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

In this paper the behaviour of a weakly controlled BLDC motor is investigated using the tools proper of dynamical system analysis and chaotic dynamics detection. Several different configurations are presented and a quantitative measurement of the chaotic trends is obtained via the Lyapunov exponents analysis. Moreover, a structural analogy with Chaotic Neural Networks is evidenced in order to introduce an equivalent only-electric reference model.

The aims of this study can be synthesized in two different main objectives. The first one is to acquire the necessary information for the design of a suitable second level controller acting in case of malfunction of the first controller, while the second one is to investigate the possible use of a BLDC motor as an electro-mechanical chaos generator displaying a wide variety of attractors only by changing few control parameters.

9

1

Analyzing a complex system [28]

Jean-Marc Ginoux & Bruno Rossetto

PROTEE Lab., Toulon University, BP 20132, 83957, LA GARDE Cedex (France) [e-mail]

Abstract The aim of this work is to explore some ways to draw out information from the solutions of a dynamical system

having two kinds of complexity, a high number of interacting freedom degrees and time varying coefficients. Some geometrical properties conferred by the system to the phase space are used to split up the system in simple elements and to analyze the symmetries. The model taken as an example concerns the dimethylsulfide (DMS) cycle. It consists of an eight-dimensional dynamical system with periodic coefficients. This problem asks more mathematical questions than it is possible to answer given what we know at the moment. But some features of the behavior of the solutions can be analyzed.

Introduction The dimethylsulfide (DMS) molecule dissolved in sea water evaporates under some conditions and helps to

supply most of the cloud condensation nuclei in the atmosphere. So the DMS cycle of ecosystems contributes to scatter and absorb incoming solar radiation and to moderate anthropogenic forcing of climate. This field gives rise to a broad interest and to a large number of papers, but the magnitude of the climate feedback of the DMS is difficult to appreciate.

The model The first part of this work is devoted to the construction of a model of the biogeochemical cycle of DMS based

on works of A. J. Gabric & al.[1]. The variables of an eight-dimensional mathematical model are concentration of phytoplankton, bacteria, zooflagellates, large protozoa, micro and mesozooplankton, dissolved inorganic nitrogen, dissolved dimethylsulfonio-propionate (DMSP) and dissolved DMS. The air-sea exchange of DMS depends in a complex way on the wind velocity and on the sea surface temperature which is a function of time.

Mathematical study At first, the asymptotic behavior of solutions is analyzed with the data of biologists and the interactions between

the populations are compared to reduce the number of dimensions of the dynamical system Then, the equation of an invariant manifold of an associated constant coefficient equivalent system is computed

in a very simple way using differential geometry results. This manifold is periodically crossed by the solutions and is involved in the structure of the attractor. On the other hand, the manifold may bring to light some symmetries of the solutions.

Conclusion The respective influence of other set of variables could be studied by this method. Reference 1. Gabric A. J., Gregg W., Najjar R., Erickson D., Matrai P., 2001. Modeling the biogeochemical cycle of

dimethylsulfide in the upper ocean: a review. Chemosph. Global Change Sc. 3: 377-392.

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2

Hyperbolic extremes and species dynamics in polychaete populations [29]

Benjamin Quiroz-Martinez1;2;3, Franc¸ois G. Schmitt1;2;3, Jean-Claude Dauvin1;2;31;2;3, & Jean-Marie Dewarume

z1

2

3Univ Lille Nord de France, F-59000 Lille, France

USTL, LOG, F-62930 Wimereux, France

9

3CNRS, UMR 8187, F-62930 Wimereux, France [e-mail]

One of the key features of environmental and geophysical field studies is their high variability at many different time and space scales. The dynamics of many natural populations involve the alternation over variable periods of time of phases of extremely low abundance and short outbreaks. The objective of this work is to characterise the dynamics of three diverse polychaete populations based on long-term benthic surveys of shallow fine sand communities in the Bay of Morlaix (Western English Channel) and in Gravelines (South of the North Sea), France. Abundance and species richness of polychaete populations display high variability, which was analysed using scaling approaches; we found that population density had heavy tailed probability density functions. We analysed the dynamics of relative species abundance in a community of trophically similar species, by estimating a diffusion coefficient which characterises its temporal fluctuations. We conclude on the necessity of using new tools to approach and model such highly variable population dynamics in coastal marine ecosystems

.

Experimental studies of defect dynamics in complex (dusty) plasmas [30]

Celine Durniak & Dmitry Samsonov

Dept. of Electrical Engineering and Electronics, The University of Liverpool, Liverpool, L69 3GJ, UK. [e-mail]

Complex plasmas consist of micron sized microspheres immersed into ordinary ion-electron plasmas. Similar to colloids, they can exist in solid, liquid or gaseous states and exhibit phase transitions. Complex plasmas are found in space: planetary rings, comets or interstellar clouds. In plasma technology, dust contamination has negative effects on the yield of semiconductor devices. The microparticles are charged negatively by the plasma. They strongly interact with each other electrostatically via a Yukawa potential. As the grains are weakly damped by gas friction and traceable individually, dynamic and nonlinear phenomena such as shocks, Mach cones, solitons, waves, elastic and plastic deformations can be observed at the kinetic level.

The experiments were performed in a capacitively coupled radio-frequency (rf) discharge. A powered lower electrode and a grounded ring upper electrode were placed in a vacuum chamber. A constant working pressure was maintained by a flow of argon. Monodisperse plastic microspheres were levitated in the sheath above the lower electrode. They were confined radially in a bowl shaped potential formed by a rim on the outer edge of the electrode and formed a monolayer hexagonal lattice. They were excited by voltage pulses applied to wires stretched above the electrode at approximately the same height as the particles. A horizontal thin sheet of laser light illuminated the particles, which were imaged by a digital video camera.

We performed a molecular dynamics (MD) simulation in order to support the experimental results. The molecular dynamics simulation code that we have developed solves the equations of motion for each microparticle moving in a global parabolic confinement potential and interacting with every other microparticle via a Yukawa potential [1]. The code is based on an object-oriented multi-threaded programming. It can be used to simulate various particle systems which can be characterised by interaction forces or potentials such as complex plasmas, colloids, granular media, plasma doping, ion beams, film growth, ion implantation. The equations of motion are solved using the fifth-order Runge Kutta method with the Cash Karp adaptive step size control. The ion-electron plasma is not explicitly included in the equations. The grains are damped by the friction force (equal to the neutral gas damping). We consider three- and two-dimensional (2D) systems of 3000 microparticles, which are first seeded randomly and the code is run until a crystalline structure is formed. Then different excitation forces are applied during a short time on the lattice.

We use an experimental model system (complex plasma) and MD simulation to study the dynamics of defects in 2D hexagonal lattices: dislocations or penta-hepta defects. We focus on their interactions with localized compressional waves in complex plasma crystals.

[1] C. Durniak, D. Samsonov, S. Zhdanov, and G. Morfill, EPL 88, 45001 (2009).

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4

Design of OPCL coupling for arbitrary lag synchronization in chaotic oscillators [31]

Prodyot Kumar Roy1, Sourav Kumar Bhowmick2, Ioan Grosu3, & Syamal Kumar Dana

21

2

3Department of Physics, Presidency College, Kolkata 700073, India

Central Instrumentation, Indian Institute of Chemical Biology, Kolkata 700032, India

Abstract We discuss the method of arbitrary lag synchronization (ALS) in chaotic oscillators under unidirectional OPCL

References: [1] D.V.Senthilkumar and M.Lakshmanan, Phys. Rev. E 71, 016211 (2005); S. Sivaprakasam, P. S. Spencer, P.

9

5Faculty of Bioengineering, University of Medicine and Pharmacy, Gr.T.Popa, Iasi, Romania [e-mail]

coupling. By ALS, we mean that any arbitrary lag time can be set between the driver and slave oscillators. The added advantage is that, one can precisely control the synchronization. LS is already reported in time-delayed systems by others [1] under unidirectional delay coupling. The limitation of such methods is their restriction on the amount of time lag. Recently, instead of using simple linear coupling other approaches [2, 3] are reported which increases the lag time for LS [2] or anticipating synchronization [3]. Although these methods enhance the lag time to an extent yet it remains restricted. In contrast, our proposed OPCL delay coupling is free from such limitation. One delay variable is introduced in the coupling term used in [4], which helps one target any ALS between the two-coupled chaotic oscillators. The delay time may be of the order of mean characteristic time scale of the system or even its multiples. Further, the method allows flexibility in controlling the lag time. We elaborate the method with numerical examples of R ¨ ossler system, a Sprott system and also with a neuron model namely the Hindmarsh-Rose model. Finally, we present experimental evidence of ALS in electronic circuit.

Rees, K. A. Shore, Optics letters 27(14), 1250 (2002). [2] K.Pyragas, T.Pyragiene, Phy.Rev.E 78, 046217 (2008). [3] J.N. Blakely, M.W. Pruitt, N.J. Corron, Chaos 18, 013117 (2008). G.Ambika and R.E.Amritkar, Phys. Rev. E 79, 056206 (2009) [4] I.Grosu, E.Padmanaban, P.K.Roy and S.K.Dana, Phy.Rev.Lett. 100, 0234102 (2008); I.Grosu, R.Banerjee, P.K.Roy and S.K.Dana, Phy.Rev.E 80, 016212 (2009)

.

Experimental study of chaos in parallel-connected DC-DC boost converter with mutually-coupled output filter-inductors [32]

Ammar Natsheh1, J. Gordon Kettleborough2, Awni Jayyousi1, & Moh’d Mothafar

31

2

3Department of Electronic and Communication Engineering, Faculty of Engineering, Al-Ahliyya Amman University, Post Code 19328 Amman, Jordan

Department of Electronic and Electrical Engineering , Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK

9

6Department of Electrical Engineering, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, Jordan ammar [e-mail]

An experimental study is presented of a modular peak current-mode controlled DC-DC boost converter. The parallel-input/parallel-output converter consists of two identical boost circuits and operates in the continuous conduction current mode. [1] investigated the small signal and transient behaviour of two-module DC-DC boost converter with mutually coupled inductors but chaotic behaviour was not addressed. This device is capable of demonstrating chaotic behaviour [2] arising as a result of period-doubling bifurcations as the main control parameter, reference current, is changed. Chaotic behaviour is undesirable since it results in increased losses together with acoustic noise, and may cause catastrophic failure of the unit. Mathematically, this controller is described by piece-wise linear differential equations under external periodic forcing [3]. To prevent chaos in it, Delayed current feedback control illustrates the effectiveness and robustness of the chaos control scheme [4]. Experimental results and FORTRAN simulations show good agreement. The effect of chaos in the presence of mutual coupling between the inductors of the constituent modules is demonstrated. Experimental results and MATLAB simulations match remarkably and correlate the presence of coupling leads the system to chaos. Results are also verified using the circuit analysis package PSPICE and COMSOL simulations

.

Single-ended chaotic Colpitts oscillator with active load [33]

Odysseus Tsakiridis1, Vassilis Stefanidis2, Evangelos Zervas, & John Stonha

m1

2Dept. of Electronics, TEI-Athens, Egaleo 12210, Athens, Greece

9

7Theta Microelectronics S.A. , Marousi 15125, Athens, Greece [e-mail]

A novel version of a single-ended microwave chaotic Colpitts oscillator is proposed. It contains two bipolar junction transistors with the one of them used as an active load which is connected as a diode, in the collector of the basic bipolar transistor. The Chaotic Colpitts oscillator with active load, compared with the classical circuit, gives different oscillator dynamics with more intense chaotic behaviour due to the high small-signal impedance and small DC voltage drop that it has. Simulations performed for two cases: the classical single-ended Chaotic Colpitts Oscillator and the novel single-ended Chaotic Colpitts Oscillator with Active Load. Results showed that the highest fundamental frequency of chaotic behaviour are about 1.4 GHz for classical chaotic oscillator and 1.67 GHz for the novel chaotic oscillator

.

Experiments in noise-enhanced propagation and related phenomena: fault-tolerant behavior and other properties [34]

Roberto R. Deza1& Mauro F. Calabria2

1

2IFIMAR (Mar del Plata Institute for Physics Research, UNMdP and CONICET), De ´ an Funes 3350, B7602AYL Mar del Plata, Argentina.

Electronics Department, Faculty of Engineering, Universidad Nacional de Mar del Plata (UNMdP), J. B. Justo 4302, B7608FDQ Mar del Plata, Argentina.

[e-mail]

We study the propagation of a low-frequency periodic signal through a chain of one-way coupled bistable oscillators, subject to uncorrelated additive noises. The system can be regarded as a mock-up of synaptic transmission between neurons. This work focuses on optimizing input SNR and switching threshold of each oscillator, to achieve maximal coherence (measured as a Hamming distance) between the last oscillator’s response and the input signal. At a further stage, we shall focus on the fault-tolerant behavior of the system [Phys. Rev. E 61, R3287 (2000)].

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8

High resolution parameter spaces for an experimental chaotic circuit [35]

Emilson R. Viana Jr1, Rero M. Rubinger2, Holokx A. Albuquerque3, Alfredo G. de Oliveira11, & Geraldo M. Ribeir

o1

2Universidade Federal de Minas Gerais, UFMG

Universidade Federal de Itajub ´ a, UNIFE

I3Universidade do Estado de Santa Catarina, UDESC [e-mail]

The interest in codimension-two bifurcations in flows, when we vary simultaneously two of the system parameters, have grown substantially in last years. This is due to the observation of complex periodic structures, immersed in chaotic regions, until recently just observed in discrete time maps. More recently, some works reported the existence of those periodic structures inside the chaotic phases in some systems described by continuous-time models. Regarding experimental data, few works reported those structures in two-dimensional parameter spaces with lowresolution. Therefore, the aim of this work is to report two high-resolution experimental parameter spaces for a chaotic circuit, in this case, a Chua´s Circuit.

The Chua´s Circuit is forced by a voltage source d.c., in series with the Chua´s Diode. Such resolution in the parameter spaces was propitiated by the use of a 0.5 mV step d.c. voltage source as the new control parameter. The voltage Vdc change the equilibrium points, defined by the intersection of the ”line charge” and the Chua´s I(V) curve. So we have different intersections points for different control parameters.

The two high-resolution codimension-two parameter-spaces presented in this work, one for the periodicity and one for the largest Lyapunov exponent, show abundance of complex periodic structures. Those complex periodic structures organize themselves in a period-adding bifurcation cascade, as (period-2)-(chaos)-(period-3)-(chaosand so on ... , that accumulates in the chaotic region, for Vdc = 0.0000 V. Numerical investigations on the dynamical model of this forced circuit were also carried out to corroborate several new features observed in those experimental high-resolution parameter-space.

This forced circuit consists in a platform for the study of this intricate periodic networks formed by periodic self-similar structures surrounded by chaotic phases. Regarding chaos based communication systems, the knowledge of what exactly is embedded in the regions of chaos, in dynamical systems, is an important question since clean and extended domains of chaos are important for applications in secure communications.

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9

Video encryption with chaotically coupled chaotic maps [36]

Emmanuel Valdes-Jaramillo1, J. Ricardo Sevilla-Escoboza1, Rider Jaimes-Reategui1, J. H. Garc ´ ia-Lopez2, Massimiliano Zanin, & A. N. Pisarchik3

11

2

3Centro Universitario de los Lagos, Universidad de Guadalajara, Enrique D ´ iaz de Le ´ on 1144, Lagos de Moreno, Jal, M ´ exico.

Universidad Autonoma de Madrid, Cantoblanco, 28049, Madrid, Spain

Centro de Investigaciones en ´ Optica, Loma del Bosque 115, Lomas del Campestre, 37150, Le ´ on, Guanajuato, M ´ exico

,[e-mail]

To encrypt video we use a secure cryptosystem for direct encryption of color images in each frame of a video, based on chaotically coupled chaotic maps, that provides good confusion and diffusion properties that ensures extremely high security because of the chaotic mixing of pixels colors, using a DMP (Digital Media Processor) we process video, separate it in 3 components RGB (Red, Green, Blue) and apply our algorithm for encryption or decryption.

10

0

Plasma confinement in tokamaks with robust torus [37]

Ricardo Egydio de Carvalho1, Caroline G. L. Martins1, Ibere L. Caldas2, & Marisa Roberto

31

2

3Univ Estadual Paulista-UNESP - Rio Claro/SP - Brazil

Universidade de S ˜ ao Paulo-USP - S ˜ ao Paulo/SP - Brazil

10

1Instituto Tecnol ´ ogico da Aeron ´ autica-ITA - S ˜ ao Jos ´ e dos Campos/SP - Brazil [e-mail]

The non-twist standard map occurs frequently in many fields of science specially in modeling the dynamics of the magnetic field lines in tokamaks. Robust tori, dynamical barriers that impede the radial transport among different regions of the phase space, are introduced in the non-twist standard map in a conservative fashion. The resulting Non-Twist Standard Map with Robust Tori (NTRT) is an improved model to study transport barriers in plasmas confined in tokamaks. The robust torus prevents the magnetic field lines to reach the tokamak wall and reduces, in its vicinity, the destruction of islands and invariant curves due to the action of resonant perturbations. Our results indicate that the RT implementation would decrease the field line transport at the tokamak plasma edge

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Pattern formation on sandy bottom: front propagation into sand ripples under the action of regular surface waves [38]

Julie Lebunetel-Levaslot1, Armelle Jarno-Druaux1, Alexander Ezersky2, & Franc¸ois Marin

11

2FRE CNRS 3102 Universit ´ e du Havre, 25 rue Philippe Lebon, 76058 Le Havre, France

10

2CNRS 6143 M2C Universit ´ e de Caen, 2-4 rue des Tilleuls, 14000 Caen, France [e-mail]

Pattern formation on a bottom under the action of surface waves is a manifestation of instability caused by relative motion of sand and water. The morphological characteristics of sand ripples patterns are important for the prediction of the dissipation of waves energy, and for the sediment transport. They also influence the biological processes occurring on the bottom and the dispersion of pollutants. We report our results of an experimental study of pattern formation on sandy bottom under the action of regular harmonic surface waves. It was found that two modes of pattern formation occurred: either from localized nucleation sites or from everywhere on the bottom as a uniform pattern. In the first regime sandy ripples appeared in the isolated regions of bottom (patches) increasing in size and front propagation speed was measured. Simple dynamical model based on Ginzburg-Landau equation was proposed to explain characteristics of patches. We have found that the propagating front characteristics depend on the direction of surface waves which generate ripples. If the velocity of front is co-directed with the surface waves propagation, the front has a larger celerity, is steeper and more irregular than the front which propagates in the opposite direction of surface wave

.

Stability analysis of turbulent boundary layer flows with adverse pressure gradient [39]

Jean-Philippe Laval1;2, Matthieu Marquillie1;2, & Uwe Ehrenstein

31

2

3CNRS, UMR 8107, F-59650 Villeneuve d’Ascq, France

Univ Lille Nord de France, F-59000 Lille, France

10

3Aix Marseille Univ, IRPHE UMR 6594, CNRS, F-13384 Marseille 13, France [e-mail]

The turbulent boundary layer flow subjugated to adverse pressure gradient coming from curvature are of crucial importance for many applications including aerodynamics of airfoils, ground vehicles or turbine blades. Significant progress are needed in understanding the near wall turbulence in order to improve the theoretical and numerical models. The available numerical models usually fail as they are based on scaling of wall turbulence which are no more valid with pressure gradient. Therefore, a careful analysis of turbulent structures generation are the only opportunity to make progress in designing accurate statistical models for turbulence. The Direct Numerical Simulation (DNS) of the Navier Stokes equations is an efficient tool to study the complete time and 3D space behaviors of the full range of turbulent structures. DNS was already used to identify and to study the cycle of generation of turbulent structures in turbulent boundary layer without pressure gradient. A large experimental and numerical database of turbulent boundary layer were generated through the European project WALLTURB in order to extract physical understanding of these flows.