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

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Synchronization of uncoupled excitable sytems induced by white and coloured noise [53]

Riccardo Meucci1, Samuel Zambrano2, Ines P. Marino2, Jesus M Seoane2, Miguel A. F. Sanjuan1, Stefano Euzzor, Andrea Geltrude1, & Tito F. Arecchi1

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2Istituto Nazionale di Ottica, Largo E. Fermi 6, 50125 Firenze, Italy

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7Universidad Rey Juan Carlos, Tulipan s/n, 28933 Mostoles, Madrid, Spain [e-mail]

We study, both numerically and experimentally, the synchronisation of uncoupled excitable systems due to a common noise. We consider two identical FitzHugh-Nagumo (FHN) systems, which display both spiking and nonspiking behaviours in chaotic or periodic regimes. An electronic circuit provides a laboratory implementation of this dynamics. Synchronisation is tested with both white and coloured noise showing that coloured noise is more effective in inducing synchronisation of the systems. We also study the effects on the synchronisation of parameter mismatch and of the presence of intrinsic (not common) noise, and we conclude that the best performance of coloured noise is robust under these distortions. Similar results are being obtained experimentally in a circuit with four uncoupled FHN with common noisy input

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Nontrivial effects of noise in excitable electronic circuits [54]

Guillermo V. Savino2, Roberto R. Deza1, & Carlos Formigli

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2IFIMAR (UNMdP and CONICET) Mar del Plata, Argentina

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8Fac. de Ciencias Exactas y Tecnolog ´ ia, Universidad Nacional de Tucum ´ an, Argentina [e-mail]

We present experimental results on noise-induced synchronization, stochastic resonance, coherence resonance and frequency matching using two non-identical weakly coupled electronic models of a neuron. Electronic neurons are always non-identical due to the value dispersion of the electronic components, and they are unavoidably coupled when using a common noise source. Our circuit can be tuned to self-oscillate so as to produce (i) single spikes at non-regular inter-spike intervals, or (ii) spikes that are interspersed with two- and three-spike bursts. The phase portrait shows a stable limit cycle and a saddle point, originating thus a stable and an unstable manifold, both necessary to get noise-induced phase synchronization according with previous theoretical models. By applying to two such “neurons” a common noise of increasing intensities, their initially very different instantaneous frequencies tend to match and the system‘s behavior to become periodic. We show that this effect is noise-mediated, rather than due to the weak coupling. The measured activation times become equal in both oscillators for a definite noise intensity, and the same occurs for excursion times. Experimental evidences support the hypothesis that the mechanisms of coherence resonance are operating.

The plot of the phase differences between the spike sequences in both circuits as function of time for different noise intensities shows plateaus with different durations, indicating phase synchronization induced by the common noise. Nevertheless, complete synchronization has not been observed.

Our experimental results are relevant for real neurons, since our circuit shares the same bifurcation scenarios, and the underlying mechanism—namely conductivity change—is the same. These experiments may thus help understand how neurons transmit, encode and process information

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Synchronization phenomena in networks of neuron models [55]

Nathalie Corson1, Stefan Balev2, & M.A. Aziz-Alaoui

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2

[e-mail]

[e-mail]

ijg(1). The neurons are coupled by nonlinear functions (2) modeling chemical synapses. 8< :_xi= ax2 i3 ix+ yizi Pn j=1cijh(x) _yi_zi= (a+  )x=  (bxi2 iy+ czh(xi;xjiii;xj) i= 1;:::;n (1)) = g( n) syn(xijV) 1 + exp( (x )) (2)

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[e-mail] [e-mail]

Synchronization phenomena arise within many natural or artificial interaction networks. In this work we consider Hindmarsh-Rose oscillators modeling an individual neuron behavior and connected in a network with adjacency matrix fc

The HR model exhibits most of the behaviors observed in the case of real neurons, such as spike firing or bursting. The bursting motion consist of successive series of action potentials separated by slow periods.

The most studied kind of synchronization is the so called complete synchronization, which means that all the nodes of the network share the same behavior at the same time. In this work we study the condition of complete synchronization in networks of coupled neuronal models. We show that in order to obtain synchronization, all the nodes must have the same in-degree. The minimal coupling force needed to synchronize a network depends only on the in-degree of the nodes as a power law.

The necessary condition for complete synchronization is quite restrictive and biologically unrealistic, that is why we consider another type of synchronization phenomenon, called burst synchronization.

An oscillator network presents burst synchronization if the noscillators of the network fire bursts starting all at the same time. If the complete synchronization is easy to detect, no algorithm exists to detect burst synchronization. Therefore, in this work, we propose an algorithm of burst synchronization detection within networks. Our algorithm can be decomposed in four main steps. In order to detect burst synchronization, bursts of different neurons must be matched. To do this, one needs to determine the start time of each burst, and before detecting bursts, spikes must be detected first. This algorithm is then applied to different kinds of networks topologies for which we study the minimal coupling force needed to obtain burst synchronization

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Bursting dynamics in a two-mode semiconductor laser with optical injection: experimental results and theoretical analysis [56]

Stephen O’Brien, Simon Osborne, David Bitauld, & Andreas Amann

Tyndall National Institute, Lee Maltings, University College, Cork, Ireland [e-mail]

In this work we describe our recent experimental and theoretical studies of bursting dynamics in an optically injected two-mode semiconductor laser. The device we consider is a specially engineered Fabry-Perot laser diode with a large (terahertz) primary mode spacing. This device can be biased such that both primary modes oscillate simultaneously with the same average power level. Where one of the primary modes is optically injected, the presence of the second lasing mode leads to a very rich dynamical scenario. In particular, we have found two distinct examples of dynamics that are associated with large amplitude bursting of the intensity of the uninjected primary mode.

The first example is characterised by irregular bursting of the intensity of the uninjected mode in regions where the dynamics of the injected mode are chaotic. In contrast, the second example is characterised by regular bursts that are in antiphase and which have variable period. These regular dynamics are found close to regions where dramatic switching between single and two-mode dynamical regimes occurs.

We have found that both of these examples of dynamics are reproduced with remarkable accuracy by a deterministic four dimensional rate equation model. The structure of the model is such that the dynamics of the well-known model of the single mode injected system are contained in an invariant submanifold of the two-mode system. Irregular bursting dynamics are then described by on-off intermittency that is associated with the transverse instability of chaotic dynamics in the injected mode submanifold. Experimentally, we have found significant departures from ideal scaling in the distribution of interburst times in this case. We are currently studying the effect of correlations on these distributions, which are in good agreement with modelling results.

On the other hand we show that the bifurcation scenario for regular bursting dynamics is organised by codimension two points at which saddle node of limit cycle and transcritical bifurcation lines tangentially intersect. At the saddle node of limit cycle bifurcation line the time interval between bursts diverges, and therefore gives rise to a dynamical behaviour which is similar to the Blue Sky Catastrophe in generic systems. We discuss the associated phase space structure, and compare with other infinite period bifurcations described in the literature.

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0

High frequency open loop control of a nonlinear oscillator like a Nd:YVO4 Q-switched laser [57]

Juan-Hugo Garcia-Lopez, Rider Jaimes Reategui, Didier Lopez-Mancilla, Edgar Sevilla, & Roger Chiu-Zarate

Universidad de Guadalajara, Centro Universitario de los Lagos, Enrique D ´ iaz de Leon, Paseos de la Monta ˜ na, 47460 Lagos de Moreno, Jalisco, M ´ exico [e-mail]

[e-mail]

Open loop control of a non-lineal oscillator like a Nd:YVO4 acoustic-optic q-switched laser at high frequency (2kHz-2MHz) is studied. The study was done by a four level transition for an ideal solid state laser, where the principal variables to consider were the population inversion and the intensity of the laser. The control parameter for this work was the modulation of the loss into the cavity of the laser, generated for the acoustic-optic modulator, using a square function. The bifurcation diagram of local maxims of the laser intensity in the interval of 1.1-1.5 MHz showed coexistent attractors and different dynamic behaviors, such as, fixed point, periodic and chaotic orbits when the control parameter was change. Words: Q-Switched, Diode Pumped, Solid State Laser.

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Experimental investigation of chaotic oscillations in DFB and FP semiconductor lasers with strong incoherent optical feedback [58]

L. Cardoza-Avenda ˜ no1, R. M. L ´ opez-Guti ´ errez1, C.A. L ´ opez-Mercado2, V. Spirin2, & C. Cruz-Hern ´ andez

21Engineering Faculty, Baja California Autonomous University (UABC), M ´ exico

2Scientific Research and Advanced Studies Center of Ensenada (CICESE), M ´ exico [e-mail]

Chaotic secure communications are the subject of many experimental and theoretical investigations, since the idea of synchronization between two chaotic oscillators was proposed by Pecora and Carroll in 1990 [1].

In particular, chaotic oscillations of laser diodes have drawn considerable attention due to their potential applications in fiber-optical secure communication systems [2]. In general, the chaotic carriers can be generated by semiconductor lasers through relatively weak optical injection, or optical feedback. There are many parameters to characterize instabilities and chaos in semiconductor lasers however one important and most useful parameter to figure out the characteristics is the reflectivity of the external mirror [2]. The optical feedback phenomena from a distant reflector longer than the laser coherence are usually attributed to the incoherent effect.

In this work, we focus on an experimental characterization and comparison of chaotic oscillations in semiconductor distributed feedback (DFB) and Fabry-Perot (FP) lasers without in-built optical isolators subjected by a strong incoherent optical feedback. Both DFB and FP standard telecommunication lasers routed to chaos exhibits a widened RF spectrum accompanied by clear optical spectrum changes. As we found the linewidth of DFB laser drastically increases up to 0.5 nm for 40 Emission in the time domain is amplitude-modulated, showing a non periodic and very complex behavior with positive maximum Lyapunov exponents for all investigated regimes. However we did not record understandable dependence of maximum Lyapunov exponent on intensity reflectivity despite the fact that standard intensity deviation strongly depends on feedback strength.

Altogether FP laser subjected by strong incoherent feedback strength demonstrates more chaotic behavior compare with DFB one for frequencies up to 1 GHz, very likely due to additional variations attributed to power switching between longitudinal FP laser modes. The obtained experimental results of chaotification in lasers are of fundamental importance for practical applications, in particular for chaos synchronization and chaotic communication in networks; see for example [3-4].

REFERENCES 1. Pecora L.M. and Carroll T.L., Synchronization in chaotic systems, Phys. Rev. Lett. 64, 1990, 821-824. 2. Junji Ohtsubo. Semiconductor Lasers. Stability, Instability and Chaos, Second, Edition, Springer-Verlag,

Berlin Heidelberg, 2005, 2008. 3. Posadas-Castillo C., L ´ opez-Guti ´ errez R.M., and Cruz-Hern ´ andez C. (2008) Synchronization of chaotic solid-

state Nd:YAG lasers: Application to secure communication, Communications in Nonlinear Science and Numerical Simulation, Vol. 13, No. 8, 1655-1667.

4. L ´ opez-Guti ´ errez R.M., Posadas-Castillo C., L ´ opez-Mancilla D., and Cruz-Hern ´ andez C. (2009). Communicating via robust synchronization of chaotic lasers. Chaos, Solitons and Fractals, Vol. 41, No. 1, 277-285.

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Feedback bandpass filter effects in the dynamics of an optoelectronic wavelength nonlinear delay system [59]

Maxime jacquot, Romain Martinenghi, Yanne Kouomou Chembo, & Laurent Larger

Optics Dept. / FEMTO-st / Besanc¸on / France [e-mail]

In a previous work [1], we studied experimentally, numerically, and analytically the response of a nonlinear optical oscillator subject to a delayed broadband bandpass filtering feedback. Its dynamical response was described by an integro?DDE that differs from Ikeda family of first order DDEs, only by the presence of an integral term. In this talk, we report on an optoelectronic wavelength nonlinear delay dynamics ruled by a feedback tunable bandpass filter. The particular influence of this filtering feedback determining the differential process of the dynamics is presented both experimentally and numerically. Multiple time scales phenomena like slow and fast periodic regime, regular or chaotic breathers, envelope dynamics, complex self pulsing, and fully developed chaos are observed ranging over several orders of magnitude, under various parameter and filtering feedback conditions. Time-frequency approach with wavelet transform is proposed in order to analyze multi-scale behaviour of the recorded time series. The influence of the characteristic delay frequency, and its location in the Fourier spectrum with respect to the filtering feedback cut-off is also reported. The observed behaviour offer attractive potential for many applications, e.g. in chaos?based communications, high spectral purity microwave generation, random number generation and chaos computing.

[1] M. Peil, M. Jacquot, Y.C. Kouomou, L. Larger, T. Erneux, ”Routes to Chaos and Multiple Time Scale Dynamics in Broadband Bandpass Nonlinear Delay Electro-Optic Oscillators”, Physical Review E, Vol.79, 026208, February 2009.

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Experimental evidence of microwave envelope chaos using an integro-differential optoelectronic system [60]

Yanne Chembo, Kirill Volyanskiy, Maxime Jacquot, & Laurent Larger

FEMTO-ST Institute (UMR CNRS 6174), Optics department, 16 route de Gray, 25030 Besanc¸on cedex, France [e-mail]

A very wide variety of systems have been shown to display a chaotic behavior since the pioneering work of Lorentz in the early sixties. This ubiquity has been experimentally evidenced in a very wide range of frequencies, ranging from the low frequency of mechanical oscillators to the ultra-high frequencies of amplitude/phase chaos in lasers.

In this communication, we experimentally evidence a spectrally interesting chaotic dynamics, where a 3 GHz microwave is driven to a state where only its slowly varying complex envelope becomes chaotic. In the Fourier domain, the system has a quasi-white spectrum within a very narrow bandwidth (16 MHz) around the central frequency of the carrier.

This dynamics is generated using a narrow-band optoelectronic oscillator. The corresponding model is an integro-differential delay differential equation, and it enables to analyze the essential dynamical features of the system. Beyond the interest to be devoted to this oscillator for its fondamental interest, it also appears to be the idoneous tool for many applications. In particular, we will explain how it could be used to implement chaos cryptography in free-space microwave telecommunication networks, or to improve the performances of wideband radar systems.

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Synchronization and mixed mode oscillations in a network of coupled light emitting diodes [61]

Marzena Ciszak1, Sora F. Abdalah1;2, Kais Al-Naimee1;3, Francesco Marino4, Riccardo Meucci1;41, & Tito F. Arecch

i1

2

3

4CNR-Istituto Nazionale di Ottica, L.go E. Fermi 6, 50125 Florence, Italy

High Institute of Telecommunications and Post, Al Salihiya, Baghdad, Iraq

Physics Department, College of Science, University of Baghdad, Al Jadiriah, Baghdad, Iraq

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5Physics Department, University of Florence, I-50019 Sesto Fiorentino (FI), Italy [e-mail]

We present results on the synchronization in a network of coupled light emitting diodes (LED) in the presence of AC-filtered nonlinear opto-electronic feedback. Each LED can undergo a variety of dynamical behaviours like chaotic and periodic mixed mode oscillations. These scenarios are found in a simplified physical model of the experimental system. The aim of the research is to create a miniaturized LED network containing many nodes imitating a neural network. Here we present experimental and numerical results for the transition to synchronization of N 6 nodes coupled in the global configuration

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Anomalous thermalization of nonlinear wave systems [62]

Stephane Randoux1, Antonio Picozzi1, Hans Jauslin2, & Pierre Suret

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2Laboratoire de Physique des Lasers, Atomes et Molecules, UMR-CNRS 8523, Universite de Lille, France

[1] A. Picozzi, Opt. Express, 15 9063 (2007) [2] S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, Physica D, 57 96 (1992) [3] P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, Phys. Rev. Lett. 104 054101 (2010)

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6Institut Carnot de Bourgogne, UMR-CNRS 5209, Universite de Bourgogne, Dijon, France [e-mail]

In complete analogy with a system of classical particules colliding inside a gas medium, an incoherent optical field can evolve, owing to nonlinearity, towards a thermodynamic equilibrium state [1]. In this respect, the spatiotemporal dynamics of the light field is governed by the nonlinear Schrodinger equation and its equilibrium spectrum has been determined in the framework of the weak turbulence theory [1,2]. It is expected that experiments made in the field of nonlinear optics can possibly lead to the observation of turbulence or thermalization of nonlinear waves [1,2]. Here we present an experiment in which we study the equilibrium spectra reached by a set of two partially-coherent light waves copropagating inside an ultra-low birefringence single-mode fiber. The two waves have opposite circular polarizations and are coupled through optical cross-Kerr effect. Using kinetic wave theory, we show that the wave system may exhibit a process of anomalous thermalization which is characterized by an irreversible evolution of the waves towards a specific equilibrium state [3]. This equilibrium state is of a fundamental different nature than the conventional RJ equilibrium state and in particular, the tails of the equilibrium spectra do not meet the property of energy equipartition. The theoretical analysis reveals that the interaction is submitted to degenerate resonances which prevent the system to reach the usual thermodynamic Rayleigh-Jeans (RJ) equilibrium distribution. The anomalous thermalization is characterized by a process of entropy production: The novel family of equilibrium states is associated to a maximum of the nonequilibrium entropy subject to an additional constraint due to the existence of a local invariant in frequency space. In the experiments, the Raman effect induces a non negligible dissipation over only a few nonlinear interaction lengths so that only the transient nonlinear regime leading to anomalous thermalization is experimentally accessible. However the observation of this transient regime reveals that some of the phenomenom signatures which are predicted in the kinetic regime (where linear effects dominates nonlinear effects), are robust enough to be preserved in the nonlinear interaction regime. The robustness of the behaviors found in experiments performed far from the kinetic regime opens theoretical questions about nonlinear propoagtion of incoherent waves in dispersive nonlinear media

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Hybrid chaos based communication system - a chaotically masked electronic message transduced to an optical carrier for transmission [63]

Joshua Toomey1, Deborah Kane1, Aleksandar Davidovic2, & Elanor Huntington

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2Physics Department, Macquarie University, Sydney, NSW 2109, Australia

School of Information Technology and Electrical Engineering, University College, University of NSW, Canberra 2600, Australi

[e-mail]

Synchronised chaotic systems are the basis of secure communication using a chaotic carrier for message masking. Systems demonstrated to date have used nonlinear electronic circuits or nonlinear laser systems to produce either an electronic or an optical chaotic carrier to which to add the data signal for masked transmission. The message can be recovered by virtue of a synchronised receiver only producing a match to the chaotic carrier, not the message. We have demonstrated a hybrid electronic/optical secure communication system for chaotic signal masking. We use an electronic circuit to generate a chaotic current signal in which a small message signal is added and masked. The combined chaos/message signal is added to the DC injection current of a semiconductor laser. The chaotic carrier plus message is reproduced as the output power variations of the laser which is transmitted optically. Transmission is by line of sight, free space propagation to an optical detector, although it is also possible to transmit via an optical fibre. A matched receiver electronic circuit synchronises only to the chaotic part of the photodetected signal. We show the successful transmission and recovery of a chaos masked message. We will present the advantages and disadvantages that this system has compared to all-optical or all-electronic chaos secure communication systems and also the prospects for further research and development.