Table of contents

This volume of Journal of Physics: Conference Series contains selected articles by the participants at the International Conference on Control and Synchronization of Dynamical Systems (CSDS-2005) organized by Centro de Investigaciones en Optica (CIO) in cooperation with the Society for Industrial and Applied Mathematics (SIAM), which was held in Leon, Guanajuato, Mexico on 4–7 October 2005.

CSDS-2005 featured the latest research in nonlinear dynamics concentrating on the theory of control and synchronization of complex systems and its applications in different areas of science and engineering, including optics, electronics, mechanics, chemistry, medicine, economy, communication, etc. The conference brought together leading researchers, both theoreticians and experimentalists, from different fields of science and provided an excellent opportunity for sharing ideas and problems among specialists in controlling dynamical systems and synchronization. The meeting served a dual purpose: to expose the scientific community to the cutting edge of forefront research done by leaders in this area from as many as 25 countries and to attract the attention of Mexican researchers to this field of science. These proceedings are intended to be a record of this conference and to serve as a reference for future research which the conference hopes to have initiated.

After the pioneering work on controlling chaos of E Ott, C Grebogi and J Yorke appeared in Physical Review Letters in 1990, the number of works on this topic grew tremendously. Our extensive bibliographic search among 110 peer reviewed journals yielded more than 1500 papers on controlling chaos and more than 2500 on synchronization of chaotic systems published during the last decade. The numbers of publications are still at their peaks that began to saturate, in 1998 and 2003, respectively.

It is my pleasure to acknowledge the contribution of the program and organizing committees, the funding agencies and cooperating organizations, and the staff of CIO in organizing CSDS-2005. I am very grateful to all plenary and invited speakers for their acceptance of our invitation. Their contributions enhanced significantly the prestige of our conference. I would like to express my special thanks to Prof. Jürgen Kurths for his agreement to give a tutorial course on synchronization after the meeting. My sincere thanks are also due to Dr Fernando Mendoza-Santoyo, the General Director of CIO, and to Dr Ramón Rodriguez-Vera, the Research Director, for facilitating the use of institutional infrastructure, as well as for their interest and help in organization of the conference. I record my special thanks to Annette Torres for her invaluable secretarial work, to Bernardo Ruiz-Olvera for his help in visa processing, and to Carolina Arriola for designing the conference website and printed materials. I would also like to thank all my students for their assistance in holding the meeting.

In the publication of the conference proceedings, I am grateful to our referees (V Aboites, V Afraimovich, F T Arecchi, A D'Anjou, C Grebogi, V Erofeyev, Hongjie Yu, L Kalyakin, A Karami, S K Dana, A K Das, J Kurths, A Loskutov, G Luna, R Meucci, C R Mirasso, C L Pando, P Parmananda, Qinhua Hu, R Rechtman, P K Roy, R Roy, N Rulkov, E A Sataev, A Shil'nikov, S Sinha, V M Somsikov, N N Verichev, U E Vincent, A Volkovskii, J Zebrowski) for their knowledgeable and careful comments, and to Dr Rider Jaimes-Reátegui and Dr Hugo Garcia-Lopez for helping me with the editorial process. Finally, I thank the Institute of Physics Editorial team for publishing these proceedings very rapidly and efficiently.

International Program Committee V Afraimovich (Universidad de San Luis Potosi, Mexico) T F Arecchi (University of Firenze, Italy) C Grebogi (Universidad de Sao Paulo, Brazil) J Haus (University of Dayton, USA) (OSA representative) J Kurths (University of Potsdam, Germany) F Mendoza-Santoyo (Centro de Investigaciones en Optica, Mexico) (General Director) A N Pisarchik (Centro de Investigaciones en Optica, Mexico) (Chair) R Roy (University of Maryland, USA) (SIAM representative) S Sinha (Institute of Mathematical Sciences, India)

Local Organizing Committee V Aboites (Centro de Investigaciones en Optica, Mexico) R Espinosa-Luna (Centro de Investigaciones en Optica, Mexico) J H García-López (Universidad de Guadalajara, Mexico) R Jaimes-Reátegui (Universidad de Guadalajara, Mexico) A V Kir'yanov (Centro de Investigaciones en Optica, Mexico) E Kourmychev (Centro de Investigaciones en Optica, Mexico) V J Pinto-Robledo (Centro de Investigaciones en Optica, Mexico) R Rodriguez-Vera (Centro de Investigaciones en Optica, Mexico) (Research Director) G V Vazquéz-García (Centro de Investigaciones en Optica, Mexico)

Funding Agencies Consejo de Ciencia y Tecnología de Estado Guanajuato, Mexico Abdus Salam International Centre for Theoretical Physics, Trieste, Italy International Mathematical Union, USA Centro Latinoamericano de Física, Brazil Academia Mexicana de Ciencias, Mexico Fundación México—Estados Unidos para la Ciencia, Mexico

Cooperating Organizations International Society for Optical Engineering American Physical Society Optical Society of America Institute of Physics

Bio-oscillation or biorhythms is one of the most important properties of living organism for maintaining life. There have been many studies on electrical phenomena associated with excitation or oscillation of biological membranes. Most of these studies have been carried out with aim of increasing our understanding of the mechanism of oscillations in bio membranes. In the present study we report a new membrane bilayer system generated by the surfactants, which is bipolar in nature and shows electrical excitability in the absence of any channel former. This new surfactant bipolar liquid membrane shows the promise of being a model for neuronal excitation.

An explanation of the mechanism of irreversible dynamics in systems with mixing is offered. The procedure of splitting a system into subsystems in equilibrium and studying the dynamics of one of them as it interacts with the other subsystems is the basis of the approach to the analysis of nonequilibrium systems. The problem of `coarse-grain' dynamics is used. The `coarse-grain' problem on the phase space is eliminated with this method. The formula, that expresses the entropy through the work of forces between systems, is submitted. The essential link between thermodynamics and classical mechanics was found.

Autoresonance is a phase locking phenomenon occurring in nonlinear oscillatory system, which is forced by oscillating perturbation. Many physical applications of the autoresonance are known in nonlinear physics. The essence of the phenomenon is that the nonlinear oscillator selfadjusts to the varying external conditions so that it remains in resonance with the driver for a long time. This long time resonance leads to a strong increase in the response amplitude under weak driving perturbation. An analytic treatment of a simple mathematical model is done here by means of asymptotic analysis using a small driving parameter. The main result is finding threshold for entering the autoresonance.

The elementary physical theory of cluster dynamics is proposed. It is shown that cluster formation of dynamical processes in latticing structures lies within the framework of classic representations on a synchronization of dynamical systems. It is shown that any cluster structure is just a simple synchronization of certain number so-called cluster oscillators. The full set of types of cluster oscillators in onedimensional homogeneous chain and two-dimensional homogeneous lattice of elementary oscillators are established. The principles of coupling of C-oscillators in cluster structures and principles of their possible transformation are explained.

An electronic circuit intended to simulate the nonlinear dynamics of a simplified 3-cell model of the pyloric central pattern generator in California spiny lobster stomato gastric ganglion is presented. The model employs the synaptic phase locked loop (SPLL) concept where the frequency of oscillations of a postsynaptic cell is mainly controlled by the synaptic current which depends on the phase shift between the oscillations. The theoretical study showed that the system has a stable steady state with correct phase shifts between the oscillations and that this regime is stable when the frequency of the pacemaker cell is varied over a wide range. The main bifurcations in the system were studied analytically, in computer simulations, and in experiments with the electronic circuit. The experimental measurements are in good agreement with the expectations of the theoretical model.

The biological systems are opened and are kept far from thermodynamics equilibrium. For these reasons, biological systems are always exposed to external perturbations, which may produce alterations on these rhythms as a consequence of coupling synchronization of the autonomous oscillator with perturbation. Coupling of therapeutic perturbations, such as drugs and radiation, on biological systems delivery to biological rhythms is known as chronotherapy. We used the Rössler system as a theoretical model for chronotherapy, generalized this formalism for chaotic behaviour. We found that when the Rössler is more dissipative, such as c increase, the systems become more robust to the perturbations.

Period-one and period-two oscillations in a diode laser subject to optical injection are experimentally investigated. The changes in the modulation frequency are studied as a function of the detuning frequency and the injection signal strength.

An excitation of ultra-high frequency (100 MHz - 1 GHz) nonlinear envelope solitary acoustic waves, propagating along the interface between a solid film and a solid substrate, is theoretically analyzed. Both the quadratic nonlinearity and the cubic one are important in the case of the envelope waves. When generation of higher harmonics is reduced due to essential waveguide dispersion and the cubic nonlinearity due to the induced zero harmonic is dominating, a possibility of the envelope solitary pulse propagation and the spatial-temporal wave collapse exists, as demonstrated. When the cubic material nonlinearity reduces the associated cubic nonlinear term, there also exists a possibility to observe a wave collapse, if the initial focusing of the input pulse at the first harmonic is applied.

It is proposed two mechanisms to explain the formation of periodic and non periodic bands and spirals as thin films of gelatinous aqueous solutions of mercury (II) chloride are dried. The first mechanism supposes an homogeneous drying, where the height of the film decreases at constant rate, forming Liesegang bands. The second mechanism implies a non homogeneous drying where an evaporation front drives the formation of periodic bands and spirals.

We analyzed databases with gait time series of adults and persons with Parkinson, Huntington and amyotrophic lateral sclerosis (ALS) diseases. We obtained the staircase graphs of accumulated events that can be bounded by a straight line whose slope can be used to distinguish between gait time series from healthy and ill persons. The global Hurst exponent of these series do not show tendencies, we intend that this is because some gait time series have monofractal behavior and others have multifractal behavior so they cannot be characterized with a single Hurst exponent. We calculated the multifractal spectra, obtained the spectra width and found that the spectra of the healthy young persons are almost monofractal. The spectra of ill persons are wider than the spectra of healthy persons. In opposition to the interbeat time series where the pathology implies loss of multifractality, in the gait time series the multifractal behavior emerges with the pathology. Data were collected from healthy and ill subjects as they walked in a roughly circular path and they have sensors in both feet, so we have one time series for the left foot and other for the right foot. First, we analyzed these time series separately, and then we compared both results, with direct comparison and with a cross correlation analysis. We tried to find differences in both time series that can be used as indicators of equilibrium problems.

A new topological invariant (Lorenz-manuscript) leading to the existence of uncountable set of topologically various attractors is proposed. A new definition of the hyperbolic properties of the Lorenz system close to singular hyperbolicity is introduced. This definition gives the opportunity to prove that small non-autonomous perturbations do not lead to the appearance of the stable solutions.

A study on non-linear Laser resonators is presented; these are produced when chaos generating elements are introduced within the resonator. The analysis of a laser resonator using ABCD matrix formalism is showed for the case where these elements are introduced in the resonator. For the first time to our knowledge the matrix of a chaos generating element is obtained. Chaos regions depend on the resonator parameters, and are therefore reported and presented as a chaos bifurcation diagram.

This paper presents a systematic approach to develop artificial neural network (ANN) models to predict the performance of a heat exchanger operating in real mechanical ventilation and air-conditioning (MVAC) system. Two approaches were attempted and presented. Every detailed components of the MVAC system have been considered and we attempt to model each of them by one ANN. This study used the neural network technique to obtain a static and a dynamic model for a heat exchanger mounted in an air handler unit (AHU), which is the key component of the MVAC system. It has been verified that almost all of the predicted values of the ANN model were within 95% - 105% of the measured values, with a consistent mean relative error (MRE) smaller than 2.5%. The paper details our experiences in using ANNs, especially those with back-propagation (BP) structures. Also, the weights and biases of our trained-up ANN models are listed out, which serve as good reference for readers to deal with their own situations.

In addition to the well-known Rössler funnel that consists in near-homoclinic orbits, perfect homoclinic orbits have been found numerically and experimentally in a simplest piecewise linear Rössler-like electronic circuit. The evolution of the system in the homoclinic range exhibits period-bubbling and period-adding cascades when a control parameter is changed. A scaling law in the period-adding cascade between the period of a homoclinic orbit and the bifurcation parameter is evaluated. Other phenomena, such as the coexistence of two homoclinic orbits, homoclinic chaos, symmetry breaking and phase bistability are also demonstrated. The results of numerical simulations are in a good agreement with experiments.

Influence of the gain saturation on the output performance of quantum-well heterostructures with modified distributed-feedback cavities is considered. Taking into account symmetrical boundary conditions, distribution of the electromagnetic field in the active region of the heterostructure laser diode with a phase-amplitude grating is determined. It is shown that both the output performance of quantum-well heterostructures and distribution of the electromagnetic field in the active region depend on the pump current and optical nonlinearity characteristics.

It is not difficult to construct chaotic Coupled Map Lattices (CMLs) that display the type of synchronization usually associated to Phase Transitions (PTs) in equilibrium statistical models. These CMLs show the emergence of an order parameter that signals longrange order in the lattice, without losing its chaotic character (the existence of a positive sector in the Lyapunov spectrum). This order parameter remains null for small couplings, and changes to non-zero values for couplings above some critical value. Around this critical coupling one also has strong fluctuations in both long- and short-range correlations. In this work two examples are reviewed, the Miller-Huse CML, which is a dynamical analogue to the Ising model, and a recently constructed dynamical analogue to the q = 3 Potts Model. Both models are constructed using fully hyperbolic continuous piece-wise linear mappings, and a diffusive coupling. It is shown for both examples how the symmetries of the models allows one to connect to the corresponding equilibrium classes. The critical behaviour of Miller-Huse in two dimensions (2D) under simultaneous updating is revisited, and new results for this model in a triangular lattice are shown, including a careful evaluation of the effects of the leading correction to scaling. It is concluded that this model does belong to the 2D Ising Universality Class. For the dynamical analogue to the q = 3 Potts Model a well defined order-disorder PT is also found. The phase diagrams for both simultaneous and sequential updating of the model are obtained. For simultaneous updating a line of continuous PTs is found. The corresponding critical exponents—again, including a leading correction to scaling- are consistent with those of the equilibrium Potts Model.

The study of a laser including a saturable absorber is presented. The non-linear system describing the complex dynamics of the laser is presented. The laser is shown to operate in several regimes depending on the parameters used. It is also shown how the control of the laser is possible depending on the operating regime parameters.

We study possibilities of controlling instabilities in a diode laser with external cavity by adding a second external cavity and adjusting its length and feedback strength. This method is approved numerically with a model of Lang-Kobayashi equations for the laser with two external cavities. We find that chaotic behaviour of the laser output can be completely stabilized to periodic orbits of different periods.

This paper deals with the control of chaotic economic motion. We show that very complicated dynamics arising, e.g., from an overlapping generations model (OLG) with production and an endogenous intertemporal decision between labour and leisure, which produces chaos, can in fact be controlled with relative simplicity. The aperiodic and very complicated motion that stems from this model can be subject to control by small perturbations in its parameters and turned into a stable steady state or into a regular cycle. Therefore, the system can be controlled without changing of its original properties. To perform the control of the totally unstable equilibrium (both eigenvalues with modulus greater than unity) in this economic model we apply the pole-placement technique, developed by Romeiras, Grebogi, Ott and Dayawansa (1992).

The application of control methods to chaotic economic dynamics may raise serious reservations, at least on mathematical and logical grounds, to some recent views on economics which have argued that economic policy becomes useless in the presence of chaotic motion (and thus, that the performance of the economic system cannot be improved by public intervention, i.e., that the amplitude of cycles can not be controlled or reduced). In fact, the fine tuning of the system (that is, the control) can be performed without having to rely only on infinitesimal accuracy in the perturbation to the system, because the control can be performed with larger or smaller perturbations, but neither too large (because these would lead to a different fixed point of the system, therefore modifying its original nature), nor too small because the control becomes too inefficient.

Poor control of steam generator water level is the main cause of unexpected shutdowns in nuclear power plants. Particularly at low powers, it is a difficult task due to shrink and swell phenomena and flow measurement errors. In addition, the steam generator is a highly complex, nonlinear and time-varying system and its parameters vary with operating conditions. Therefore, it seems that design of a suitable controller is a necessary step to enhance plant availability factor. The purpose of this paper is to design, analyze and evaluate a water level controller for U-tube steam generators using wavelet neural networks. Computer simulations show that the proposed controller improves transient response of steam generator water level and demonstrate its superiority to existing controllers.

This paper is intended to explain the basic ideas related to designing digital algorithms using unmeasured mechanical variables (but observed without sensors), to control electromechanical systems digitally. The main idea is to simplify algorithms by using a linear discrete-time model without any variable limitations. Original references limiter based on the digital sliding mode for the exception of the variable limitations influences is proposed and designed. The simulation confirmed high dynamic accuracy, simplification of the digital control algorithm, and reduction of the computing capacity requirements of the controller.

On the basis of the FitzHugh-Nagumo-type model we investigate the possibility of suppression of the spiral wave turbulence by weak pacemaker excitations. We consider different ways of media stabilization and study the dependence of the suppression efficiency on the excitation shape and the media parameters. Also, we analyze the frequency of target waves in the unperturbed media as a function of the external force frequency. Applications of the obtained results to cardiac rhythm pathologies are considered.

A new original method of information processing and secure communications based on the coding of alphabet symbols by stabilized cycles of certain perturbed one-dimensional dynamical systems is proposed. The foundation of the proposed method is ciphering by the one-to-one correspondence between periods of such cycles and certain alphabet symbols. It is shown that for some maps perturbations which lead to the stabilization of cycles of the given period, form some domain in the parametric space. This fact is used for coding identical symbols via random selection of parameters from this domain, that ensures that the probability of decoding the transmitting information by an external observer is zero. Analytic estimations of the admissible noise level in the communication channel and the randomness degree of transmitting signals are made. Some variants of the ciphered sequences are presented.

A method is developed for producing deterministic chaotic motion from the linear superposition of a bi-infinite sequence of randomly polarized basis functions. The resultant waveform is also formally a random process in the usual sense. In the example given, a threedimensional embedding produces an idealized version of Lorenz motion. The one-dimensional approximate return map is piecewise linear; a tent or shift, depending on the Poincaré section. The results are presented in an informal style so that they are accessible to a wide audience interested in both theory and applications of symbolic dynamics communication.

In this paper using an analogy with a mechanical dynamic system we have assigned to a Lorenz chaotic system an energy like measure that has allowed us to analyse the flows of energy that appear when two identical Lorenz systems are coupled via feedback, and their relationships with some salient features of the interaction between both systems. We study the whole range of values of the coupling strength ranging from no interaction at all to identical synchronization. Before the coupling strength reaches the level of identical synchronization it induces local changes in the stability of the synchronization manifold that have a reflect in the Lyapunov exponents and phase synchronization measures and that can be easily detected using quantitative measures of energy.

Using light-controlled oscillators (LCOs) and a mathematical model of them introduced in [1], we have analyzed a population of LCOs arranged in chains with nonperiodic (linear configuration) and periodic (ring configuration) boundary conditions in which we have solved numerically the corresponding equations for a broad interval of coupling strength values and for chains between 2 and 25 LCOs. We have considered three different situations, viz. identical LCOs, identical LCOs with simplifications (LCOs considered as integrate-and-fire (IF) oscillators), and finally nonidentical LCOs. We study synchronization under two criteria: the first takes into account the simultaneity of flashing events (phase difference criterion), and the second considers period-locking as a criterion for synchronization. For each case, we have identified regions of synchronization in the plane coupling strength versus number of oscillators. We observe different behaviors depending on the values of these variables.

A novel form of multiplexing information-bearing chaotic waveforms is demonstrated experimentally. This scheme dramatically increases the information carrying capacity of a chaotic communication system. In the transmitter, information is encoded in the chaotic waveforms of two electronic circuits using small perturbations to induce the symbolic dynamics to follow a prescribed symbol sequence. Waveforms from each of the drive oscillators are summed to form a single scalar signal that is transmitted to the receiver. Identical oscillators in the receiver synchronize to their counterparts in the drive system, effectively de-multiplexing the transmitted signal. The transmitted information in each channel is extracted from simple return maps of the receiver oscillators.

In this work, a modified chaos-based communication scheme is presented. In particular, we use the modified scheme proposed by López-Mancilla and Cruz-Hernández (2005), that improves the basic scheme for chaotic masking using a single transmission channel proposed by Cuomo and coworkers (1993). It is extended for a special class of Generalized Hamiltonian systems. Substantial differences that signi.cantly affect the reception quality of the sent message, with or without considering noise effect in the transmission channel are given. We use two Hamiltonian Lorenz systems unidirectionally coupled, the first like a master/transmitter system and the other like a slave/receiver system in order to illustrate with numerical simulations the effectiveness of the modified scheme, using chaos synchronization with Hamiltonian forms and observer.

We present a novel synchronization scheme for secure communication with two chaotic unidirectionally coupled Rössler circuits. The circuits are synchronized via one of the variables, while a signal is transmitted through another variable. We show that this scheme allows more stable communications. The system dynamics is studied numerically and experimentally in a wide range of a control parameter. The possibility of secure communications with an audio signal is demonstrated.

We present two algorithms to synchronize second order systems that can be described by phase state variables, we call them phase planar systems. We use the master slave configuration and the synchronization objective is to obtain identical synchronization between the master and slave systems in spite of the existence of external perturbations and parametric variations. We use discontinuous control techniques to design the coupling signal that produce sliding modes of first and second order. These discontinuous controllers render the closed loop system robust with respect to matched bounded disturbances and to terms produced by parametric variations. The performance of the proposed synchronization techniques are illustrated experimentally.

Existence of different kinds of synchronizations, namely anticipatory, complete and lag type synchronizations (both exact and approximate), are shown to be possible in timedelay coupled piecewise linear systems. We deduce stability condition for synchronization of such unidirectionally coupled systems following Krasovskii-Lyapunov theory. Transition from anticipatory to lag synchronization via complete synchronization as a function of coupling delay is discussed. The existence of exact synchronization is preceded by a region of approximate synchronization from desynchronized state as a function of a system parameter, whose value determines the stability condition for synchronization. The results are corroborated by the nature of similarity functions. A new type of oscillating synchronization that oscillates between anticipatory, complete and lag synchronization, is identified as a consequence of delay time modulation with suitable stability condition.

We present an algorithm to synchronize, under the master/slave configuration, a class of piecewise linear chaotic systems known as Sprott systems. The synchronization objective is to obtain identical synchronization between the master and slave systems in spite of the existence of external perturbations and parametric variations. The sliding control technique is used to design the coupling signal. This discontinuous controller renders the closed loop system robust with respect to matched bounded disturbances and to terms produced by parametric variations. The performance of the proposed controlled synchronization is illustrated numerically and experimentally.

We discuss the relation between phenomena of chaos synchronization in coupled systems and creation of hyperchaotic attractors (attractors with at least two positive Lyapunov exponents. Such attractors are common in higher-dimensional dynamical systems (at least two-dimensional maps or four-dimensional flows). Riddling bifurcation i.e., the bifurcation in which one of the unstable periodic orbits embedded in a chaotic attractor located on the invariant manifold becomes unstable transversely to the attractor leads to the loss of chaos synchronization in coupled identical systems.

We show that generalized riddling bifurcation defined as the bifurcation in which one of the unstable periodic orbits embedded in a higher-dimensional chaotic attractor (not necessarily located on the invariant manifold) becomes unstable transversely to the attractor explains mechanism of the creation of hyperchaotic attractors. Additionally we show that the generalized riddling bifurcation can give physical mechanism explaining interstellar journeys described in science-fiction literature.