Differential Geometry Applied to Dynamical Systems (World Scientific Series on Nonlinear Science, Series a)

341 Pages·2009·36.06 MB·English

by  Jean-marc Ginoux

DIFFERENTIAL GEOMETRY APPLIED
TO DYNAMICAL SYSTEMS
WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE
Editor: Leon O. Chua
University of California, Berkeley
Series A. MONOGRAPHS AND TREATISES*
Volume 48: Bio-Inspired Emergent Control of Locomotion Systems
M. Frasca, P. Arena & L. Fortuna
Volume 49: Nonlinear and Parametric Phenomena
V. Damgov
Volume 50: Cellular Neural Networks, Multi-Scroll Chaos and Synchronization
M. E. Yalcin, J. A. K. Suykens & J. P. L. Vandewalle
Volume 51: Symmetry and Complexity
K. Mainzer
Volume 52: Applied Nonlinear Time Series Analysis
M. Small
Volume 53: Bifurcation Theory and Applications
T. Ma & S. Wang
Volume 54: Dynamics of Crowd-Minds
A. Adamatzky
Volume 55: Control of Homoclinic Chaos by Weak Periodic Perturbations
R. Chacón
Volume 56: Strange Nonchaotic Attractors
U. Feudel, S. Kuznetsov & A. Pikovsky
Volume 57: A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science
L. O. Chua
Volume 58: New Methods for Chaotic Dynamics
N. A. Magnitskii & S. V. Sidorov
Volume 59: Equations of Phase-Locked Loops
J. Kudrewicz & S. Wasowicz
Volume 60: Smooth and Nonsmooth High Dimensional Chaos and
the Melnikov-Type Methods
J. Awrejcewicz & M. M. Holicke
Volume 61: A Gallery of Chua Attractors (with CD-ROM)
E. Bilotta & P. Pantano
Volume 62: Numerical Simulation of Waves and Fronts in Inhomogeneous Solids
A. Berezovski, J. Engelbrecht & G. A. Maugin
Volume 63: Advanced Topics on Cellular Self-Organizing Nets and Chaotic
Nonlinear Dynamics to Model and Control Complex Systems
R. Caponetto, L. Fortuna & M. Frasca
Volume 64: Control of Chaos in Nonlinear Circuits and Systems
B. W.-K. Ling, H. H-C. Lu & H.K. Lam
Volume 65: Chua’s Circuit Implementations: Yesterday, Today and Tomorrow
L. Fortuna, M. Frasca & M. G. Xibilia
*To view the complete list of the published volumes in the series, please visit:
http://www.worldscibooks.com/series/wssnsa_series.shtml
Lakshmi - Diff Geometry Applied.pmd 2 9/23/2009, 1:03 PM
NOWONRLLDI NSCEIEANTRIFI CS SCERIIEESN OCNE Series A Vol. 66
Series Editor: Leon O. Chua
DIFFERENTIAL GEOMETRY APPLIED
TO DYNAMICAL SYSTEMS
Jean-Marc Ginoux
Université du Sud, France
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
World Scientific Series on Nonlinear Science, Series A — Vol. 66
DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS
(With CD-ROM)
Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN-13 978-981-4277-14-3
ISBN-10 981-4277-14-2
Printed in Singapore.
Lakshmi - Diff Geometry Applied.pmd 1 9/23/2009, 1:03 PM
September11,2009 8:55 WorldScientificBook-9inx6in DGeometry
`a Christine...
v
September11,2009 8:55 WorldScientificBook-9inx6in DGeometry
vi DifferentialGeometry Applied to Dynamical Systems
“...every time the system absorbs energy the curvature of its
trajectory decreases and viceversa ...”.
— N.Minorsky1
1N.Minorsky(1967,p.108).
September11,2009 8:55 WorldScientificBook-9inx6in DGeometry
Preface
The study of dynamical systems, i.e. of systems of differential equations
finds roots in the works of L. A. Cauchy (1835) and the so-called calculus
of limits which gave rise to an analytic approach1. Thus, many methods
basedonregularexpansions enabledtodeducelocal behaviorsofdynamical
systems.
Then, a geometric approach was initiated by Henri Poincar´e (1881-
1886) in his famous memoirs: Sur les courbes d´efinies par une ´equation
diff´erentielle whichrepresentthe foundationsofthequalitative2 orgeomet-
ric theory of differential equations. Continued during the XXth century
with the works of G. Valiron (1950), S. Lefschetz (1957), V. V. Nemytskii
& V. V. Stepanov (1960), N. Minorsky (1962-1967), F. Brauer & J. A.
Nohel (1969), ... it seems nevertheless that Differential Geometry had
been rarely used for dynamical systems3 study.
The aim of this book is to present a new approach which consists of
applying Differential Geometry to Dynamical Systems and is called Flow
Curvature Method. Thus, while considering the trajectory curve, integral
of any n-dimensional dynamical system, as a curve in Euclidean n-space,
the curvature of the trajectory curve, i.e. curvature of the flow may be an-
alytically computed. Then, the location of the points where the curvature
of the flow vanishes defines a manifold called: flow curvature manifold.
It will be stated that, since such a manifold is defined starting from the
1SeeJ.Molk(1910)andE.L.Ince(1926)p. 529foraHistoryofdifferentialequations.
2See C. Gilain (1977) for details about Poincar´e’s geometric approach of differential
equations.
3The oldest reference which has been indicated to me by Prof. C. Mira is: M. Haag
(1879).
vii
September11,2009 8:55 WorldScientificBook-9inx6in DGeometry
viii DifferentialGeometry Applied to Dynamical Systems
time derivatives of the velocity vector field and so, contains information
aboutthe dynamics ofthe system, its onlyknowledgeenables tofindagain
the main features of the dynamical system studied. These features may
be considered as the foundations of Dynamical Systems Theory. There are
six of them: differential equations, dynamical systems, invariant sets, local
bifurcations, slow-fast dynamical systems, integrability and to eachof these
concepts corresponds a chapter. Thus, this manuscript has been designed
inasymmetricmannerandconsistsofthreepartseachofthemcomprising
these six chapters.
The first part which may be regarded independently of the two others
is a detailed presentation of these six chapters from the analytic point of
view of Dynamical Systems Theory accompanied by references4, anecdotes
and many examples. Chapter 1, Introduction, is an historical presentation
of differential equations used to modelize natural phenomena. In Ch. 2,
Dynamical systems, state space and flow definitions, existence and unique-
ness and Liapounoff stability theorems are summarized and emphasized
with original references and significant examples as well as the notion of
Poincar´e index, the concept of limit cycle or strange attractor. Then, def-
initions of first integral and Lie derivative which will be extensively used
in this book are presented. Hamiltonian integrable systems and K.A.M.
theorem are also recalled. Chapter 3, Invariant sets, consists of definitions
of global (resp. local) invariant manifolds and stable manifold theorem for
a fixed point. Chapter 4 entitled local bifurcations is devoted to the Center
Manifold Theorem and Normal Form Theorem which are presented with
originalproofsandhighlightedthroughexamplesaswellaslocalbifurcations
suchassaddle-node, transcritical,pitchfork orHopf bifurcations. InCh. 5,
Slow-Fast Dynamical Systems,definitionsofsingularly perturbed dynamical
systems andslow-fast dynamical systems areproposed. Then,theso-called
Geometric Singular Perturbation Theory andthe conceptofslow invariant
manifold are recalled and emphasized with paradigmatic Van der Pol and
Chua systems. In Ch. 6, Integrability, integrability conditions, integrating
factor and multiplier of dynamical systems are reminded. Then, Darboux
Theory of Integrability ispresentedforthefirsttimewithitsoriginalproofs
and examples applied to dynamical systems.
4Historicalreferencestooriginalworksaremadebypage.
September11,2009 8:55 WorldScientificBook-9inx6in DGeometry
Preface ix
The second part is exactly symmetric5 to the first one since it involves
thesamechaptersandconceptsasthosepreviouslydefinedbutthenconsid-
ered from the Differential Geometry point of view. Chapter 7, Differential
Geometry, is a presentation of the concepts inherent to Differential Ge-
ometry such as curves, osculating plane and curvatures. By considering
the trajectory curves integral of any n-dimensional dynamical systems as
curves in Euclidean n-space which possess local metrics properties of cur-
vatures enables to define a manifold called: flow curvature manifold. Let’s
note that the point of view is completely different from the previous one
since it deals with curvature of trajectory curves insteadof vector field, i.e.
one substitutes a manifold to a differential equation, to a dynamical sys-
tem. Thus, the Flow Curvature Method is based on the idea that if it is
generally impossible to have a closed form of the trajectory curve it is still
possibletoanalyticallycomputeitscurvature sinceitonlyinvolvesitstime
derivatives.
Then, it will be stated in chapters 8, 9, 10 & 11 that all the results
foundinchapters2,3,4,5&6suchasfixed points stability, invariant sets,
centre manifold, normal forms, local bifurcations, slow invariant manifold
andintegrability ofdynamical systems maybefoundagainaccordingtothe
Flow Curvature Method, i.e. starting from the flow curvature manifold.
In Ch. 8, it is stated that the Flow Curvature Method enables to find
again stability theorems for fixed points of low-dimensional two and three
dynamical systems according to a theorem due to Henri Poincar´e.
In Ch. 9, concepts of global invariance and local invariance, which are
of great importance since all the proofs are based on them, are (re)defined
from Darboux invariance theorem. Then, it will be stated that flow cur-
vature manifold also enables to “detect” linear invariant manifolds of any
n-dimensional dynamical systems whichmaybeusedtobuildfirstintegrals
of these systems. For nonlinear invariant manifolds identity between flow
curvature manifold and the so-called extatic manifolds is also stated.
In Ch. 10, it is established that the Flow Curvature Method enables
to easily compute the coefficients of the centre manifold approximation of
anyn-dimensional dynamical systems accordingto global invariance ofthe
5Chaptersofparttwo(three)arechapters ofpartone(two)incrementedof6.