. Even if deterministic models were already found from experimental data, they are very rarely obtained from biological data [1]. If proving chaos is more or less out of scope, it remains possible to detect the action of a nonlinearity in the processes governing the dynamics under investigation. The noise titration technique developed by Mauricio Barahona and Chi-Sang Poon [2] is based on the comparison of one-step-ahead predictions using linear and nonlinear models, respectively. We show that this technique has to be used in right conditions, that is, to be applied on well sampled data and using models with appropriate structures. Moreover, the noise titration technique is shown to depend on the choice of the observable with the R ¨ ossler system used as a test case.

[1]. U. S. Freitas, E. Roulin, J.-F. Muir & C. Letellier, Identifying determinism underlying heart rate: the right task ?, Chaos, 19, 028505 (2009).

[2]. C.-S. Poon & M. Barahona, Titration of chaos with added noise, Proceedings of the National Academy of Sciences (USA), 98, 7107-7112, 2001.

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0

Regional predictability variations [17]

Reason Machete

Department of Mathematics, P. O. Box 220, Whiteknights, Reading, RG6 6AX, UK [e-mail]

It is traditionally thought that regional losses in predictability are an evidence of the instability of the underlying flow. While this may be appealing on the surface, a deeper analysis indicates that this could be a signature of other factors, which may be even more dominant. It is evident from Takens’ theorem that in the absence of model error, model state space versus system state are contributing factors. Appealing to an experimental circuit, it is demonstrated that model error and model state space play crucial roles. It is also found that model state space contribution may dominate model error. The tool used is the time for initial uncertainty orientations to increase by a factor of q, called q-pling times. One cannot be too careful not to confuse the map with the territory.

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1

Connecting curves for dynamical systems [18]

Timothy Jones1, Robert Gilmore1, Jean-Marc Ginoux2;3, Christophe Letellier3, & U. S. Freitas

31

2

3Physics Department, Drexel University, Philadelphia, Pennsylvania 19104, USA

UMR 7586 - Institut de Mathematiques de Jussieu, Universite Pierre et Marie Curie, Paris VI,

8

2CORIA UMR 6614 - Universite de Rouen, BP 12 Av. de l’Universite, Saint-Etienne du Rouvray cedex, France [e-mail]

We introduce one dimensional sets to help describe and constrain the integral curves of an n dimensional dynamical system. These curves provide more information about the system than the zero-dimensional sets (fixed points) do. In fact, these curves pass through the fixed points. Connecting curves are introduced using two different but equivalent definitions, one from dynamical systems theory, the other from differential geometry. We describe how to compute these curves and illustrate their properties by showing the connecting curves for a number of dynamical systems. If one can determine the vector field associated with a flow, then our algorithm can be applied to locate vortex filaments. These lines define regions around which the flow circulates

.

The stability of adaptive synchronization of chaotic systems [19]

Adam Cohen, Bhargava Ravoori, 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]

2In order to achieve identical synchronization of a network of Ncoupled chaotic oscillators, each node must be set to have nominally identical parameters and the Nelements of the adjacency matrix must be fine-tuned to ensure that the synchronous solution is admitted and stable. The analytic tool for determining whether a particular network configuration can maintain a synchronous state is given by the master stability function formulation. Recently, an adaptive strategy was presented [1] that can maintain a globally synchronous state even when the coupling strengths are unknown and time-varying. This is a distributed technique that runs at each node and employs only local information, i.e. an internal signal and an aggregate signal representing the superposition of transmitted signals from the other nodes. This adaptive synchronization strategy has been demonstrated with experiments on a network of chaotic optoelectronic oscillators [2] and with numerical simulations of large networks. In this talk, the stability of this scheme is addressed through an extension of the master stability function technique to include adaptation [3]. The results of the stability study are compared with experimental measurements.

References: [1] F. Sorrentino and E. Ott, Phys. Rev. Lett. 100, 114101 (2008). [2] B. Ravoori et al., Phys. Rev. E 80, 056205 (2009). [3] F. Sorrentino et al., Chaos 20, 013103 (2010). This work was supported by DOD MURI grant (ONR N000140710734).

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3

Synchronization analysis in complex networks with identical structural parameters [20]

Jose Benjamin Mercado Sanchez1, Mar ´ ia Teresa Rodr ´ iguez Sahag ´ un1, Didier L ´ opez Mancilla2, Rider Jaimes Re ´ ategui, & Juan Hugo Garc ´ ia L ´ opez2

21

2Centro Universitario de Ciencias Exactas e Ingenier ´ ias, Universidad de Guadalajara (CUCEI-UdeG) C.P. 44420, Guadalajara, Jal., M ´ exico

Centro Universitario de los Lagos, Universidad de Guadalajara (CULagos-UdeG) C.P 47460, Lagos de Moreno, Jalisco, M ´ exic

[e-mail]

Abstract: In this work, we present a network synchronizability analysis in networks with identical structural parameters and chaotic dynamic nodes. Each network is configured with Lorenz system, R ¨ ossler system or Chua circuits. Results of the simulation in MATLAB are shown for networks with different indexes of synchronizability and identical structural parameters. The main idea of this work consists of contributing to the study of the relationship between the characteristics of the synchronizability and the structural parameters of a network.

Keywords: Synchronization, Complex networks, Lorenz system, Synchronizability, Chaos.

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4

Physics of age-related macular degeneration [21]

Fereydoon Family

Physics Department, Emory University, Atlanta, GA 30322, USA [e-mail]

Age related macular degeneration (AMD) is the leading cause of blindness in the adult population. Choroidal neovascularization, which is the abnormal growth of blood vessels in the choroidal region, is the most common cause of AMD. CNV is produced with age by accumulation of residual material in the retinal pigment epithelium cells (RPE). With time, incompletely degraded membrane material build up in the RPE in the form of lipofuscin, cause abnormal growth of blood vessels that break through the Bruchs membrane, and raise the macula and eventually lead to blindness. The fact that a number of far from equilibrium dynamical processes are involved in the formation and growth of AMD makes this a rich field for application of many techniques of statistical and nonlinear physics. I will give some examples of the open problems and discuss the results of a kinetic Monte Carlo simulation of a deposition and aggregation model of lipofuscin formation in the RPE cells, as well as both two and three-dimensional simulations of the formation of CNV, that we have recently carried out.

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5

Oscillations in the expression of a self-repressed gene: interaction of a transport delay with transcriptional response [22]

Jingkui Wang1;2, Quentin Thommen1;2, & Marc Lefranc1;

21Universit ´ e Lille 1, Laboratoire de Physique des Lasers, Atomes, Mol ´ ecules, F-59655 Villeneuve d’Ascq, France

2Universit ´ e Lille 1, Institut de Recherche Interdisciplinaire, F-59655 Villeneuve d’Ascq, France [e-mail]

Mathematical models of gene networks often assume that transcription reacts instantaneously to variations in regulatory protein concentrations. However, some experiments have evidenced a slow transcriptional dynamics at time scales comparable to other biochemical processes. It is thus important to understand how trancriptional response can modify the dynamical behavior of gene circuits.

Recently, we have revisited the dynamics of a self-repressed gene where transcription rate is not a function of protein concentration but a dynamical variable converging to the usual equilibrium value over a finite time, playing the role of a delay. To understand the interplay of this delay with nonlinearity in the degradation processes, we considered arbitrary degradation mechanisms for RNA and protein. Remarkably, the oscillation threshold of this model can be computed analytically, and depends only on normalized gene response time and degradation rates. We also found that when gene response time is equal to a characteristic time whose expression can also be computed analytically, oscillations can be induced by degradation mechanisms much less nonlinear than for infinitely fast regulation.

To determine if this behavior is robust, we have studied a model including an additional delay, describing cellular transport or transcription/translation. We considered both the case of an explicit delay and of a delay resulting from an extra reaction step, to understand the influence of the modeling choice. Again, we could find analytical criteria for the appearance of oscillations.

These results allow us not only to characterize quantitatively the interplay of delay and nonlinear degradation, but also to study how two delays interact. In particular, we found that two delays in sequence can be more destabilising than a single delay of equivalent duration, and that a small delay added on top of a large delay can suffice to trigger oscillations.

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6

Exploring the dynamics of postural sway in humans using recurrence quantification analysis [23]

Sofiane Ramdani1, Beno ˆ it Seigle1, Julien Lagarde1, Fr ´ ed ´ eric Bouchara2, & Pierre Louis Bernard

11EA 2991 Efficience et D ´ eficience Motrices, Universit ´ e de Montpellier I, Montpellier, France

2UMR CNRS 6168 LSIS, Universit ´ e du Sud Toulon-Var, La Garde, France [e-mail]

In humans, postural sway during quiet standing can be measured through the fluctuations of the center of pressure (COP) by means of a force platform (Winter, 1995). COP time series are irregular, non-stationary and exhibit high variability. The complexity of such data has motivated human movement scientists to go beyond the classical kinematical measures derived from COP signals (Collins & DeLuca, 1993; Yamada, 1995; Riley et al., 1999; Costa et al, 2007; Ramdani et al., 2009).

Recurrence Quantification Analysis (RQA) is a nonlinear tool for the characterization of the underlying dynamics of time series (Eckmann et al., 1987; Zbilut & Webber, 1992, 1994; Marwan et al., 2007). It can be applied to non-stationary data. RQA has been first applied to COP by Riley et al. (1999). Others have used RQA to explore the effect of disease or aging on postural dynamics (Schmit et al., 2006; Seigle et al., 2009). Generally, the high level of percentage of determinism (DET) output of RQA is implicitly associated to the presence of nonlinear determinism in COP time series. The nature of their dynamics is still discussed in the literature (Pascolo et al., 2005, 2006; Ramdani et al., 2009). Here, we propose to test the hypothesis of the presence of nonlinear determinism by combining the computation of DET with the Monte-Carlo based approach of phase randomized surrogates (Theiler et al., 1992).

We recruited 10 young and healthy adults who were tested in two visual configurations. The data were analyzed in both anteriorposterior (AP) and mediolateral (ML) directions. The recordings lasted for 51.2 sec. The sampling frequency was 40 Hz, leading to 2048-points time series.

After extracting 1800-points subsequences minimizing the end-to-end mismatch, we generated 39 iteratively refined amplitude adjusted Fourier transform (iAAFT) surrogates (Schreiber & Schmitz, 1996, 2000) for each recorded time series. iAAFT surrogates are designed to test the null hypothesis Hof a linear stochastic underlying process. RQA was then performed on both original subsequences and their surrogate counterparts (with time delay 6, embedding dimension 8, radius 0 :25 of mean distance and lmin0= 4). The DET measure was used as a discriminating statistic.

0The recurrence rates were 0 :0749  0 :0435 (AP) and 0 :0564  0:0238 (ML). The DET values were 0:9650   0:0289 and 0 :9761  0:0117. The null hypothesis Hwas rejected for only 4 of the 40 analyzed time series. Our conclusion is that the high COP DET values are not the result of a nonlinear determinism but probably

caused by the correlations characterizing these data. Indeed, it has been reported that DET is not a measure of determinism and that it can be related to the correlations observed in the analyzed time series (Marwan & Kurths, 2009). This result is in accordance with the stochastic modeling of COP time series (Collins & DeLuca, 1993, 1995; Bosek, 2008).

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Direct observation of spontaneous veins formation and thickness oscillations in Physarum polycephalum [24]

Paul Dely1, Christophe Szwaj1, Serge Bielawski1, Olivier Hugon2, Olivier Jacquin2, Eric Lacot32, & Toshiyuki Nakagak

i1

2Lab. PhLAM, Universit ´ e de Lille 1 (France)

Lab. Spectro, Universit ´ e J. Fourier, Grenoble (France

)3RIES, Hokkaido University (Japan) [e-mail]

Physarum polycephalum (or slime mold) is a giant biological cell of the myxomycete family, which size is typically in the several cm range. Though primitive, this system displays complex spatiotemporal behaviors. In particular, this organism exhibits thickness oscillations (with temporal period around 1 min.) that generate cytoplasmic movement and a structuration of the cell with channels and veins [1].

Here we focus on experimental analysis (by infrared microscopy) of velocity fields in regions where a transition occurs from liquid cytoplasm to gel state, i.e., at places of vein formation. In addition, we present study of the thickness oscillations by the laser imaging technique LOFI [2].

The main objective is to obtain time-resolved, quantitative data, against which microfluidic theories of vein formation will be developed and tested.

[1] T. Nakagaki, Nature 417, 470 (2000); Yamada et al. PRE 59, 1009 (1999); Nakagaki et al., J. Theor Biol. 197, 497 (1999).

[2] E. Lacot, O. Hugon, Applied Optics, 2004, 43, 4915

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Automatic classification of sleep stages from one EEG measurement using nonlinear DDEs [25]

Claudia Lainscsek1;2, Dounia Bounoiare3;4, Adriana Portmann3, Antoine Cuvelier3, Jean-Franc¸ois Muir43, & Christophe Letellie

r1

2

3

4_x= f(x

 jInstitute for Theoretical Physics, University of Technology, Graz, Austria

INC, University of California at San Diego, La Jolla, CA, USA

GRHV EA 3830, Rouen Universitary Hospital, France

Our novel method is based on nonlinear DDE analysis. A DDE is an equation

1;x 2

= x(x

 CORIA UMR 6614, University of Rouen, France [e-mail]

Quantifying sleep fragmentation is central in assessment of sleep quality. The graphic representation of sleepstage sequences across the night is called a hypnogram and derived by visual scoring of 20–30 s pieces of EEG (electroencephalogram), EOG (electrooculogram), and EMG (electromyogram) recordings. Visual scoring is labor intensive, time consuming, and subject to errors between different scorers of around 20 %. Therefore a tool to automatically produce a hypnogram would be very helpful.

Here a method for automatic classification of sleep stages from one single EEG measurement using nonlinear delay differential equations (DDEs) is presented. The so obtained hypnograms are then compared to visual scorings by a neurologist.

;:::) (1) where xj ) and that relates the derivative at a data point to previous data points of the signal. The linear terms of such a DDE correspond to the main frequencies of the treated signal while the nonlinear terms are related to the phase couplings between its harmonic parts. This framework therefore can be seen as a time-domain analysis equivalent to a Fourier analysis that is very robust against noise contamination and fast.

In this study, 35 polysomnographies were extracted from our data base. They were recorded in patients who received noninvasive mechanical ventilation. In this work, the manually scored hypnograms were compared to scorings automatically obtained from the single time series of EEGs from the C3/A2 area. This was done by using the coefficients of the nonlinear term of a three-term DDE. The correlation between the manual and automatic scorings was around 80% for all patients.

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Soft iron impellers: induction mechanism and dynamo [26]

Gautier Verhille1, Nicolas Plihon1, Mickael Bourgoin2, Philippe Odier1, & Jean-Franc¸ois Pinton

11

2Laboratoire de Physique, CNRS & Ecole Normale Sup ´ erieure de Lyon, UMR5672, Universit ´ e de Lyon, 46 All ´ ee d’Italie, F69007, Lyon, France

Laboratoire des ´ Ecoulements G ´ eophysiques et Industriels, CNRS/UJF/INPG UMR5519, BP53, F38041 Grenoble, Franc

[e-mail]

The VKS experiments have shown a remarkable variety of dynamo regimes in a von K ´ arm ´ an (VK) flow of liquid sodium, with following main characteristics which we want to address: i) dynamo action has only been observed when soft iron impellers are used to drive the fluid motion, ii) for exact counter-rotation of the impellers, the magnetic field generated is an axial dipole whereas numerical simulation which do not include ferromagnetic boundaries predict a transverse dipole, iii) when the forcing is asymetric, dynamical regime may occur and can be described by a low dimensional involving only 2 magnetic modes.

In order to understand the role of soft iron, we have studied induction processes in a gallium von K ´ arm ´ an flow, with impellers made of different materials (stainless steel, soft iron and copper). Our results show that the soft iron promotes induction processes localized near the impellers. Extending our results to VK flows in liquid sodium (at significantly higher magnetic Reynolds numbers), we propose a mechanism for dynamo generation in VKS. This mechanism successfully accounts for the 3 points mentioned above.

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Chaotic oscillations in a weakly controlled brushless DC motor [27]

Domenico Porto

STMicroelectronics S.r.l. , Stradale Primosole 50, 95121 Catania, Italy [e-mail]

Brushless Direct Current motors (or BLDC motors) are widely used as actuators in a lot of industrial applications, and in particular in automotive field, for their robustness and for the safety-compliant absence of brushes, which are dangerous for the possible generation of sparks, quite undesired in these environments. Without brushes, motor currents commutation is obtained through an external control unit which also provides the achievement of the desired torque. In case of insufficient control, usually due to wrong parameters or malfunction, chaotic oscillations of the state variables (currents and speed) can be observed.