Nonlinear Phenomena and Complex Systems – serie

This text contains the lectures given at the Fourth School on Statistical Physics and Cooperative Systems, held in Santiago, Chile, December 12-16, 1994. This school brings together scientists working on subjects related to recent trends in complex systems. Subjects dealt with include dynamical systems, ergodic theory, cellular automata, symbolic and arithmetic dynamics, spatial systems, large deviation theory and neural networks.

This book contains the courses given at the Fifth School on Complex Systems held at Santiago, Chile, from 9th .to 13th December 1996. At this school met researchers working on areas related with recent trends in Complex Systems, which include dynamical systems, cellular automata, symbolic dynamics, spatial systems, statistical physics and thermodynamics. Scientists working in these subjects come from several areas: pure and applied mathematics, physics, biology, computer science and electrical engineering. Each contribution is devoted to one of the above subjects. In most cases they are structured as surveys, presenting at the same time an original point of view about the topic and showing mostly new results. The paper of Bruno Durand presents the state of the art on the relationships between the notions of surjectivity, injectivity and reversibility in cellular automata when finite, infinite or periodic configurations are considered, also he discusses decidability problems related with the classification of cellular automata as well as global properties mentioned above. The paper of Eric Goles and Martin Matamala gives a uniform presentation of simulations of Turing machines by cellular automata. The main ingredient is the encoding function which must be fixed for all Turing machine. In this context known results are revised and new results are presented.

I am very pleased and privileged to write a short foreword for the monograph of Dean Driebe: Fully Chaotic Maps and Broken Time Symmetry. Despite the technical title this book deals with a problem of fundamental importance. To appreciate its meaning we have to go back to the tragic struggle that was initiated by the work of the great theoretical physicist Ludwig Boltzmann in the second half of the 19th century. Ludwig Boltzmann tried to emulate in physics what Charles Darwin had done in biology and to formulate an evolutionary approach in which past and future would play different roles. Boltzmann's work has lead to innumerable controversies as the laws of classical mechanics (as well as the laws of quan­ tum mechanics) as traditionally formulated imply symmetry between past and future. As is well known, Albert Einstein often stated that "Time is an illusion". Indeed, as long as dynamics is associated with trajectories satisfy­ ing the equations of classical mechanics, explaining irreversibility in terms of trajectories appears, as Henri Poincare concluded, as a logical error. After a long struggle, Boltzmann acknowledged his defeat and introduced a probabil­ ity description in which all microscopic states are supposed to have the same a priori probability. Irreversibility would then be due to the imperfection of our observations associated only with the "macroscopic" state described by temperature, pressure and other similar parameters. Irreversibility then appears devoid of any fundamental significance. However today this position has become untenable.

This sixth Volume of the International Workshop on Instabilities and Nonequilibrium Structures is dedicated to the memory of my friend Walter Zeller, Professor of the Universidad C'at6lica df' Valparaiso and Vice-Director of the Workshop. Walter Zeller was much more than an organizer of this meeting: his enthusiasm, dedication and critical views were many times the essential ingredients to continue with a task which in occasions faced difficulties and incomprehensiolls. It is in great part due to him that the workshop has adquired to-day tradition. maturity and international recognition. This Volume should have been coedited by Walter and it is with df'ep emotion that I learned that his disciples Javier Martinez and Rolando Tiemann wanted as a last hommage to their Professor and friend to coedit tfus book. No me seria posible terminal' estas lineas sin pensar en la senora Adriana Gamonal de Zelln. qUf' ella encuentre en este libro la admiraci6n y reconocimiento hacia su marido de quiPIlf's [l\Prall sus discipulos, colegas y amigos.

This volume contains the courses given at the Sixth Summer School on Complex Systems held at the Faculty of Physical and Mathematical Sciences, University of Chile at Santiago, Chile, 14-18 December 1998. The contributions, which in some cases have been structured as surveys, treat recoding Sturmian sequences on a subshift of finite type chaos from order; Lyapunov exponents and synchronization of cellular automata; dynamical systems and biological regulations; cellular automata and artificial life; Kolmogorov complexity; and cutoff for Markov chains. This book should be of interest to graduate students and researchers whose work involves mathematical modelling and industrial mathematics, statistical physics, thermodynamics, algorithms and computational theory, statistics and probability, and discrete mathematics.

This text contains the lectures given at the Conference on Dynamics and Randomness held at the Centro de Modelamiento Matematico of the Universidad de Chile, on December 11-15, 2000. This meeting brought together mathematicians, theoretical physicists, and theoretical computer scientists, and graduate students interested in fields related to probability theory, ergodic theory, and symbolic and topological dynamics. Each chapter is devoted to one of these subjects. Some papers are structured as surveys, presenting at the same time an original point of view and showing mostly new results.

The contents of this book correspond to Sessions VII and VIII of the International Workshop on Instabilities and Nonequilibrium Structures which took place in Vifia del Mar, Chile, in December 1997 and December 1999, respectively. We were not able to publish this book before and we apologize for this fact to the authors and participants of the meeting. We have made an effort to actualize the courses and articles which have been reviewed by the authors. Both Workshops were organized by Facultad de Ciencias Fisicas y Matematicas, Universidad de Chile, Instituto de Fisica of Universidad Cat61ica de Valparaiso and Centro de Fisica No Lineal y Sistemas Complejos de Santiago. We are glad to acknowledge here the support of the Facultad de Ingenieria of Universidad de los Andes of Santiago which also be from now on one of the organizing Institutions of future Workshops. Enrique Tirapegui PREFACE This book is divided in two parts. In Part I we have collected the courses given in Sessions VII and VIII of the Workshop and in Part II we include a selection of the invited Conferences and Seminars presented at both meetings.

We present here the lectures and a selection of the seminars given at the Ninth International Workshop on Instabilities and Nonequilibrium Structures which took place in Vifiadel Mar, Chile, in December 2001. The Workshop was organized by Facultad de Ciencias Fisicas y Matematicas, Universidad de Chile, Instituto de Fisica of Universidad Cat6lica de Valparaiso, Centro de Fisica No Lineal y Sistemas Complejos de Santiago and Facultad de Ingenieria, Universidad de los Andes, which starting from this year joins the other institutions in the coorganization ofthe Workshop. The organizers would like to express their gratitude to the following sponsors: Facultad de Ciencias Fisicas y Matematicas de la Universidad de Chile, Instituto de Fisica de la Universidad Cat6lica de Valparaiso, Facultad de Ingenieria de la Universidad de los Andes, Centro de Fisica No Lineal y Sistemas Complejos de Santiago, Academia Chilena de Ciencias, Ministere Francais des Affaires Etrangeres, CONICYT (Comisi6n Nacional de Investigaci6n Cientifica y Tecno16gicade Chile) and Departamento Tecnico de Investigaci6n y de Relaciones Internacionales de la Universidad de Chile.Enrique Tirapegui PREFACE This book consists of two parts, the first one has three lectures written by Professors H. R. Brand, M. Moreau and L. S. Tuckerman. H. R. Brand gives an overview about reorientation and undulation instabilities in liquid crystals, M. Moreau presents recent results on biased tracer diffusion in lattice gases, finally, L. S. Tuckerman summarizes some numerical methods used in bifurcation problems.

This book contains the lectures given at the II Canference an Dynamics and Randamness held at the Centro de Modelamiento Matematico of the Universidad de Chile, from December 9th to 13th, 2002. This meeting brought together mathematicians, theoretical physicists, theoretical computer scientists, and graduate students interested in fields related to probability theory, ergodic theory, symbolic and topological dynamics. We would like to express our gratitude to an the participants of the conference and to the people who contributed to its orga- nization. In particular, to Pierre Collet, BerIiard Rost and Karl Petersen for their scientific advise.We want to thank warmly the authors of each chapter for their stimulating lectures and for their manuscripts devoted to a various of appealing subjects in probability and dynamics: to Jean Bertoin for his course on Some aspects of random fragmentation in con- tinuous time; to Anton Bovier for his course on Metastability and ageing in stochastic dynamics; to Steve Lalley for his course on AI- gebraic systems of generat ing functions and return probabilities for random walks; to Elon Lindenstrauss for his course on Recurrent measures and measure rigidity; to Sylvie Meleard for her course on Stochastic particle approximations for two-dimensional N avier- Stokes equations; and to Anatoly Vershik for his course on Random and universal metric spaces.

This book contains the courses given at the Fourth School on Statistical Physics and Cooperative Systems held at Santiago, Chile, from 12th to 16th December 1994. This School brings together scientists working on subjects related to recent trends in complex systems. Some of these subjects deal with dynamical systems, ergodic theory, cellular automata, symbolic and arithmetic dynamics, spatial systems, large deviation theory and neural networks. Scientists working in these subjects come from several aeras: pure and applied mathematics, non linear physics, biology, computer science, electrical engineering and artificial intelligence. Each contribution is devoted to one or more of the previous subjects. In most cases they are structured as surveys, presenting at the same time an original point of view about the topic and showing mostly new results. The expository text of Roberto Livi concerns the study of coupled map lattices (CML) as models of spatially extended dynamical systems. CML is one of the most used tools for the investigation of spatially extended systems. The paper emphasizes rigorous results about the dynamical behavior of one dimensional CML; i.e. a uniform real local function defined in the interval [0,1], interacting with its nearest neighbors in a one dimensional lattice.

This book contains the courses given at the Fifth School on Complex Systems held at Santiago, Chile, from 9th .to 13th December 1996. At this school met researchers working on areas related with recent trends in Complex Systems, which include dynamical systems, cellular automata, symbolic dynamics, spatial systems, statistical physics and thermodynamics. Scientists working in these subjects come from several areas: pure and applied mathematics, physics, biology, computer science and electrical engineering. Each contribution is devoted to one of the above subjects. In most cases they are structured as surveys, presenting at the same time an original point of view about the topic and showing mostly new results. The paper of Bruno Durand presents the state of the art on the relationships between the notions of surjectivity, injectivity and reversibility in cellular automata when finite, infinite or periodic configurations are considered, also he discusses decidability problems related with the classification of cellular automata as well as global properties mentioned above. The paper of Eric Goles and Martin Matamala gives a uniform presentation of simulations of Turing machines by cellular automata. The main ingredient is the encoding function which must be fixed for all Turing machine. In this context known results are revised and new results are presented.

I am very pleased and privileged to write a short foreword for the monograph of Dean Driebe: Fully Chaotic Maps and Broken Time Symmetry. Despite the technical title this book deals with a problem of fundamental importance. To appreciate its meaning we have to go back to the tragic struggle that was initiated by the work of the great theoretical physicist Ludwig Boltzmann in the second half of the 19th century. Ludwig Boltzmann tried to emulate in physics what Charles Darwin had done in biology and to formulate an evolutionary approach in which past and future would play different roles. Boltzmann's work has lead to innumerable controversies as the laws of classical mechanics (as well as the laws of quan­ tum mechanics) as traditionally formulated imply symmetry between past and future. As is well known, Albert Einstein often stated that "Time is an illusion". Indeed, as long as dynamics is associated with trajectories satisfy­ ing the equations of classical mechanics, explaining irreversibility in terms of trajectories appears, as Henri Poincare concluded, as a logical error. After a long struggle, Boltzmann acknowledged his defeat and introduced a probabil­ ity description in which all microscopic states are supposed to have the same a priori probability. Irreversibility would then be due to the imperfection of our observations associated only with the "macroscopic" state described by temperature, pressure and other similar parameters. Irreversibility then appears devoid of any fundamental significance. However today this position has become untenable.

This book contains the lectures given at the II Canference an Dynamics and Randamness held at the Centro de Modelamiento Matematico of the Universidad de Chile, from December 9th to 13th, 2002. This meeting brought together mathematicians, theoretical physicists, theoretical computer scientists, and graduate students interested in fields related to probability theory, ergodic theory, symbolic and topological dynamics. We would like to express our gratitude to an the participants of the conference and to the people who contributed to its orga- nization. In particular, to Pierre Collet, BerIiard Rost and Karl Petersen for their scientific advise.We want to thank warmly the authors of each chapter for their stimulating lectures and for their manuscripts devoted to a various of appealing subjects in probability and dynamics: to Jean Bertoin for his course on Some aspects of random fragmentation in con- tinuous time; to Anton Bovier for his course on Metastability and ageing in stochastic dynamics; to Steve Lalley for his course on AI- gebraic systems of generat ing functions and return probabilities for random walks; to Elon Lindenstrauss for his course on Recurrent measures and measure rigidity; to Sylvie Meleard for her course on Stochastic particle approximations for two-dimensional N avier- Stokes equations; and to Anatoly Vershik for his course on Random and universal metric spaces.

We present here the lectures and a selection of the seminars given at the Ninth International Workshop on Instabilities and Nonequilibrium Structures which took place in Vifiadel Mar, Chile, in December 2001. The Workshop was organized by Facultad de Ciencias Fisicas y Matematicas, Universidad de Chile, Instituto de Fisica of Universidad Cat6lica de Valparaiso, Centro de Fisica No Lineal y Sistemas Complejos de Santiago and Facultad de Ingenieria, Universidad de los Andes, which starting from this year joins the other institutions in the coorganization ofthe Workshop. The organizers would like to express their gratitude to the following sponsors: Facultad de Ciencias Fisicas y Matematicas de la Universidad de Chile, Instituto de Fisica de la Universidad Cat6lica de Valparaiso, Facultad de Ingenieria de la Universidad de los Andes, Centro de Fisica No Lineal y Sistemas Complejos de Santiago, Academia Chilena de Ciencias, Ministere Francais des Affaires Etrangeres, CONICYT (Comisi6n Nacional de Investigaci6n Cientifica y Tecno16gicade Chile) and Departamento Tecnico de Investigaci6n y de Relaciones Internacionales de la Universidad de Chile.Enrique Tirapegui PREFACE This book consists of two parts, the first one has three lectures written by Professors H. R. Brand, M. Moreau and L. S. Tuckerman. H. R. Brand gives an overview about reorientation and undulation instabilities in liquid crystals, M. Moreau presents recent results on biased tracer diffusion in lattice gases, finally, L. S. Tuckerman summarizes some numerical methods used in bifurcation problems.

This volume contains the courses given at the Sixth Summer School on Complex Systems held at Facultad de Ciencias Fisicas y Maternaticas, Universidad de Chile at Santiago, Chile, from 14th to 18th December 1998. This school was addressed to graduate students and researchers working on areas related with recent trends in Complex Systems, including dynamical systems, cellular automata, complexity and cutoff in Markov chains. Each contribution is devoted to one of these subjects. In some cases they are structured as surveys, presenting at the same time an original point of view and showing mostly new results. The paper of Pierre Arnoux investigates the relation between low complex systems and chaotic systems, showing that they can be put into relation by some re­ normalization operations. The case of quasi-crystals is fully studied, in particular the Sturmian quasi-crystals. The paper of Franco Bagnoli and Raul Rechtman establishes relations be­ tween Lyapunov exponents and synchronization processes in cellular automata. The principal goal is to associate tools, usually used in physical problems, to an important problem in cellularautomata and computer science, the synchronization problem. The paper of Jacques Demongeot and colleagues gives a presentation of at­ tractors of dynamical systems appearing in biological situations. For instance, the relation between positive or negative loops and regulation systems.

This book contains the lectures given at the Conference on Dynamics and Randomness held at the Centro de Modelamiento Matematico of the Universidad de Chile from December 11th to 15th, 2000. This meeting brought together mathematicians, theoretical physicists and theoretical computer scientists, and graduate students interested in fields re­ lated to probability theory, ergodic theory, symbolic and topological dynam­ ics. We would like to express our gratitude to all the participants of the con­ ference and to the people who contributed to its organization. In particular, to Pierre Collet, Bernard Host and Mike Keane for their scientific advise. VVe want to thank especially the authors of each chapter for their well­ prepared manuscripts and the stimulating conferences they gave at Santiago. We are also indebted to our sponsors and supporting institutions, whose interest and help was essential to organize this meeting: ECOS-CONICYT, FONDAP Program in Applied Mathematics, French Cooperation, Fundacion Andes, Presidential Fellowship and Universidad de Chile. We are grateful to Ms. Gladys Cavallone for their excellent work during the preparation of the meeting as well as for the considerable task of unifying the typography of the different chapters of this book.

This sixth Volume of the International Workshop on Instabilities and Nonequilibrium Structures is dedicated to the memory of my friend Walter Zeller, Professor of the Universidad C'at6lica df' Valparaiso and Vice-Director of the Workshop. Walter Zeller was much more than an organizer of this meeting: his enthusiasm, dedication and critical views were many times the essential ingredients to continue with a task which in occasions faced difficulties and incomprehensiolls. It is in great part due to him that the workshop has adquired to-day tradition. maturity and international recognition. This Volume should have been coedited by Walter and it is with df'ep emotion that I learned that his disciples Javier Martinez and Rolando Tiemann wanted as a last hommage to their Professor and friend to coedit tfus book. No me seria posible terminal' estas lineas sin pensar en la senora Adriana Gamonal de Zelln. qUf' ella encuentre en este libro la admiraci6n y reconocimiento hacia su marido de quiPIlf's [l\Prall sus discipulos, colegas y amigos.

This book contains the lectures and a selection of the seminars gi ven in the Fifth International Workshop on Instabilities and Nonequilibrium Structures which took place in Santiago, Chile, in December 1993. The Workshop was organized by Facultad de Ciencias Fisicas y Matematicas, Universidad de Chile, Instituto de Fisica of Universidad Cat6lica de Valparaiso and Centro de Fisica No Lineal y Sistemas Complejos de Santiago. This volume is the first of a new series of Kluwer on Nonlinear Phenomena and Complex Systems which will be edited by the Centro de Fisica No Lineal y Sistemas Complejos de Santiago. We thank Dr. David Lamer of Kluwer for his encouragements and support for this project. ix LIST OF SPONSORS OF THE WORKSHOP * Academia Chilena de Ciencias * Facultad de Ciencias Fisicas y Mathematicas de la Univ. de Chile * Instituto de Fisica de la Univ.Cat6lica de Valparaiso * Centro de FIsica No Lineal y Sistemas Complejos de Santiago (CFNL) * CONICYT (Chile) * Ministere Francais des Affaires Etrangeres * International Centre for Theoretical Physics (Trieste) * UNESCO * Fundaci6n Andes (Chile) * Departamento Tecnico de Investigaci6n y de Relaciones Internationa- cion ales de la Universidad de Chile * IDIEM (Fac. Cs. FIs. y Mat., Univ. de Chile) * CHILGENER S.A.