Complex Dynamics and Morphogenesis: An Introduction to Nonlinear Science

475 Pages·2017·13.929 MB·English

by  Chaouqi Misbah (auth.)

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Chaouqi Misbah
Complex Dynamics and Morphogenesis
Grenoble Sciences
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Grenoble Sciences is a department of the Grenoble Alpes University
supported by the ministère de l’Éducation nationale,
de l’Enseignement supérieur et de la Recherche
and the région Auvergne-Rhône-Alpes.
Complex Dynamics and Morphogenesis is an improved version of the
original book Dynamiques complexes et morphogenèse by Chaouqi
Misbah, Springer-Verlag France, 2011,ISBN 978 2 8178 0193 3.
The reading committee included the following members:
Bernard Billia, CNRS Senior Researcher,IM2NP, Marseille;
François Charru, Professor, Toulouse University;
Stéphane Douady, CNRS Researcher, Laboratoire MSC, Paris;
Hamid Kellay, Professor, Bordeaux University;
Paul Manneville, CNRS Senior Researcher, LADHYX, Palaiseau;
Christian Wagner, Professor, Saarland University.
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Sylvie Bordage; cover illustration: Alice Giraud, from fig. 1.4 [(cid:2)c NOAA],
13.5a [(cid:2)c forcdan Fotolia], 15.2 [all rights reserved], 15.3d [from [78],
reprinted with permission from Macmillan Publishers Ltd] and elements
provided by the author.
Chaouqi Misbah
Complex Dynamics
and Morphogenesis
An Introduction to Nonlinear Science
123
ChaouqiMisbah
CNRS
Laboratoire Interdisciplinaire dePhysique
Grenoble
France
Co-published by Springer International Publishing AG, Gewerbestrasse 11, 3330 Cham,
Switzerland, and Editions Grenoble Sciences, Université Joseph Fourier, 230 rue de la
Physique, BP5338041Grenoble cedex09, France.
ISBN978-94-024-1018-1 ISBN978-94-024-1020-4 (eBook)
DOI 10.1007/978-94-024-1020-4
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To Florence, Hisham and Sammy
To my parents, sisters and brothers
Foreword
As stated by its subtitle, this book is an introduction to nonlinear sciences.
Thissubtitlehasthemeritofemphasizingthemultiplicityofthisbookwhich
deals with biology, chemistry, and of course physics and mathematics.
If someone thought to divide science into linear and nonlinear science, no
doubt that the latter would form the main part. However, the range of
linear physics, i.e. physics described by first order equations, is surprisingly
extended. For instance, linear elasticity is a good approximation in many
cases. Maxwell’s equations of electromagnetism are linear with respect to
theelectromagneticfield.Schrödinger’sandDirac’sequationsarelinearwith
respect to the wavefunction. These partial differential equations allow the
use of sophisticated mathematical techniques such as Green’s functions and
retarded potentials, making linear physics efficient, but also cumbersome.
Nonlinear science is quite different: analytic methods are of limited use.
Computers may of course provide a workaround, but numerical results do
not give the same insight as analytic arguments. Thus, a texbook on non-
linear science has to be less ambitious in its accuracy of description than a
book on quantum mechanics, electromagnetism or elasticity – but it may be
more pleasantto read.The presentbookby ChaouqiMisbahhas this virtue.
Sinceonecannotgomoredeeplyintodetails,itpresentstheessentialfeatures,
sometimes in a purely qualitative way.These features have an extraordinary
diversity, in contrast to the rather monotonous landscape of linear physics.
In this book, the reader finds a variety of phenomena, many familiar but
not understood – for example, a swing. It is unfortunate that children at
school do not learn how to use a swing, because they would discover what
parametric resonance is. As explained in the book, the greatestamplitude is
reached if the child lowers their center of mass when the swing goes up, and
raisesitwhentheswinggoesdown,creatinganexcitationfrequencyattwice
the natural frequency of the swing. This is typical of parametric resonance.
VIII Complex Dynamics and Morphogenesis
Swings are not the only entertainment the reader may find in this book.
Anotherexampleis thetransitiontochaosbysubharmoniccascade.Isitnot
engaging to play with a pocket computer and observe how the change of a
parameter (a in formula (8.8)) changes the trivial fixed point into a discrete
limit cycle of2 points,then 4 points,8 points,etc., before reachingcomplete
chaos?Thiswillbesurprisingforthenewcomertononlinearscience;another
surprise awaits (section 8.5) with the discovery that the continuous version
of the same model is not chaotic at all but has a perfectly stable, nontrivial
fixed point.
All the same, nonlinear science cannot be reduced to entertainment, as
the reasonable number of equations and formulae in this book testify.
Mathematical formulae are certainly necessary for the precise description
of physical phenomena. They can also be a didactic tool for clarifying an
argument. For example, in chapter 4 of the book, catastrophes are classified
for cases of up to 4 control parameters. As it turns out, there are 7 catas-
trophes (as Snow White is accompanied by 7 dwarfs, as the golden Hebrew
lampstand has 7 branches!). As noted in the book, this classification can
be done without writing the equations governing the systems, but the au-
thor’s argument does rely on mathematical formulae. Although not strictly
necessary, they certainly simplify the proof. A merit of this book lies in its
clever use of mathematical formulae to help one understand the mechanism
but avoiding complications when not absolutely necessary. Sometimes they
are necessary: chapters 10 and 12 are really tough. However,as a whole, the
reader has before them an enjoyable source of discoveries which mixes rigor
and entertainment harmoniously.
Jacques Villain
Emeritus CEA Senior Researcher
Member of the French Academy of Sciences
Acknowledgments
I am indebted to Christiane Caroli and Philippe Nozières for their sup-
portive attitudes which encouraged me to undertake the project of writing
this book. I am also grateful to Michel Saint Jean who encouraged me from
the outset of this project and Jacques Villain for many enlightening discus-
sions, and I warmly thank the members of my research team in Grenoble,
Dynamique des Fluides Complexes et Morphogenèse (DYFCOM), for the
critical reading of the manuscript. I thank especially Pierre-Yves Gires who
has gone into many details of the derivations of the model equations and
their solutions, Christian Wojcik for preparing many plots, Richard Michel
who produced from scratch many numerical results presented in this book
and Alexander Farutin for checking the solutions to the exercises. I express
my gratitude to Marc Rabaud for having redone the Saffman–Taylor exper-
iment and providing me with the photos presented in chapter 15. During
the elaboration of this project I have greatly benefitted from references
[6,9,16,23,29,32,42,58,75,77,80,87,91,99,101].
Chaouqi Misbah
Contents
Chapter 1 – Presentation of Main Ideas 1
Chapter 2 – Basic Introduction to Bifurcations in 1-D 15
2.1. A Simple Mechanical Example ......................................... 16
2.1.1. Potential Energy and Equilibrium ............................ 16
2.1.2. An Intuitive Explanation of the Bifurcation ................ 18
2.1.3. Analysis of the Bifurcation ..................................... 19
2.1.4. Universality in the Vicinity of a Bifurcation Point ......... 20
2.2. An Analogy between Pitchfork Bifurcations
and Second Order Phase Transitions .................................. 22
2.3. Dynamical Considerations .............................................. 22
2.3.1. Analysis of Linear Stability .................................... 23
2.3.2. Critical Slowing Down .......................................... 24
2.3.3. Adiabatic Elimination of Fast Modes:
Reducing the Number of Degrees of Freedom ............... 25
2.3.4. Amplitude Equation ............................................ 27
2.3.5. Canonical Form of the Amplitude Equation ................ 28
2.3.6. Attractors ......................................................... 29
2.3.7. Lyapunov Function .............................................. 30
2.3.8. Symmetry Breaking ............................................. 30
2.4. Dynamical Systems and the Importance
of the Pitchfork Bifurcation ............................................ 31
2.5. Exercises ................................................................... 31