Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, M~Jnchen, K. Hepp, Z0rich R. Kippenhahn, M0nchen, H. A. Weidenm~Jller, Heidelberg and J. Zittartz, KSIn

179

Dynamical Systems and Chaos Proceedings of the Sitges Conference on Statistical Mechanics Sitges, Barcelona/Spain September 5-11, 1982

Edited by L. Garrido

Springer-Verlag Berlin Heidelberg New York Tokyo 1983

Editor Luis Garrido Departarnento de Fisica Te6rica, Universidad de Barcelona Diagonal 647, Barcelona-28, Spain

ISBN 3-540422764 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-3874 22764 Springer-Verlag New York Heidelberg Berlin Tokyo

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Weft", Munich. © by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruclq Hemsbach/Bergstr, 2153/3140-543210

ACKNOWLEDGEMENT

I would like to take this opportunity to thank all those who col-

laborated in the organization of this Conference. In particular,

my deep appreciation goes to Professor D. Jou from the Department

of Thermology of the Autonomous University of Barcelona for his great

enthusiasm in solving the numerous problems involved in preparing

and running the Conference, and to Professor M. San Miguel and

J.M. Sancho from the Department of Theoretical Physics of the

University of Barcelona for their efforts and cooperation.

I also extend my warmest thanks to the City of Sitges for allowing

us to use the museum "Maricel" as a lecture hall, and to the Inter-

national University "Menendez Pelayo" for its economic support.

My final thanks go to my wife for her patience and unremitting

cooperation.

L. Garrido

IIJ

Participants

Dr. AGUIPJtE, E., Universidad Complutense, Facu!tad de Matem~ticas, Ciudad Universitaria, Madrid-3, Spain

Mr. ALSEDA I SOLER, LI., Universitat Aut6noma de Barcelona, Facultat de Ciencies Economiques, Be!laterra, Barcelona, Spain

Prof. ATTEN, P., C.N.R.S., Laboratoire d'Electrostatique, Grenoble~ France

Prof. AUBANELL POU, A., Universitat de Barcelona, Pacultat de Mate- m~tiques, Barcelona-9, Spain

Prof. AUGE, J.~ Universigat de Barcelona, Facultat de Matem~tiques ~ Barcelona-9, Spain Gran Via j8~,

Prof. d'AURiAC, A., C.N.R.S., C.R.T.B.~.~ Grenoble, France

Miss BAESENS, C., Universit6 Libre de Bruxelles, Facult$ des Sciences; Campus Plaine, Blvd. du T~iomphe, I050 Bruxelles, Belgium

Dr. BARCONS, P., Universidad de Santander, Pacultad de Ciencias, Santander, Spain

Dr. BEN-JACOB, E., Institute for Theoretical Physics, Santa Barbara CA 93106, USA

Mr. BENSENY, A., Pacultat de Matem~tiques, Universitat de Barce]on~ Barcelona, Spain

Prof. BOUN~IS, T.C., Clarkson College of Technoiogy, Potsdam, NY 13696, USA

Prof. BRANNER-J~RGENSEN, B., Maghematical Institute, Lyngby, Denmark

Prof. BP~EITENECKER, M., Institute for Theoretical Physics, University of Vienna, Vienna, Austria

Dr. van den BROECK, C. Vrye Universigeit Brussel, i0~0 Brussel, Belgium

° r ~ r ° . .

Mr. CALS!NA, A., Seccmo ~e Matematmques, Pacultat de Cmencmes, Uni- versitat Aut6noma de Barcelona, Bellaterra, Barcelona, Spain

Mr. CALVO HERNANDEZ, A°, Pacultad de Ciencias, Dpto. Termolog{a Salamanca, Spain

Mr. CARBONELL, J~, Insti%ut des Sciences Nucl$aires, Grenoble, France

Prof. CARL, H., Institut de Physique Th$orique, Universit$ de Lausanne, I01~ Dorigny, Switzerland

Prof. CASARTELLI, M., Sezione Teorica, Universit~ di Parma, Parma, ~3!00, Italy

Miss CASASAYAS I MAS, J., Universidad de Barcelona, Facultad de Matem~ticas, Barcelona, Spain

Prof. CASA~I, G., Istituto di Scienze Pisiche, Unmversmta degli studi di Mi!ano, 20133 Milano, Italy

Prof. CAWLEY, R., US Naval Surface Weapons Center, Silver Springs, MD 20910, USA

Prof. COSNA~D, M., C.N.~.S., Laboratoire iMAG, Grenoble 380g!, Prance

Mr. DANA, I., ~echnion, Dept. of Physics, Technion City, 32000 Haifa, Israel

V

Mr. DELSHAMS, A., Facultat de Matem~tiques, Uniwersitat de Barcelona, Barcelona-7, Spain

i " " • • " Mr. DE SOI~-MORALES, J., Facultat de C encmes, Secclo de Matematlques~ Universitat Aut$nomade Barcelona, Bellaterra, Barcelona, Spain

Prof. DIII~0, R., Centro de Fisica de Materia C0ndensada , 1699 Lisboa, Portugal

Dr. EDERY, D., C.E.A., STGI-Fusion~ Fontenay-Aux-Roses 92260, France

Prof. DIAS DE DEUS, J.V.C., Centro de F~sica de Materia Condensada, 1699 Lisboa, Portugal

Mr. ELGIN, J., Imperial College, Dept. Physics, London SW 7, Great Britain

Prof. EU, B. Ch., McGill University, Dept. of Chemistry, Montreal PQ Canada H3A 2K6

Prof. FALQU~S SERRA, A., Universidad Polit6cnica Barcelona (E.T.S. Enginyers Camins~, Barcelona-34, Spain

Prof. FARQUHAR, I.E., University of St. Andrews, School of Physical Sciences, North Haugh, St. Andrews, Fife, Scotland

Prof. FEIGENBAUM, M. J., Los Alamos Laboratory, Los Alamos NM 37545, USA

• J •

Mr. FONTICH JULIa, E.~ Universitat Polltecnlca de Barcelona, Barcelona-34, Spain

Mr. FONT, J., Facultat de Matem~tiques, Universitat de Barcelona, Barcelona-7, Spain

Prof. FRISCH, U., CNRS, 0bservatoire de Nice, 06007 Nice, France

Prof. FROYLAND, J., Institute of Physics, University of Oslo, Blindern, Oslo 3, Norway

Dr. GAST, 0.W., Theoretical Physics I, University of MGnster, 44 MGnster~ Germany

Prof. GARRIDO, L., Universidad de Barcelona, Facultad de F{sica, Diagonal 647, Barcelona-28, Spain

Prof. GEISEL, T., Universit[t Regensburg, Institut fGr Physik I, 8400 Regensburg, Germany

Prof. GIACHETTI, R., I.N.F.N. Sezione di Firenze, 50125 Firenze, Italy

Prof. GIGLIO, M., CISE S.P.A. - 20100 Milano, Italy

Prof. GILBERT, A.D., Dept. of Mathematics, University of Edinburgh Edinburgh EH9 3JZ, Scotland

Dr. GILLOT, Ch., Institut National des Sciences Appliqu$es, 31077 Toulouse, Franc~

Mr. GLENDINNING, P., University of Cambridge, DAMTP, Cambridge CB3 9EW, Great Britain.

Prof. GOMEZ MUNTAN~, G., Universitat Aut6noma de Barcelona, Facultat de Ciencies, Bellaterra, Barcelona, Spain

Prof. GONZALEZ-GASCON, F., Universidad de Salamanca, Facultad de Ciencias, Salamanca, Spain

Mr. GRAU, M., Facultad de Matem~ticas, Universidad de Barcelona, Barcelona-7, Spain

Mr. GRUNBAUM, K., University of Roskjilde, IMFUFA, 4000 Roskjilde Denmark

Vl

Prof. GUCKENHEIMER, J°, University of California, Dept. of Physics Santa Cruz 95064, USA

Prof. GUMOWSKI, I., Universit$ Paul Sabatier, 31062 Toulouse, France

Mr. HAUBS, G., Institut fNr Theoretische Physik, I, Stuttgart-80, Germany

Dr. HENTSCHEL, H. G. E., Weizmann Institute of Science, Rehovot, Israel

Miss HERNANDEZ, A., Universidad de Barcelona, Facultad de Fisica, Diagonal 647, Barcelona-28, Spain

Prof. HUBERMAN, B.A., Xerox Corporation, Palo Alto Research Center, Palo Alto, CA 94304~ USA

Dr. JAUSLIN, H.R., Universit$ de Gen~ve, Dept. de Physique Th$o- rique, 32, Blvd. d'ivoy, 1211 Gen~ve, Switzerland

~rof. J0U MIRABENT, D., Universidad Aut6noma, Facultad de Fisiea, Bellaterra, Barcelona, Spain

Prof. KAPLAN, H., Syracuse University, Dept. of Physics, Syracuse, NY 13210, USA

Prof. KATOK, A., University of Maryland, Dept. of Mathematics, College Park, MD 20742, USA

Dr. KELLER, G., Institut f~r angewandte Mathematik, Universit~t Heidelberg, Im Neuenheimer Feld, 69 Heidelberg, Germany

Prof. KRUSKAL, M.D., Princeton University, Physics Dept. Princeton NJ 08544, USA

Dr. KUNiCK, A., Rechenzentrum/ZOD 52, 8520 Er!angen, Germany

Dr. KURLAND, H.L., Boston Univer~y, Dept. of Mathematics, Boston Mass. 02215, USA

Prof. LIBCHABER, A., Groupe de Physqique des Solides, Ecole Normale Superleure, 24 rue Lhomond, 95231 Paris, France

Prof. L~IBRE, J., Universitat Aut6noma de Barcelona, Facultad de Ciencias, Bellaterra, Barcelona, Spain

s . Mr. LOPEZ DESA, J.M., Facultad de Matematlcas, Universidad Complutense de Madrid, Ciudad Universitaria, Madrid-3, Spain

Prof. LOVESEY, S.W., Rutherford Appleton Laboratory, Chilton, Oxford- shire OXll 0QX~ Great Britain

Mr. LLEBOT, J.E., Dpto. de Termologla, Universidad Aut6noma de Barcelona, Bellaterra, Barcelona, Spain

Prof. LUENGO PASCUAL, ~.M., Facultad de Inform~tica, Jordi Girona Salgado 31, Barcelona-34, Spain

Miss LUIS, M.A., Facultad de Qulmicas, Universidad Complutense de Madrid, Ciudad Universitaria, Madrid-3, Spain

Dr. MAGNENAT, P., 0bservatoire de Gen~ve, CH 1290 Sauverny, Switzer- land

Mr. MALAGRIDA, I., Depto. de Paleontologla, Facultad de Geologla, Universidad de Barcelona, Spain

Mrs. MARTINEZ BARCHINO, R., Universitat Aut6noma~ Secci6 de Mate- m~tiques, Bellaterra, Barcelona, Spain

Dr. MARTINEZ, J., Universidad de ~alencia~ Faeultad de Matem~ticas, Valencia, Spain

Prof. MASSAGUER, J.M., E.T.S.I., Universidad Polit$cnica de Barce- • lona, Jorge Girona Salgado 31, Barcelona-34~ Spain

VII

Mr. MAYER-KRESS, G., Institut fGr Theoretische Physik I, 7 Stuttgart- 80, Germany

Prof. MAYNARD, R., Universit~ de Grenoble, C.N.R.S. - CRTBT, Grenoble 38042, France

Prof. MIRA, Ch., Inst. National des Sciences Appliqu6es, Av. de Rangueil, 31077 Toulouse, France

Dr. MISGUICH, J., C.E.A., STGI - Fusion, 92260 Fontenay-Aux-Roses, France

Miss MOHEDANO, M.V., Facultad de Ciencias Matem&ticas, Universidad Comp!utense, Ciudad Universitaria, Madrid-3, Spain

~. MORA, X., Secclo de Matematlques, Facultat de Clencmes, Univer- i s dad Autonoma de Barcelona, Bellaterra, Barcelona, Spain

Dr. MOROZ, I.M., School of Mathematics, University of Leeds, Leeds L52 9JT, Great Britain

Miss MULLENBACH, S., Inst. national des Sciences AppliquSes, Av. de Rangueil, 31097 Toulouse, France

Dr. NAGASHIMA, T., Institute of Precission Mechanics, Faculty of Engineering, Hokkaido University, Japan

Prof. NEWHOUSE, S., University of North Carolina, Phillips Hall 039-A, Chapel Hill, NC 27514, USA

Dr. NIERWETBERG, J., Institut I, Theoretische Physik, Universit~t Regensburg, D 84 Regensburg, Germany

Prof. NOGUERA BATLLE, M., Facultat d'Informatica, U.P.B., Jordi Girona Salgado 31, Barcelona-34, Spain

Prof. NOHONHA DA COSTA, A., CMFC, Av. Prof. Gama Pinto, 2 1699 Lisboa, Portugal

Prof. OBERMAIR, G.M., Universit~t Regensburg, Fakult~t fGr Physik, 84 Regensburg, Germany

Dr. PACKARD, N.H., IHES, 35 Route de Chartres, 91540 Bures-sur- Yvette, France

Prof. PERELL0 i VALLS, C., Universitat Aut6noma, Facultat de Ciencies Bellaterra, Barcelona, Spain

Dr. PERINI, U., CISE S.P.A., P0B 12061, 20100 Milano, Italy

Dr. PESQUEPA GONZALEZ, L., Universidad de Santander, Fac. de Ciencias, Santander, Spain

Miss PIGNATAR0, T., Physics Dept., Princeton University, POB 908 Princeton, nJ 08550, USA

Prof. PISMEN, L.M., Institute of Applied Chemical Physics, CCNY, Convent Av. and l$Oth St. new York, NY 10031, USA

Prof. PROCACCIA, I., Dept. of Chemical Physics, Weizmamm Institute of Science, Rehovot, Israel 76100

Dr. RAMASWAMY, R., Tara Institute of Fundamental Research, Homo Bhabha Rd., Bombay 400 005, India

Prof. RAMADA, A., Universidad Complutense, Facultad de Ciencias Fisicas, Ciudad Universitaria, Madrid-3, Spain

Mr. REICHERT, P., Institut fHr Physik, Universit~t Basel, Klingel- bergstr. 82, $056 Basel, Switzerland

Dr. RENZ, W., Technical University, RWTH Aachen, Templergraben 55, 5100 Aachen, Germany

VIIB

Prof. RiELA, G., Universit~ di Palermo, Istituto di Fisiea, Via Archirafi, Palermo 90123, Italy

Prof. HOD, D.L., University of Calgary, Dept. Mathematics, Calgary Alberta T2N IN4, Canada

Mr. RODRIGUEZ DIAZ, M.A., Universidad de Santander, Faeultad de Ciencias, Santander, Spain

Dr. RODRIGUEZ MENDEZ, J.A., Facultad de Matem~tieas, Universidad de Santiago de Composte!a, Spain

Prof. ROJAS, E., Dpto. de Termologla, Facultad de Cieneias, Univer sidad de Salamanca, Spain

Mr. ROGERS, J.B., University of Melbourne, Mathematics Dept. Park- rifle, Victoria, Australia 3052

Miss ROS, R.M., Universidad de Barcelona, Facultad de Fisica, Diagonal 647, Barcelona-28, Spain

Prof. ROSSLER, 0.E., Universit~t THbingen, Auf der Morgenstelle 8, 74 THbingen~ Germany

Dr. ROSSLER, (Mrs.), Universit~t THbingen, Auf der Morgenstelle 8, 74 T~bingen, Germany

~. RUBi, J.M., Dpto. Termologla, Universidad Aut6noma, Bellaterra Barcelona, Spain

Prof. RUDNICK~ J., University of California at Davis, Dept. of Physics, Davis, CA 95616, USA

Prof. SAENZ, A.W., Catholic University of America, Physics Dept., Washington, D.C. 20064, USA

Mr. SAGU~S, F., Universidad de Barcelona, Facultad de Fisica, Diagonal 647, Barcelona-28, Spain

Prof. SANCHO, J.M., Universidad de Barcelona, Facultad de Fisica~ Diagonal 647, Barcelona-28, Spain

Prof. SANDERS, J.A., Wiskundig Seminarium~ Vr~e Universlteit, PB 9161, I007 MC Amsterdam, Holland

Prof. SAN MIGUEL, M., Universidad de Barcelona, Facultad de Fisiea, Diagonal 647, Barcelona-28, Spain

Dr. SARAMIT0, B., C.E.A., D.R.F.C. - S.T.G.I., Centre d'Etudes Nucl$aires, BP 92260, Fontenay-aux-Roses, France

Dr. SCHILLING, R., Institute of Physics, University of Basel, Klingelbergstr. 82, CH 4056 Basel~ Switzerland

Dr. SCHOLL, E., Institut fHr Theoretische Physik, HWTH Aachen~ 5100 Aachen, Germany

Prof. SCOTTI, A., Joint Research Center of the European Communities and GNSM-CNR, Istituto di Fisica, Universit~ di Parma, Italy

Mr. SERRA, R., INB "Sant Cugat", Rambla Ribatallada 56, Sant Cugat del VallSs, Barcelona, Spain

Dr. SHIETER, J.S., Biophysik. Chemie, Biozentrum, Universit~t Basel 4056 Basel, Switzerland

Dr. SIEGBERG, H.W., Universit~t Bremen, FB Mathematik, D 28 Bremen, Germany

Prof. SIGGIA, E., Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA

Prof. SIMO, C., Universidad de Barcelona, Facultad de Matem~ticas Gran Via 585, Barcelona-7, Spain

IX

Dr. SOUSA RAMOS, J., Centro de Fisica da Materia Condensada, Av. Prof. Gama Pinto 2, 1699 Lisboa, Portugal

Dr. STEEB, W.H., Universit~t Paderborn, Theoretische Physik, 4790 Paderborn, Germany

Dr. STEPHAN, W., Fakult~t fGr Bio!ogie, Universit~t Konstanz, PF 5560~ 7750 Konstanz, Germany

Prof. TEBALDI, C., Istituto di Matematica Applicata, Facolt~ di Univers!ta di Bologna, Bologna 40136, Italy Ingegneria, " " ~

Mr. THUAL, 0.~ Observatoire de Nice, BP 525, 06007 Nice, France

Prof. TIRAPEGUI, E., Depto. de F~sica, Universidad de Chile, Avda. Blanco Encalada 2008, Santiago de Chile, Chile

Mr. TISHBY, N., Dept. of Theoretical Physics~ The Hebrew University of Jerusalem~ Jerusalem 91904, Israel

Mr. VALL~S, J., Universidad de Barcelona, Facultad de F~sica, Diagonal 647, Barcelona-28, Spain

Mr. VAZQUEZ MARTINEZ, L., Universidad Complutense~ Facultad de F~sica Ciudad Universitaria, Madrid-3, Spain

Prof. VELASCO MAILLO, S., Universidad de Salamanca, Facultad de Ciencias, Salamanca, Spain

Prof. WEISS, N., University of Cambridge, DAMPT, Silver St. Cambridge, CB3 9EW, Great Britain

Prof. WEST, B.J., La Jolla Institute~ P0B 1~34, La Jolla, CA 92038, USA

Prof. WILCOX, C.H., University of Utah~ Mathematics Dept. Salt Lake City~ Utah 84112, USA

Prof. WILLIAMS, M., Virginia Polytechnic Institute, Dept. of Mathema- tics, Blacksburg, VA 2~061~ USA

Dr. WOLFF, W., Institut fNr Theoretische Physik~ Universit~t zu KSln 5000 KSln-41~ Germany

CONTENTS

PROLOGUE: SOME IDEAS ABOUT STRANGE ATTRACTORS

1. Introduction

2. On the definition of strange attractors

3. Mechanisms explaining the strange attractors

4. Examples of systems with strange attractors

5. Conclusions

CHAOTIC DYNAMICS IN HAMILTONIAN SYSTEMS WITH DIVIDED

PHASE SPACE

1. Introduction

2. Poincar@'s problem

3. Solution of the Poincar@ problem

4. On structure of the chaos border

PERIODIC AND QUASI-PERIODIC ORBITS FOR TWIST MAPS

1. Introduction

2. Twist maps

3. Motivating examples

4. Integrable twist maps and invariant circles

5. Special orbits

6. Continuity of special orbits

7. The Lagrangian and variational arguments

8. Concluding remarks

MACROSCOPIC BEHAVIOR IN A SIMPLE CHAOTIC

HAMILTONIAN SYSTEM

1. Introduction

2. Collision with a frictionless trap door

3. A collision model

4. A smooth model

5. Numerical results

6. Discussion

QUANTUM DYNAMICS

L. GARHID 9 C. SIMO

1

3

9

14

17

B.V. CHIRIKOV

29

3O

31

36

A. KATOK

47

48

50

52

53

57

58

62

O.E. R~SSLER

67

69

7O

71

73

76

B.A. HUBERMAN

79

×I

A UNIVERSAL TRANSITION FROM QUASI-PERIODICITY

TO CHAOS - ABSTRACT

E.D. SIGGIA

91

SELF-GENERATED DIFFUSION AND UNIVERSAL CRITICAL

PROPERTIES IN CHAOTIC SYSTEMS

T. GEISEL J. NIERWETBERG

1. Introduction

2. Discrete dynamical systems and period-doubling

3. Fine structure of the chaotic regime in

period-doubling systems

4, Theoretical analysis of the fine structure

5. Self-generated diffusion - an example

6. Universal critical behavior at the onset of diffusion

93

94

99

102

107

108

SUBHARMONICS AND THE TRANSITION TO CHAOS J. RUDNICK

1. Introduction

2. Simple subharmonic response: The parametrically

driven pendulum

3. The spectral signature of period doubling and the

strucutre of the universal map

115

116

122

LOW DIMENSIONAL DYNAMICS AND THE PERIOD

DOUBLING SCENARIO

M. FEIGENBAUM

1, Introduction

2. Minimal models and mappings

3. Period doubling

4 Trajectory scaling

5, Power spectrum

131

132

134

139

144

STRANGEATTRACTORS IN FLUID DYNAMICS J. GUCKENHEIMER

149

EXPERIMENTAL ASPECTS OF THE PERIOD DOUBLING SCENARIO A. LIBCHABER

1. The ~eriod doubling scenario

2. The interrupted cascade

158

162

XII

ENTROPY AND SMOOTH DYNAMICS S.E. NEWHOUSE

1. Introduction

2. Entropy and }U-equilibrium states

3. Topological entropy and volumes of submanifolds

~65

167

172

SEMINARS AND COMMUNICATIONS

MIRA, C.: "Imbedding of a one-dimensional endomorphism into a two-dimensional diffeomorphism. Implications" 180

SIMO, C.: "Strange attractors for differential delay equations" 188

KELLER, G.: "Stochastic perturbations of some strange attractors" 192

WEST, B.J.: " Solutions of stochastic differential equations and fractal trajectories" 194

SCH~LL, E.: "Continuous bifurcation and dissipative structures associated with a soft mode recombination instability in semiconductors" 204

PROCACCIA, I., et al.: "On the characterization of chaotic motions" 212

BAESENS, C.: "Complex bifurcations in a periodically forced normal form" 223

DIAS DE DEUS, J. et al.: "Topological entropy and scaling behaviour" 225

BOUNTIS, T.: "On the analytic structure of chaos in dynamical systems" 227

MAYER-KRESS, G., HAKEN, H.: "Type-lll-intermittency in a smooth perturbation of the logistic system" 237

MARTINEZ, S., TIRAPEGUI, S.: "Irreversible evolution of dynamical dynamical systems" 239

GOMEZ, G., SIMO, C.: "Homoclinic and heteroclinic points in the H~non map" 245

ALSEDA, L., SERRA, R.: "The simple periodic orbits in the unimodal maps" 248

NAGASHIMA, T.: "Modulation properties in decaying processes of the correlation function in a family of 1-D maps" 250

CASARTELLI, M.: "Relaxation times and randomness for a nonlinear classical system" 252

GILLOT, C.: "Topological entropy on rotation sequences" 254

FRISCH, U.: "The Taylor-Green vortex: Fully developed turbulence and transition to spatial chaos" 257

STEEB, W.-H., et al.: "Anharmonic systems in external periodic fields with chaotic behaviour" 260

COSNARD, M., EBERHARD, A.: "Renormaiization of non-analytical unimodal maps" 262

VAN DEN BROECK, C.: "Critical fluctuations in a thermo-chemical instability" 264

AUBANELL, A.: "The second order Melnikov integral applied to detect quasi-randomness" 266

XlII

MUNOZ SUDUPE,A., ALVAREZ-ESTRADA, R.F.: The Fokker-Planck equation as a dynamical system 268

FONTICH, E.: "On integrability of quadratic area preserving mappings in the plane" 270

GUMOWSKI, I.: "Resonances: Key elements to the understanding of nonlinear oscillations" 272

DELSHAMS, A.: "On systems passing through resonances" 280

MAGNENAT, P.: "The Lyapunov characteristic numbers and the number of isolating integrals in galactic models" 282

GRAU, M., et al.: "On the periodic orbits of the Contopoulos Hamiltonian" 284

PIGNATARO, T.: "Feasibility of calculating dimension and topological entropy" 287

WILLIAMS, M.: "Diffusions generated from dynamical systems" 290

SANDERS, J.A.: Report on the driven Josephson equation" 297

XIV