One service mathematics has rendered the 'Ht moi, ...* Ii j'avait so comment en revenir, je ny _ais point aile':' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf neJll to the dusty canister labelled 'discarded non- The series is diwrgent; therefore we may be sense' . * ble to do something with it. Eric T. Bell O. H eniside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, alI kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d't!tre of this series.

We have classified the articles presented here in two Sections according to their general content. In Part I we have included papers which deal with statistical mechanics, math ematical aspects of dynamical systems and sthochastic effects in nonequilibrium systems. Part II is devoted mainly to instabilities and self-organization in extended nonequilibrium systems. The study of partial differential equations by numerical and analytic methods plays a great role here and many works are related to this subject. Most recent developments in this fascinating and rapidly growing area are discussed. PART I STATISTICAL MECHANICS AND RELATED TOPICS NONEQUILIBRIUM POTENTIALS FOR PERIOD DOUBLING R. Graham and A. Hamm Fachbereich Physik, Universitiit Gesamthochschule Essen D4300 Essen 1 Germany ABSTRACT. In this lecture we consider the influence of weak stochastic perturbations on period doubling using nonequilibrium potentials, a concept which is explained in section 1 and formulated for the case of maps in section 2. In section 3 nonequilibrium potentials are considered for the family of quadratic maps (a) at the Feigenbaum 'attractor' with Gaussian noise, (b) for more general non Gaussian noise, and (c) for the case of a strange repeller. Our discussion will be informal. A more detailed account of this and related material can be found in our papers [1-3] and in the reviews [4, 5], where further references to related work are also given. 1.

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.

We have classified the articles presented here in two Sections according to their general content. In Part I we have included papers which deal with statistical mechanics, math ematical aspects of dynamical systems and sthochastic effects in nonequilibrium systems. Part II is devoted mainly to instabilities and self-organization in extended nonequilibrium systems. The study of partial differential equations by numerical and analytic methods plays a great role here and many works are related to this subject. Most recent developments in this fascinating and rapidly growing area are discussed. PART I STATISTICAL MECHANICS AND RELATED TOPICS NONEQUILIBRIUM POTENTIALS FOR PERIOD DOUBLING R. Graham and A. Hamm Fachbereich Physik, Universitiit Gesamthochschule Essen D4300 Essen 1 Germany ABSTRACT. In this lecture we consider the influence of weak stochastic perturbations on period doubling using nonequilibrium potentials, a concept which is explained in section 1 and formulated for the case of maps in section 2. In section 3 nonequilibrium potentials are considered for the family of quadratic maps (a) at the Feigenbaum 'attractor' with Gaussian noise, (b) for more general non Gaussian noise, and (c) for the case of a strange repeller. Our discussion will be informal. A more detailed account of this and related material can be found in our papers [1-3] and in the reviews [4, 5], where further references to related work are also given. 1.

One service mathematics has rendered the 'Ht moi, ...* Ii j'avait so comment en revenir, je ny _ais point aile':' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf neJll to the dusty canister labelled 'discarded non- The series is diwrgent; therefore we may be sense' . * ble to do something with it. Eric T. Bell O. H eniside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, alI kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d't!tre of this series.

1m Rahml'll der Technisch-Wissenschaftlichen Veranstaltungen de; Berliner Bezirksvereins Deutscher Ingenieure wurde im]'riihjahr 1938 ge meinsam mit dem AuBeninstitut der Technischen Hochschule Berlin und dem FachausschuB fiir Larmminderung des VDI eine Vortragsreihe iiber: "Schallabwehr im Bau- und Maschinenwesen" veranstaltet. Die groB ' Zahl der Teilnehmer lieB das Interesse an den behandelten Gegem;tanden erkennen. AuBerdem wurde vielfach del' Wunsch geauBert, das Material, welches in den Vortragen an dem Horer doch schnell voriiberzog, zum Nachschlagen zur Verfiigung zu haben, um einzelne FaIle der SchaIl abwehr innerhalb des eigenen Wirkungskreises mit Erfolg angreifen zu konnen. Da z. B. keine moderne deutsche Zusammenfassung auf diesem fiir dic Gesundheit eines sehr groBen Teiles der Bevolkerung ungemein wichtigen Gebiet vorhanden iHt, habe ich es mit groBziigiger Unterstiit zung des Verlages iibernommen, die einzelnen Vortrage gegeneinander ab zustimmen, Wichtiges, was in der beschrankten Zeit des Vortrages au gelassen werden muBte, einzufiigen, weiterfiihrende Literatur anzugeben und eine Formelsammlung anzufiigen, die dem auf dem Gebiet del' Schallabwehr tatigen Ingenieur die Bearbeitung bestimmter Aufgaben erleichtern solI. Damit die Zusammenstellung ihre Aufgabe als Nach t-lchlagewerk besser erfiillen kann, ist ein ausfiihrliches Inhalts- lind Saeh verzeichnis eingefiigt. Ich hoffe, daB damit, wenn auch nicht etwas Voll kommenes, so doch fUr den praktischen Gebrauch Niitzliches entstanden ist. Das Buch soil nicht allein eine kurze Einfiihrung in dies Sonder gebiet sein, sondern soll dem Praktiker zeigen, welche Allfgaben bereit ' gelOst sind, wo noch besondere Schwierigkeiten zu erwarten und wiC' }1 inzelfalle gemeistert sind."