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

12

7

Identification of multiple folding mechanisms of chaos generation by topological analysis applied to a highly dissipative system [64]

Juan Carlos Mart ´ in1& Javier Used2

1

2Department of Applied Physics, University of Zaragoza, C/ Pedro Cerbuna, 12, E-50009 Zaragoza, Spain

Department of Physics, Univ. Rey Juan Carlos, C/ Tulip ´ an s/n, E-28933 M ´ ostoles, Madrid, Spain [e-mail]

The chaotic emission of an erbium-doped fiber laser with sine-wave pump modulation has been analyzed for different modulation frequencies and modulation indexes. For each working condition considered, the template which summarizes the corresponding chaotic attractor has been determined by means of topological analysis techniques. The interest of the work is double: on the one hand, because of the procedure employed for the analysis, which is not the conventional one; and on the other hand, because of the diversity of templates obtained, much wider than in any other experimental systems previously studied, and particularly because of the novelty of some of these templates.

As the system is highly dissipative, it is possible to complement the usual topological analysis procedure (1) with a different technique (2): the high dissipation causes that the Poincare sections obtained are thin enough to be considered as a line. A continuous parameterization along this one-dimensional object can be defined so that the first-return map with regard to the parameter chosen is an application. Maxima and minima of the first-return map obtained determine a generating partition and, therefore, the number of branches of the template, the parity of each branch and the symbolic names of the unstable periodic orbits identified are easily obtained. This way, the procedure of analysis is considerably simplified. Concerning the templates found, apart from horseshoes, reverse horseshoes or jellyroll structures with different global torsions, two more kinds of structures have been observed. One of them presents three branches folded the same way than a staple. The other one, also with three branches, presents the folding mechanism of an S, which is especially notable as it does not fit the rolling scheme valid for all templates found in former experimental studies.

The variety of topological structures obtained strengthens the usefulness of templates as significant objects for characterization of chaotic attractors of three-dimensional dynamical systems.

1 R. Gilmore, M. Lefranc, The Topology of Chaos (Wiley, New York, 2002). 2 J. Used, J.C. Martin, Phys. Rev. E 79, 046213 (2009).

12

8

The scaling behavior of oscillations arising in delay-coupled optoelectronic devices [65]

Lucas Illing, Greg Hoth, & Lauren Shareshian

Reed College, Portland, USA [e-mail]

We study the effect of asymmetric coupling strength on the onset of oscillations in an experimental system of nonlinear optoelectronic devices with delayed feedback and wide-band bandpass filtering. Specifically, we consider a network consisting of two Mach-Zehnder modulators that are cross-coupled optoelectronically. We find that oscillations appear in the system when the product of the coupling strengths exceeds a critical value. We also find a scaling law that describes how the amplitude of the oscillations depends on the coupling strengths. The observations are in good agreement with predictions from normal form theory.

12

9

Influence of Bragg-gratings-induced third-order dispersion on the optical power spectrum of Raman fiber lasers [66]

Pierre Suret, Nicolas Dalloz, & Stephane Randoux

Laboratoire Phlam / Universite de Lille 1 / bat. P5 / 59655 Villeneuve d’Ascq cedex [e-mail]

6Raman fiber lasers (RFLs) are light sources made with long cavities in which a very large number of modes (up to 10) interact through linear (dispersive) and nonlinear effects. They are good candidate to observe turbulent-like behaviors [1]. The generation of the multiple cavity modes in RFLs is now commonly described from a complex Ginzburg-Landau equation which has been analyzed from the weak-turbulence theory [1].

In particular, it is now admitted that the interplay between second-order dispersion and nonlinear optical Kerr effect inside the laser cavity leads to the generation of an optical spectrum with a symetric hyperbolic secant shape [1]. Recent works have been devoted to the study of the influence of the sign of the second-order dispersion [2] and of the mirrors reflectivity spectra on the optical spectrum of RFLs [3].

Here, we show from experiments that the third order dispersion cannot be neglected even when the RFL is operated in a strongly normal dispersion regime. In particular, in our experiments, the optical spectrum of a RFL oscillating near threshold is shown to be asymmetric. From a mean-field model (generalized Ginzburg-Landau equation), we use numerical simulations to show that the observed behaviors arises from higher-order dispersive effects (third-order dispersion) breaking the symmetry of the laser spectra.

We show precisely that third-order dispersion effects arise from reflexions at the fiber Bragg gratings (FBGs) mirrors used to close the laser cavity. Our experimental setup is a very common configuration and the dispersion of the FBGs is always high on the side of the reflectivity spectra. This means that the phenomena presented here will arise in most of the experimental setups because the optical spectrum of RFLs is generally broader than the FBGs spectral width.

From the theoretical point of view, we explore how third order dispersion influences the optical spectrum of RFLs. Simple phase-matching arguments explain the origin of the asymetry in the optical spectrum. We show that these results may have connection with anomalous thermalization recently described in nonlinear wave systems [4].

[1] S. A. Babin et al. J. Opt. Soc. Am. B (24), 8, (Aug 2007) [2] E.G. Turitsyna et al. Phys. Rev. A. (80), 031804(R) (2009) [3] E. G. Turitsyna et al. Opt. Express. (18), 5, p. 4469 (2010) [4] P. Suret et al. Phys. Rev. Lett. (104), 054101 (2010)

13

0

Pleating tori, a way to bifurcate toward chaos in a spatio-temporal laser [67]

Dalila Amroun-Aliane1, Luc Pastur2, & Christophe Letellier

31

2LEQ, Universit ´ e des Sciences et Technologie Houari Boumediene, BP 32, Bab Ezzouar, 16111 Algiers, Algeria

LIMSI-CNRS, Universit ´ e de Paris Sud , BP 133, B ˆ at 508, 91403 Orsay cedex, Franc

e3CORIA-UMR 6614, Universit ´ e de Rouen, BP 12, 76801 Saint-Etienne du Rouvray cedex, France amroun [e-mail]

Homogeneously broadened single-mode lasers are known to produce quite complicated spatio-temporal dynamics [1,2]. Most of the time, they are investigated either by using a temporal approach with phase portraits and first-return maps, or by using spatio-temporal diagrams. But, to the best of our knowledge, there is no investigation trying to combine both to provide a better understanding of the bifurcations that may be observed when a parameter is varied. In our case, the observed dynamics is interpreted in terms of non-trivial (pleated and/or folded) toro ¨ idal structures. For instance, in a certain domain of the parameter space, the chaotic behavior occurs after three Hopf bifurcations, followed by “pleating” requiring an additional dimension. The chaotic behavior is observed once the torus is sufficiently pleated, then inducing a folding as invoked in the Curry-Yorke scenario (foldings on the torus) [3]. The road to chaos is thus a combination between the Ruelle-Takens scenario [4] and the Curry-Yorke scenario. An unexpected “pleating” making a link between them. The corresponding spatio-temporal diagrams show changes that may be linked with each of the bifurcations identified in the temporal approach. In particular, the defects are observed only when a pleated torus or a toro ¨ idal chaos is identified in the phase portrait. The advantage of combining the temporal and the spatio-temporal approaches is therefore demonstrated.

Bibliography

[1] D. AMROUN, M. BRUNEL, C. LETELLIER, H. LEBLOND AND F. SANCHEZ, Complex intermittent dynamics in large-aspect-ratio homogeneously broadened single-mode lasers, Physica D, 203, 185-197 (2005).

th[2] D. AMROUN ALIANE, C. LETELLIER AND L. PASTUR, Dynamiques toro ¨ idales non triviales dans un laser spatio-temporel, Proceedings of the 13Rencontre du Non-Lin ´ eaire, 7-12, Paris, March 11 12th(2010).

[3] J. H. CURRY AND J. A. YORKE, The structure of attractors in dynamical systems, Lecture Notes in Mathematics, 668, 48-66 (1978).

[4] D. RUELLE AND F. TAKENS, On the nature of turbulence, Communications in Mathematical Physics, 20, 167-192 (1971) .

13

1

Foams, hyperbolic kaleidoscopes, and chaotic scattering [68]

Adriana Pedrosa Biscaia Tufaile1, Alberto Tufaile1, & G ´ erard Liger-Belair

21

2Escola de Artes, Ci ˆ encias e Humanidades, Soft Matter Laboratory, Universidade de S ˜ ao Paulo, 03828-000 S ˜ ao Paulo, SP, Brazil

Laboratoire d’OEnologie et Chimie Appliqu ´ ee, UPRES EA 2069, URVVC, Facult ´ e de Sciences de Reims, Moulin de la Housse, B. P. 1039, 51687 Reims, Cedex 2, Franc

[e-mail]

Foams have been intensively investigated for many years as natural phenomena that are prominent in everyday life [1]. The propagation of light in foams has received attention, and a number of patterns are observed in foams spontaneously due to the reflection and refraction of light. Some of these patterns bear a resemblance to those observed in some systems involving chaotic scattering and multiple light reflections between spheres [2]. These patterns can be obtained using mirrored spheres, and basically the image obtained is due to the fact that the different spheres are mirrored in each of the other spheres giving rise to multiple mirror images [3]. We have studied the analogy between chaotic scattering and the effects of light rays in foams observed in Hele-Shaw cells [4]. The goal of this work is to describe the existence of these triangular patterns in foams and their relation with the images obtained from the chaotic scattering of light in spheres and spherical shells. We discuss some aspects of the patterns obtained by the light scattering in a cavity formed by the three spherical shells, and compared them to the case of hyperbolic kaleidoscopes using the Poincar ´ e disk model. The analogy between chaotic scattering in dynamical systems and light scattering in foams (liquid bridges) is based on the fact that, in both cases, there is a bounded area in which light rays or particles bounce back and forth for a certain number of iterations. In that way, the incoming wave undergoes successive Moebius transformations, such as translations, rotations, inversions, and dilations. We have obtained some patterns related to Sierpinsk gaskets. In addition to that, the effects of the refraction and reflection of the light rays were studied using some properties of soft billiards. The existence of finite positive values of Kolmogorov-Sinai entropy is an indicative that light can be channeled through the network of Plateau borders. This work was supported by Conselho Nacional de Desenvolvimento Cient ´ ifico e Tecnol ´ ogico (CNPq), and Instituto Nacional de Ci ˆ encia e Tecnologia de Fluidos Complexos (INCT-FCx). [1] Chaotic bubbling and non-stagnant foams, A. Tufaile, J.C. Sartorelli, P. Jeandet, G. Liger-Belair, Phys. Rev. E 75, 066216 (2007). [2] Topology in chaotic scattering, D. Sweet, E. Ott, J. A. Yorke, Nature 399, 315 (1999). [3] Three-dimensional optical billiard chaotic scattering, D. Sweet, B. W. Zeff, E. Ott, D. P. Lathrop, Physica D 154, 207-218 (2001). [4] Simulating and interpretating images of foams with computational ray-tracing techniques, A. van der Net, L. Blondel, A. Saugey, W. Drenckhan, Journal of Colloids and surfaces A: physicochemical and engineering aspects, 309, 1-3, 159-176 (2007).

13

2

Secure optoelectronic communication using laser diode driving by chaotic R ¨ ossler oscillators [69]

Rider Jaimes-Re ´ ategui1, J. Ricardo Sevilla-Escoboza1, A. N. Pisarchik2, J. H. Garc ´ ia-Lopez1, Guillermo Huerta-Cuellar, Didier Lopez-Mancilla1, & Flavio Ruiz-Oliveras2

11

2Universidad de Guadalajara, Centro Universitario de los Lagos, Enrique D ´ iaz de Leon, Paseos de la Monta ˜ na, 47460 Lagos de Moreno, Jalisco, M ´ exico

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

,[e-mail]

Secure optical communication has been realized with two semiconductor lasers driven by two chaotic R ¨ ossler oscillators. The communication system contains two channels: optical and electronic; the information is transmitted through an optical fiber, while the R ¨ ossler oscillators are synchronized via electronic channel. One of the outputs of the R ¨ ossler oscillators serves for modulating the laser pump current, and another for coupling the oscillators. The results of numerical simulations are in good agreement with experiments which demonstrate high communication quality.

13

3

Nonlinear dynamics of extended cavity Ti:sapphire modelocked oscillator [70]

Tomasz Karda ´ s, Wojciech Gadomski, & Bo ˙ zena Gadomska

Department of Chemistry, University of Warsaw, ˙ Zwirki i Wigury 101, 02-089 Warszawa

[e-mail]

We present the results of our studies on the stability of Ti:sapphire oscillator with low repetition rate. The oscillator repetition rate is reduced by extending its cavity with Herriot cell, which consists of a stable two-mirror resonator with beam injection and the extraction mechanism. Lowering of the repetition rate, while keeping oscillator output power constant, results in the increase of a pulse energy. We have found the areas of the laser stability as a function of two order parameters: intracavity dispersion and the pump power. It appears that for certain values of order parameters the laser output exhibits two types of instabilities. The first one is the automodulation, which is caused by the competition between the laser light intensity and the population inversion. The second one is related to the cavity geometry. Moreover we provide the theoretical four-level model describing the dynamics of the laser system, in which the multimode approach is considered.

13

4

Regularization of tunneling rates with quantum chaos [71]

Louis Pecora1, Hoshik Lee2, Dong-Ho Wu1, Ed Ott3, Thomas Antonsen3, & Ming-Jer Lee

31

2

3US Naval Research Laboratory, Washington, DC, USA

College Of William and Mary, Williamsburg, VA, USA

13

5University of Maryland, College Park, MD, USA [e-mail]

We study tunneling in various shaped, closed, two-dimensional, flat potential, double wells by calculating the energy splitting between symmetric and anti-symmetric state pairs. We use the boundary and finite element methods for the calculations. For shapes that have regular or nearly regular classical behavior (e.g. rectangular or circular wells) we find that tunneling rates for nearby energy states vary over wide ranges. Rates for energetically close quantum states can differ by several orders of magnitude. As we transition to well shapes that admit more classically chaotic behavior (e.g. the stadium, the Sinai billiard) the range of tunneling rates narrows, often by an order of magnitude. For well shapes in which the classical behavior appears to be fully chaotic (as determined from numerical bounce maps) the tunneling rates’ range narrows to about a factor of 4 or so between the smallest and largest rates in a wide range of energies. This dramatic narrowing appears to come from destabilization of periodic orbits in the regular wells that produce the largest and smallest tunneling rates. It is in this sense that we say the quantum chaos regularizes the tunneling rates. We have devised a theory based on a random plane wave approximation that yields tunneling rates in the chaotic systems that match our calculations with no adjustable parameters. These results suggest that it may be possible to control the distribution of tunneling rates as a function of energy in quantum dots and other systems by changing the shape of the dot thereby providing a design tool for nanodevices

.

Image encryption based on trigonometric chaotic maps for secure communications [72]

Mar ´ ia Teresa Rodr ´ iguez Sahag ´ un1, Jos ´ e Benjam ´ in Mercado S ´ anchez2, Didier L ´ opez Mancilla4, Rider Jaimes Re ´ ategui, & Juan Hugo Garc ´ ia L ´ opez5

31

2

3

4

5Centro Universitario de Ciencias Exactas e Ingenier ´ ias, Universidad de Guadalajara (CUCEI-UdeG) C.P. 44420, Guadalajara, Jal., M ´ exico

Centro 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 ´ exico

Centro Universitario de los Lagos, Universidad de Guadalajara (CULagos-UdeG) C.P 47460, Lagos de Moreno, Jalisco, 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 modification for encryption scheme based on the trigonometric chaotic map of Jafarizadeh (2001) and Sohrab (2008). These maps are defined as polynomial quotients of N degrees. They have properties, such as: variable chaotic region, bifurcation from a stable state to a chaotic one (and viceversa) without presenting the usual scenario of double period or n period in route to chaos, and the possibility of building composition maps. With the objective of achieving image encryption, a Composition of Trigonometric Chaotic Maps (CTCM) is applied to permutate the image pixels. Another CTCM is used in the diffusion process. In this work, we propose a color image encryption of variable sizes applying CTCM in the permutation, and a new algorithm in the diffusion process using a second map. The encryption and decryption algorithm presented can fulfill high-level security requirements, big key space, and an acceptable encryption speed for a color image. Numerical simulations and graphic representations are executed for image encryption and decryption using MatLab software.

Keywords: Trigonometric Chaotic Maps, image encryption, secure communications.

13

6

Comparative statistical analysis of encrypting methods using discrete chaotic systems in imaging transmission [73]

Didier L ´ opez-Mancilla1, Mar ´ ia Teresa Rodr ´ iguez-Sahag ´ un2, Jos ´ e Benjam ´ in Mercado-S ´ anchez1, Juan Hugo Garc ´ ia-L ´ opez, & Rider Jaimes-Re ´ ategui1

21

2Centro Universitario de los Lagos, Universidad de Guadalajara (CULagos-UdeG) C.P. 47460, Lagos de Moreno, Jalisco, M ´ exico.

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

[e-mail]

In this work, a comparative statistical analysis of some image encrypting methods using discrete-time chaotic systems (logistic map, Henon map, Chen system and trigonometric chaotic map) is proposed. For each one of the methods, a process of permutation, followed by a diffusion process is considered. For statistical analysis, some histograms for the encrypted and plane image are developed. For correlation, the behavior of two adjacent pixels on horizontal, vertical and diagonal directions are evaluated. Also is analyzed the performance of these algorithms for the most commons cryptographic attacks.

13

7

A matched filter for chaos: the missing piece for chaos communications [74]

Ned Corron, Mark Stahl, & Jonathan Blakely

U. S. Army RDECOM, Redstone Arsenal, Alabama 35898, USA [e-mail]

In conventional communication systems, a matched filter provides optimal receiver performance in the presence of noise. As such, matched filters are highly desirable, yet they are practical only when a relatively small number of basis functions are used to encode information. For communications using chaotic waveforms, it is generally assumed that the unpredictable and nonrepeating nature of chaos precludes the use of a matched filter; consequently, it is widely accepted that using chaos for communications results in lower performance capabilities compared to conventional, nonchaotic systems. Here, we show this assumption is not necessarily true. We report the construction and operation of a novel chaotic electronic oscillator that admits a simple matched filter. The audio-frequency circuit, which contains both analog and digital components, is modeled by a hybrid dynamical system including both a continuous differential equation and a discrete switching condition. Surprisingly, an exact analytic solution for the system can be written as the linear convolution of a symbol sequence and a fixed basis function, similar to conventional communications waveforms. Waveform returns sampled at switching times are conjugate to a shift map, effectively proving the circuit is chaotic, and the analytic solution accurately reconstructs a measured waveform, thereby validating the circuit model. A matched filter for the basis function is derived in the form of a delay differential equation. An experimental realization of the matched filter is implemented in a simple analog circuit. The filter is used to detect the symbolic dynamics of the oscillator waveform, and an analytic bit-error rate is found to be comparable to binary phase-shift keying (BPSK). Scaled to higher frequencies, this oscillator has potential application in Hayes-type chaos communications where a message signal is encoded in the symbolic dynamics via small perturbation control. The discovery of a practical matched filter finally provides a coherent receiver to complement the elegant encoding in such systems.

13

8

Experimental transition to chaos in low-temperature plasma [75]

Dan-Gheorghe Dimitriu

Faculty of Physics, Alexandru Ioan Cuza University, 11 Carol I Blvd., RO-700506 Iasi, Romania [e-mail]

Experimental results are reported on the transition to chaos in plasma by way of two scenarios: type I intermittency and cascade of spatio-temporal sub-harmonics generations, respectively. Both of these scenarios develop in connection with the generation and dynamics of patterns in plasma, in form of simple or multiple concentric fireballs.

It is well known that a very luminous, almost spherical structure (fireball) appears in front of a positively biased electrode immersed into low-temperature plasma up to a threshold value of the potential applied on the electrode. Up to a second threshold value of the potential applied on the electrode, this structure passes into a dynamic state, in which the double layer at its border periodically disrupts and re-aggregates. In certain experimental conditions, regular oscillations interrupted by random bursts were observed in the time series of the current collected by the electrode. By increasing the voltage applied on the electrode, the random bursts appear more frequently, the final state of plasma being a chaotic one. By applying the modern methods of the nonlinear dynamics, we identified a scenario of transition to chaos by type-I intermittencies. The recorded time series were also analyzed by recurrence plot quantification.

In certain experimental conditions, a more complex pattern appears in front of electrode, in form of multiple concentric fireballs (like an onion shape). By gradually increasing the voltage applied on the electrode, we have observed that each new luminous sheet appears simultaneously with the appearance of a new sub-harmonic in the power spectrum of the complex structure dynamics. After a cascade of such sub-harmonics generation (both spatial and temporal ones), the final state of the plasma system is a chaotic one. This seems to be a new scenario of transition to chaos, being different from quasi-periodic or Feigenbaum scenarios. A further experimental and theoretical analysis of this new scenario of transition to chaos will be necessary.

All experimental data were analyzed by the methods of the nonlinear dynamics, including the reconstruction of the states space by time delay method and recurrence plot quantification.

13

9

Modified extended active control for tracking control and synchronization of chaotic and hyperchaotic systems [76]

A. N. Njah

(Nonlinear Dynamics Research Group ), Department of Physics, University of Agriculture Abeokuta (UNAAB), Ogun State, Nigeria.

[e-mail]

The active control which is outstanding for its robustness and ease of design has limitation on practical implementation partly due to the fact that the number of control functions, which is usually equal to the dimension of the system, are too many and the fact that its control signals are fixed and too large. In this paper a modified extended active control technique suitable for practical implementation is proposed. By applying the Lyapunov stability theory (LST) and the Rourth-Hurwitz criteria (RHC) to the extended active control technique, single active control functions are designed for the effective control and synchronization of chaotic and hyperchaotic systems. The single controller design, which could be achieved in different ways (via a manipulation of the LST and RHC, or a suitable choice of the control matrix, or a suitable choice of the control strength matrix) leads to a significant reduction in controller complexity. By varying the control strength matrix the control signal can be made as low as desired. The reduction in both controller complexity and the strength of the control signal in the proposed modified active control technique makes it suitable for practical implementation. Numerical result are provided for certain classes of chaotic and hyperchaotic systems to demonstrate the effectiveness of the technique.

14

0

Influence of pulse power to dynamics of laser droplet generation [77]

Blaz Krese & Edvard Govekar

University of Ljubljana, Faculty of Mechanical Enginineering, Laboratory of Synergetics, SI 1000, Ljublana Slovenija [e-mail]

A metal droplet can be used in various industrial applications [1]. Due to this different droplet generation processes are subject of intensive investigations. The laser droplet generation is a process where a laser pulse is used to melt the tip of the vertically fed metal wire [2]. The process phenomenologically consists of two phases. In the first phase from the molten tip of the wire a pendant droplet is formed due to the surface tension and gravity force. The second phase represents the detachment of the pendant droplet from the solid tip of the wire. To achieve this, the surface tension force needs to be overcome. In order to stimulate the detachment of the droplet we append an additional short pulse, i.e., detachment pulse at the end of the pendant droplet formation phase. In the paper we characterize experimentally the influence of the power of the detachment pulse on dynamics of the laser droplet generation. For that purpose a set of experiments were performed with a selected fixed laser pulse frequency rate while stepwise changing the detachment pulse power from 0 kW to 8kW. For the characterization of the process dynamics, scalar time series were generated from the snapshots of high speed infrared camera. Based on time series analysis we are able to observe qualitatively different dynamics regimes of droplet generation, from spontaneous chaotic [3] to forced periodic dripping when changing the power of detachment pulse from 0 kW to 8kW. Different linear and nonlinear characteristics [4, 5] are used to detect and quantitatively characterize observed dynamical regimes. The transition between observed regimes presumably resembles an intermittency scenario.

References: [1] GOVEKAR, Edvard, JERIC, Anze. Laser droplet generation: Application to droplet joining. CIRP ann., 2009, vol. 58, iss. 1, 205-208. [2] KOKALJ, Tadej, KLEMENCIC, Jure, MUZIC, Peter, GRABEC, Igor, GOVEKAR, Edvard. Analysis of a laser droplet formation process. J. manuf. sci. eng., 2006, vol. 128, iss. 1, 307-314. [3] KRESE Blaz,PERC Matjaz, GOVEKAR Edvard; Dynamics of laser droplet generation. Accepted in Chaos, March 2010 issue. [4] KANTZ Holger, SCHREIBER Thomas. Nonlinear time series analysis. Cambridge University Press, second edition, 2004. [5] MARWAN Norbert, ROMANO M. Carmen, THIEL Marco, KURTHS J ¨ urgen. Recurrence plots for the analysis of complex systems. Physics Reports, 2007, vol. 438, iss. 5-6, 237-329.

14

1

Conditions for the synchronization of bandlimited discrete-time chaotic systems [78]

Renato Fanganiello1, Marcio Eisencraft2, & Luiz Monteiro1;

31

2Escola de Engenharia, Universidade Presbiteriana Mackenzie, S ˜ ao Paulo, Brazil

Centro de Engenharia, Modelagem e Ci ˆ encias Sociais Aplicadas, Universidade Federal do ABC, Santo Andr ´ e, Brazi

l3Escola Polit ´ ecnica, Universidade de S ˜ ao Paulo, S ˜ ao Paulo, Brazil [e-mail]

Since Pecora and Carroll’s seminal work [1], much has been written about the potential usefulness of chaotic synchronization in communication systems (e.g. [2, 5]). Much of the impetus for chaotic communications has been the rationale whereby both analog and digital chaotic modulations would have the same properties as conventional spread spectrum techniques [6]. However, the inherent wideband characteristic of chaotic signals becomes a problem when the communication channel imposes bandwidth limitations. Because of the receiver’s nonlinear nature, all spectral components at the receiver become affected if any spectral component is amiss. Even minute gain or phase changes are enough to fully hinder synchronism [3]. A method for synchronizing both transmitter and receiver using chaotic signals under bandwidth limitations was independently proposed by [3] and [7]. The basic idea is to apply an identical filter on both transmitter and receiver in order to circumvent channel impairments. An analog circuit implementation was proposed by [3]. In [8] we have extended this method to discrete-time dynamical systems [7]. Much of the interest in this approach lies in the ease of employing Digital Signal Processors for their implementation. Although this approach has worked satisfactorily, numerical experiments have shown that depending on the filters employed, the generated signals could cease to be chaotic or diverge. In the current work we provide an analytical demonstration that synchronization is not affected when identical finite impulse response filters are included in both the transmitter and receiver. Furthermore, we numerically investigate for which filter’s orders and cut-off frequencies it is possible to obtain chaotic signals. References [1] L. Pecora, T. Carroll, ”Synchronization in chaotic systems,” Physical Review Letters, v. 64, n. 8, p. 821-824, 1990 [2] T. Carroll, L. Pecora, ”Synchronizing chaotic circuits,” Circuits and Systems, IEEE Transactions on, v. 38, n. 4, p. 453-456, Apr 1991 [3] N. Rulkov and L. Tsimring, ”Synchronization methods for communication with chaos over band-limited channels,” International Journal of Circuit Theory and Applications, v. 27, p. 555-567,1999 [5] L. Torres, ”Discrete-time dynamic systems synchronization: information transmission and model matching,” Physica D: Nonlinear Phenomena, v. 228, n. 1, p. 31-39, 2007 [6] W. Tam, et al, Digital Communications with Chaos: Multiple Access Techniques and Performance. New York, NY, USA: Elsevier Science Inc., 2006 [7] M. Eisencraft, M. Gerken, ”Comunicacao utilizando sinais caoticos: influencia de ruido e limitacao em banda,” in Anais do XVIII Simposio brasileiro de Telecomunicacoes, Gramado, Brasil, 2001 [8] M. Eisencraft, R. Fanganiello, L. Baccala, ”Synchronization of discrete-time chaotic systems in bandlimited channels,” Mathematical Problems in Engineering, 12 pages, 2009. [Online]. Available: http://www.hindawi.com/journals/mpe/2009/207971.cta.html

14

2

The ”lost” first international conference on nonlinear science [79]

Jean-Marc Ginoux1& Lo ¨ ic Petitgirard2

1

[e-mail]

14

[e-mail] [e-mail]

In a famous article entitled The nonlinear theory of electric oscillations published in 1934 in the Proceedings of the Institute of Radio Engineers Balthazar Van der Pol ended his introduction by this sentence: “...a special international conference dedicated solely to the problems arising in the nonlinear oscillation theory was recently held in Paris, on January 28-30, 1933”. Celebrating the centenary of the birth of Papaleksi in 1981, the Russian Vladimir Vasil’evich Migulin told that during the first international Conference on Nonlinear Oscillations which took place in January 1933 in Paris, Nikola ¨ i Dimitrievich Papaleksi presented two papers on the studies being conducted in the USSR along this line. Twenty five years later, the Russian Academician Evgenu L’vovich Feinberg, still celebrating Papaleksi wrote: “It is not surprising that, when the first international conference on nonlinear oscillations was convened in Paris in 1932 (among its participants were such pioneers in this field as B. Van der Pol, L. Brillouin, and others), it was Papaleksi who represented the Moscow school of Mandel’shtam and Papaleksi, their closest disciples and colleagues Andronov, A. A. Vitt, Khaikin, and others, reporting on its achievements”. The problem is that, apart from these references, there was no trace of this conference: no announcement, no location (in Paris), no proceedings, no list of participants and no program. So, has it really happened? The aim of this article is to clarify this question. Thus, it will be shown that the first (lost) international conference on nonlinear do happened in Paris at the Institut Henri Poincar ´ e under the presidence of Balthazar Van der Pol and Nikola ¨ i Dimitrievich Papaleksi and in presence of Alfred Li ´ enard, Elie and Henri Cartan, Ernest Esclangon, Henri Abraham, L ´ eon Brillouin, Philippe Le Corbeiller, Yves Rocard, Camille Gutton, ...Then, the importance of such a meeting on the emergence of Non-Linear Mechanics in France during this period as well as the full list of participants and the thematic of the discussions will be analyzed

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Supercritical and subcritical period doubling bifurcations - influence of near-resonant and resonant perturbations [80]

Martin Diestelhorst, Sebastian Lemm, Kay Barz, & Horst Beige

Martin-Luther-University Halle-Wittenberg, Institute of Physics, von-Danckelmann-Platz 3, 06120 Halle, Germany [e-mail]

Using different ferroelectrics as nonlinear capacitors in a series resonance circuit gives rise to different kinds of bifurcations. Both supercritical and subcritical period doubling bifurcations could be observed depending on the choice of the ferroelectric. Whereas triglycine sulphate (TGS) in the circuit caused supercritical period doubling bifurcations, we observed subcritical period doubling bifurcations when we used a relaxor ferroelectric lead magnesium niobate-lead titanate (PMN-PT). In both systems we investigated the influence of both near resonant and resonant perturbations on the bifurcations experimentally. We observed the shift of the bifurcation points under the influence of perturbation compared to the unperturbed bifurcation. The phenomena are discussed in the framework of the corresponding center manifold. It was predicted earlier that tuning the resonance circuit towards a period doubling bifurcation under the action of a near resonant or resonant perturbation, may yield an amplification of the perturbation in the vicinity of the bifurcation. This effect of small signal amplification was investigated with respect to its applicability as a detector for signals, which may be coupled into the circuit via the special properties of the ferroelectric materials.

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Detachment regimes in laser droplet generation [81]

Andrej Jeromen & Edvard Govekar

University of Ljubljana, Faculty of Mechanical Engineering, Laboratory of Synergetics, A ˇ sker ˇ ceva 6, SI-1000 Ljubljana, Slovenia

[e-mail]

The laser droplet generation is a process where the tip of the vertically fed metal wire is melted by a laser pulse. The outcome and the dynamics of the process of sequential droplet generation is governed by detachment of pendant droplet that can be influenced by numerous process variables. The complexity of this engineering process is additionaly increased by the interaction between the laser pulse frequency and the dynamics of a pendant droplet.

A series of experiments is presented where the frequency of the square laser pulses was varied keeping both the average laser power and the feeding speed of the metal wire constant. Depending on the decreasing laser pulse frequency from 300 Hz to 50 Hz three different detachment regimes accompanied by different dynamics have been identified: a) dripping, caused by the force of gravity alone, b) resonant detachment, caused by a combination of the gravity force and the laser induced normal oscillation modes of the pendant droplet, and c) break-up caused by the Rayleigh-Plateau instability. The observed regimes are characterized based on the geometrical properties of the generated droplets and the time series generated from the high speed IR camera records. Dripping can experimentally be characterized as a periodic droplet detachment with larger droplet volume of low scatter. Decreasing the laser pulse frequency leads to a decrease of the periodically detached droplets volume and a transition to resonant detachment which is observed at 150 Hz. At this frequency a periodic detachment with the lowest scattering of the detached droplets volume is identified. Further decreasing the frequency leads to the transition to break-up droplet detachment regime where the smallest droplets are observed while the droplet detachment and corresponding droplet volume become very irregular. The frequency of 150 Hz that corresponds to the lowest observed droplet volume scattering presumably coincides with the half of the normal droplet oscillation mode frequency fl=2.

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A new experimental probe for investigating the spatiotemporal dynamics of relativistic electrons in storage rings [82]

Serge Bielawski1;2, Christophe Szwaj1;2, Cl ´ ement Evain7, Marc Le Parquier2, Masahito Hosaka4, Miho Shimada, Masahiro Adachi5, Heishun Zen5, Masahiro Katoh5, Yoshifumi Takashima33, Shin-ichi Kimura6, Toshiharu Takahasahi, Naoto Yamamoto3, & Takanori Tanikawa5

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6PhLAM Bat. P5, Universit ´ e des Sciences et Technologies de lille, 59655 Villeneuve d’Ascq (France)

CERLA, Universit ´ e des Sciences et Technologies de lille, 59655 Villeneuve d’Ascq (France)

Graduate School of Engineering, Nagoya University 464-8603 Nagoya, Japan.

High Energy Accelerator Research Organization, KEK 305-0801, Tsukuba, Japan

UVSOR Facility, Institute for Molecular Science, National Institutes of Natural Sciences, Okazaki 444-8585, Japan

Research Reactor Institute, Kyoto University, 590-049 Osaka, Japa

n7Synchrotron SOLEIL, Saint Aubin, BP 34, 91 192 Gif-sur-Yvette, France [e-mail]

In an electron storage ring, relativistic electrons are ”trapped” during a long time (typically several hours). This type machine is of particular interest for producing producing synchrotron radiation, as various wavelengths for users. However the operation of such machines involves complicated nonlinear dynamics issues.

From the theoretical point of view, the electron bunch experiences spatiotemporal dynamics, in a phase space (in the thermodynamical sense) with 6 dimensions. As an ubiquitous feature, a perturbation with wavenumber k will experience both rotation in space space at a slow frequency, typically in the 10 KHz range for our accelerator (UVSOR-II, Japan) called the synchrotron frequency, and a diffusion process. An important point is that these processes provide only a slow damping of perturbations. Therefore instabilities of the system are likely to occur easily. A important destabilizing ingredient is the interaction between electrons of the bunch, via the so-called wakefield effect. This leads to the so-called microwave and CSR (coherent synchrotron radiation) instabilities.

Although theoretical desriptions exists since a long time, few direct comparisons between theory and experiments have been performed up to now, essentially because of the high difficulty to observe in real time the space space evolution of the electrons. Moreover, though of major importance for the dynamics, theoretical and experimental investigations of the electron wakefield is a difficult task.

In this work, we adopt an alternate strategy. We have constructed an experimental setup allowing to perturb selectively the electron bunch using various wavenumbers, and to study the transient following the perturbations. This uses an external laser, as already presented in the last ECC conference [1], and an additional setup for analyzing the damping/growth of perturbations from the terahertz emission analysis. This allows to compare new features of theoretical models against experiments. In particular we will make comparison with the Fokker-Planck-Vlasov equations, and show that characteristic features of the dynamics are due to the presence of wakefields, and thus interactions between electrons.

[1] Tunable narrowband terahertz emission from mastered laser-electron beam interaction S. Bielawski, C. Evain, T. Hara, M. Hosaka, M. Katoh, S. Kimura, A. Mochihashi, M. Shimada, C. Szwaj, T. Takahashi, and Y. Takashima Nature Physics 4, 390 (2008)

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Pulse splitting effects in short wavelength seeded free-electron lasers [83]

Nicolas Joly1, Marie Labat2, Serge Bielawski3, Christophe Szwaj3, Christelle Bruni24, & Marie-Emmanuelle Coupri

e1

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3University of Erlangen Nuremberg, G ¨ unther-Scharowsky 1 / Bau 24. D-91058 Erlangen, Germany

Synchrotron SOLEIL, Saint Aubin, BP 34, 91192 Gif-sur-Yvette, France

PhLAM/CERLA, Bat. P5, Universit ´ e des Sciences et Technologies de lille, 59655 Villeneuve d’Ascq, Franc

e4Laboratoire de l’Accelerateur Lineaire, Orsay, France [e-mail]

The present state of the art in electron accelerators allows to realize optical amplifiers in the VUV and X range, with very high gain. As a consequence, powerful emission can be obtained at very short wavelengths, using a single pass in the amplifier. To achieve temporal coherence of the output light, a strategy consists of injecting a low power coherent seed pulse from a classical (table-top) source. Experimental feasibility using harmonics generated in gases has been shown recently by part of the authors [1].

The way opened by these feasibility studies motivates systematic studies of the dynamics of the pulse propagation. In addition, the complexity of the experiments requires preliminary numerical and theoretical studies, before testing new setups, or operation in new conditions.

With this purpose, we present a theoretical and numerical study of the process, and show that a complex dynamics affects pulse propagation. In particular a pulse-splitting effect [2] is shown to affect propagation inside the FEL. We describe here the modeling of the effect and the numerical results. In particular, we use the FEL equations (the so-called Colson-Bonifaccio FEL pendulum equations) in an adimentional form in which relevant reduced parameters appear. Inspection of the reduced parameters should allow to anticipate the dynamical behavior of FELs prior to the design of new injection experiments.

[1] Injection of harmonics generated in gas in a free-electron laser providing intense and coherent extremeultraviolet light, G. Lambert, T. Hara, D. Garzella, T. Tanikawa, M. Labat, B. Carre, H. Kitamura, T. Shintake, M. Bougeard, S. Inoue, Y. Tanaka, P. Salieres, H. Merdji, O. Chubar, O. Gobert, K. Tahara, and M.-E. Couprie, Nature Physics 4, 296 - 300 (2008)

[2] Pulse splitting in short wavelength seeded Free Electron Lasers, M. Labat, N. Joly, S. Bielawski, C. Szwaj, C. Bruni, and M. E. Couprie, Phys. Rev. Lett. 103, 264801 (2009)

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Structural heterogeneity of detonation diamond-containing material [84]

Anatoly Korets1, Alexandr Krylov2, & Evgeny Mironov

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3Siberian Federal University; 26 Kirensky str.Krasnoyarsk 660074 Russia, [e-mail]

Institute of Physics, SB RAS; Akademgorodok; Krasnoyarsk 660036 Russia

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8Krasnoyarsk Institute of Railway Transport, 89 Ketshoveli str., 660028 Krasnoyarsk, Russia; [e-mail]

Synthesis of diamond-containing material (DCM) by means of detonation proceeds under non-equilibrium physical and chemical conditions. The assumption about significant influence of density fluctuations on the synthesis of this material is likely to be related to the several positions. First, scattering of the main material characteristics should be observed for this material. Second, structural heterogeneity for the particles implies constancy of the non-diamond part and appearance of the density contrast. Third, the difference between the equilibrium thermodynamics describing the diamond phase formation and the synthesis should be observed. The first position had been already examined. The goal of this work is to study the structural heterogeneity and composition of the centrifugation fractions. Detonation diamond-containing material synthesized by detonation in the different preseravation mediums were separated into fractions. Raman and infrared spectra (IR) and X-ray diffraction patterns (XRD) of the individual fractions were measured. The particles of this material were characterized by the variable ratio of the diamond sp3 and non-diamond components. It means the irregular density distribution for this material. The distribution of sp3 grains in the particles was of complicated character. The fine DCM particles contained insignificant amount of diamond [1].

1. A.Ya. Korets, A.S. Krylov, E.V. Mironov, Proceed. XXV International Conference on Equations of State for Matter, Elbrus, Russia, 2010

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A fast and robust chaos-based cryptosystem for transmitted data [85]

Safwan EL ASSAD, Hassan NOURA, & Daniel CARAGATA

´ Ecole d’ing ´ enieurs de l’universit ´ e de Nantes Site de la Chantrerie -Rue Christian Pauc B.P 50609 - 44306 NANTES CEDEX 3

- France [e-mail]

In this paper, a fast and robust chaos-based image encryption/decryption system is presented. The proposed cryptosystem includes a new perturbed chaotic generator using 32-bits finite precision with integer representation to facilitate hardware implementation and uses a variable block cipher length with different modes. The proposed chaotic generator is composed of two nonlinear digital IIR filters, connected in parallel. The non linear function used is the integer skew tent map. In the block encryption/decryption algorithms, a 2D cat-map with chaotic control parameters is used to shuffle the image pixel positions. Then, multiple rounds of substitution (confusion) and permutation (diffusion) operations, based on two of the proposed chaotic generators, are performed on every block. The perturbing orbit technique improves the dynamical statistical properties of generated chaotic sequences. This technique increases also the orbit cycle length. The problem of error propagation in various cipher block modes: Cipher Block Chaining (CBC), Cipher Feedback (CFB), Output Feedback (OFB), and Counter mode (CTR) is presented. The dependence between input and output error probability of the modes is studied. The obtained simulation results demonstrate that the proposed cryptosystem including OFB, or CTR modes, is suitable to transmit encrypted data over a corrupted digital channel. To quantify the security level of the proposed cryptosystem, we analyze the global dynamical properties of the chaotic generator using the NIST (National Institute of Standards and Technology) test, and we show that, the algorithm can resist the statistical and differential attacks; it also passed the key sensitivity test. Moreover, the algorithm has a large key space. The experimental results indicate that the scheme is secure, efficient, and faster than conventional advanced encryption standard (AES).

Keywords : Chaos-based Crypto-system, chaotic generator, chaotic permutation, Security analysis References C. Vladeanu, S. El Assad, J-C. Carlach, R. Quere. ”Chaotic digital encoding for trellis-coded mod-

ulation”, IEEE Trans on Circuits and Systems II, Vol. 56, No. 6, June 2009, pp. 509-513. Impact factor: 1.436 S. El Assad, H. Noura, I. Taralova. ”Design and analyses of efficient chaotic generators for crypto-systems”, Advances in Electrical and Electronics Engineering- IAENG Special Edition of the World Congress on Engineering and Computer Science 2008, vol. I, pp. 3-12, ISBN: 978-0-7695-3555-5. H. Noura, S. Henaf, I. Taralova, S. El Assad. ”Efficient Cascaded 1-D and 2-D Chaotic Generators”, 2nd IFAC conference on analysis and control of chaotic systems, TB2 Communication, London, UK, June 2009, 6 pages. A. Awad, S. El Assad, D. Carragata. ”A Robust Cryptosystem Based Chaos for Secure Data”, IEEE, ISIVC Conference On, Image/Video Communications over fixed and mobile networks, Bilbao Spain, July 2008, 4 pages.

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Elastic pendulum [86]

Pavel Pokorny

Prague Institute of Chemical Technology, Math Dept, Technicka 5, Prague, Czech Republic [e-mail]

Elastic pendulum is a simple mechanical system consisting of a point mass suspended on an elastic spring. Besides being an interesting physical system of its own it serves as a model for certain triatomic molecules (e.g. CO2). The conservative model of elastic pendulum has two equilibrium points. The upper equilibrium points is unstable, while the lower e.p. is stable. The vertical line going through the lower e.p. is invariant, for a given total energy there exists a periodic solution in this vertical line (for appropriate initial conditions). For certain parameter values and for certain amplitude this periodic solution is unstable. We investigate the border of stability in the parameter–amplitude space. We formulate the condition of stability, and we use the continuation technique to find the border numerically. Finally we find an analytic formula to approximate the border of stability in a wide range of parameter and amplitude values. References: P. Pokorny: Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum. Regular and Chaotic Dynamics (2008) Vol.13 No.3 pp.155-165. http://www.vscht.cz/mat/Pavel.Pokorny/rcd/RCD155-color.pdf P.Pokorny: Continuation of Periodic Solutions of Dissipative and Conservative Systems - Application to Elastic Pendulum. Mathematical Problems in Engineering doi:10.1155/2009/104547 http://www.vscht.cz/mat/Pavel.Pokorny/mpe/104547.pdf

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Nonlinear effects in complex plasmas [87]

Dmitry Samsonov1, Celine Durniak1, Paul Harvey1, Edward Hall1, Neil Oxtoby1, Jason Ralph2, Sergei Zhdanov, & Gregor Morfill2

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2Dept. of Electrical Engineering and Electronics, The University of Liverpool, Liverpool, L69 3GJ, UK

[1] D. Samsonov, A.V. Ivlev, R.A. Quinn, et. al, Phys. Rev. Lett. 88 (9), 095004 (2002) [2] C. Durniak, D. Samsonov, S. Zhdanov and G. Morfill, Europhys. Lett., 88, 45001 (2009) [3] D. Samsonov and G. E. Morfill, IEEE T. Plasma Sci. 36 , 1020 (2008) [4] D. Samsonov, A. Elsaesser, A. Edwards, et. al, Rev. Sci. Instrum. 79, 035102 (2008)

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1Max-Planck-Institute for Extraterrestrial Physics, D-85740 Garching, Germany [e-mail]

Mixtures of ion-electron plasmas with micron-sized particles or grains are called complex (dusty) plasmas. These highly charged grains can be levitated and confined in a gas discharge. They strongly interact with each other, form liquid- or solid-like structures, and exhibit a range of collective effects such as phase transitions, waves, solitons, shocks, etc. Complex plasmas are similar to colloids where the liquid medium is replaced with gaseous. Since the damping rate is many orders of magnitude lower in gases than in liquids, particle-mediated dynamic effects can be observed. Individual traceability of the grains makes complex plasmas a very useful tool for studying general phenomena in solids and liquids at a microscopic level.

We performed complex plasma experiments in a radio-frequency gas discharge, where a monolayer of monodispersed microspheres was levitated and confined. The particles were illuminated with a sheet of laser light and imaged with a high speed video camera. The monolayer was excited with electrostatic pulses applied to wires stretched at or below the layer.

The dynamic phenomena that we have studied include shock waves, solitons, their interaction with each other, with the medium and with the lattice defects. As a dispersive and nonlinear medium, crystalline complex plasmas sustain Korteveg-de Vries solitons [1]. It was shown that the soliton parameter is conserved in the presence of weak damping. We demonstrated that after two counter-propagating solitons collide, they do not change their shape but get delayed. It was observed also that the soliton amplitude grows when it propagates in a medium with decreasing density [2]. Lattice defects were affected by solitons. We found that the defects jumped across the lattice preferentially in the direction of their Burger’s vectors. Shock waves were observed to melt and compress the lattice and to induce phase transitions [3].

Complex plasma as a collection of microparticles is used for test and development of fast three-dimensional particle diagnostics. We are working on three methods, the first is based on a laser sheet scanner synchronized with the recording camera [4], the second on a gradient illumination, the third on a color coded illumination. Potential applications include flow visualization, pollution and aerosol particle tracing, and monitoring of contamination in fusion reactors. We are also developing particle tracing algorithms based on an Extended Kalman Filter with the goal of maximizing the tracking accuracy

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Experimental results and a few surprises from the Malkus waterwheel [88]

George Rutherford, Richard Martin, & Epaminondas Rosa

Department of Physics, Illinois State University, Normal, IL 61790 [e-mail]

Since its elegant demonstration by Malkus, the chaotic waterwheel has become a familiar nonlinear system, a simple mechanical analog of the Lorenz equations. Numerous theoretical and numerical investigations have appeared in the literature, but no systematic experimental data have yet been published. We will present a large collection of data taken with a research-grade waterwheel consisting of a vacuum-formed polycarbonate frame in which 36 cylindrical cells are mounted on an 18 inch (0.46 m) diameter. The wheel and its axis can be tilted, and water is fed into the top of the wheel and drains out through thin tubes at the bottom of each cell. An aluminum skirt at the wheel’s periphery passes through a variable gap magnet to provide magnetic braking that is proportional to the angular velocity. Angular time series data are collected with an absolute rotary encoder. The data are smoothed and angular velocity and acceleration are calculated via fast Fourier transforms. The data show quasi-uniform rotation as well as periodic and chaotic reversals and agree in part with computer simulations of the idealized wheel equations. A fairly detailed bifurcation plot will be shown, using the magnetic brake strength as the adjustable parameter. Preliminary results indicate some differences between the data and numerical simulations. While the first bifurcation (from uniform rotation to pendulum-like oscillations) is predicted well by the simulations when the initial angular velocity is low, there is an initial condition dependence in the real system that is not present in the model. Second, there is a disparity in the brake value corresponding to the first transition to chaos. Finally, the simulations predict a large region of periodic motion for braking values higher than those in the chaotic region, but the experimental trajectories more closely resemble noisy periodic or even chaotic motion.

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Time-of-flight estimation using synchronized chaotic systems [89]

Christian Wallinger & Markus Brandner

Institute of Electrical Measurement and Measurement Signal Processing, Graz University of Technology, Kopernikusgasse 24/IV, 8010Graz, Austria

[e-mail]

Time-of-flight (ToF) estimates are primary measurands in many metrological applications such as distance measurement, localization, and tracking. From the metrological point of view such applications are required to deliver estimates with low measurement uncertainties in the presence of bandwidth limitations, small signal-tonoise ratios, and different kinds of disturbers.

In the last two decades, the synchronization of chaotic systems has received a great deal of attention in the area of signal processing and communication engineering. In this context the beneficial properties of signals generated by chaotic systems are their unpredictability and their noise-like appearance.

In this work we investigate the use of synchronized chaotic systems in a ToF measurement system. Our setup consists of a narrow-band ultrasonic transmitter-receiver chain. We modulate the amplitude of a carrier signal with the output of a Lorenz system. The demodulated signal is used to synchronize a second Lorenz system at the receiver side. Upon synchronization of the receiver system we apply different methods to estimate the time delay between the two chaotic systems. In particular, we investigate the performance of ToF estimates for different channels using the state space representation of the systems. A comparison of these results with a standard correlation-based approach is given.

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Chaotic synchronization between Malkus’ waterwheel and the Lorenz system [90]

David Becerra-Alonso

ETEA-INSA - C/Escritor Castilla Aguayo, 4, 14004, Cordoba, Spain [e-mail]

In 1972, W.V.R. Malkus invented and constructed the waterwheel that bears his name, along with a publication on toroidal convection that presents the same dynamics. The waterwheel was intended as a lab device capable of resembling the behaviour of the equations published a decade before by E. Lorenz.

Though simple in its conception, Malkus’ waterwheel is not completely intuitive in its performance. Since it was first proposed, many experimental and real-world applications for the waterwheel where also presented. A series of lab and natural phenomena share the dynamics of Malkus’ Waterwheel: Electro-rotation (see Lemaire, 2002), Haline Oceanic Flow (see Huang, 1996), Rayleigh-Benard Convection (see Fontenele, 1999).

In this poster, the general equations of the discrete (bucket-based) waterwheel are obtained via analytical mechanics. The result is an (N+2) dimensions system that can be derived into a 3 dimensional system of differential equations. These equations in turn are simply a rescaled form of the Lorenz system, and present the same dynamics. We show to what extent the Lorenz-like equations can be synchronized with the original discrete waterwheel system, and the requirements needed for this synchronization to take place robustly under minimal driving sync forces.

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Part III

List of participants and Inde

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