Chaotic Dynamics in Nonlinear Theory

118 Pages·2014·9.063 MB·English

by  Lakshmi Burra (auth.)

Lakshmi Burra
Chaotic
Dynamics in
Nonlinear
Theory
Chaotic Dynamics in Nonlinear Theory
Lakshmi Burra
Chaotic Dynamics
in Nonlinear Theory
123
Lakshmi Burra
Department of Mathematics
Jawaharlal Nehru Technological University
Hyderabad
India
ISBN 978-81-322-2091-6 ISBN 978-81-322-2092-3 (eBook)
DOI 10.1007/978-81-322-2092-3
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I dedicate this book to my (late) father
who had constantly encouraged me
to pursue the study of mathematics
and mother for her faith in my ability
to be a student forever.
Preface
There has been a renewed interest in the study of nonlinear dynamical systems
during the last two decades. This has been partly because of “Chaotic Behavior”
whichisassociatedwiththestudyofDynamicalSystems.Researchersfromvaried
disciplines have seen the power of the various techniques invented in this period,
both geometric and qualitative. With these new techniques stunning results have
beenachieved.Theycouldvisualizeextremelydifficultproblems,fromananalytic
point of view using geometric and qualitative methods. Many nonlinear systems
haverandomnessandchaosbuiltintothem.Thisfeaturehasbeenbroughtoutvery
well using geometric techniques.
ThestudyoftheQualitativeTheoryofDynamicalsystemsbeganwithPoincaré
in his study of Celestial Mechanics (1899). For Poincaré a study of the global
behaviourwasmoreimportantthantheirlocalbehaviour.Birkhoffadoptedmanyof
Poincaré’sviewsanddevelopedthetheoryfurtherinthebeginningofthetwentieth
century. Lately, many striking results have emerged mostly because of the pio-
neering work of Smale and the “Smale Horseshoes”.
This theory is with regard to a set S which is diffeomorphic to a rectangle in a
two-dimensional manifold M and a diffeomorphism f :S!M. Starting with the
first the iterate of f on S, conclusions about all iterates of f can be made. Of
particular interest is the conclusion of Smale about the existence of a compact
invariantsetinSwhichishomeomorphictoashiftonMsymbols.Inthisbookwe
willcallMthe“crossingnumber”.1Thedefinitionofthisnumberisafundamental
concept in our analysis of chaos present in dynamical systems.
1 Kennedy, J., Yorke, J.A.: Topological horseshoes. Trans. Amer. Math. Soc. 353, 2513–2530
(2001).
vii
viii Preface
However, Smale’s assumption of hyperbolicity of f on S is in practice slightly
difficult. It is our contention that even without the hyperbolicity assumption, the
dynamics remain just as rich as before. This is with respect to the existence of a
compactinvariantsetinSthatfactorsoverashiftonMsymbols,althoughthemapis
notone-to-one.Someauthors2refertotheseas“geometrichorseshoes”andtheterm
“topological horseshoes” is also often used. (The authors discuss situations where
stableandunstablemanifoldscrossbutnotnecessarilytransversally.Theycometo
theconclusionthatthereisageometrichorseshoeforsomeiterateofthemap.)
Hence, more general and less stringent definitions of a horseshoe have been
suggested so as to reproduce some geometrical features typical of the Smale
horseshoewhileleavingoutthehyperbolicityconditionsassociatedwithit.Thisled
to the study of the so-called topological horseshoes.
Thisistheapproachusedinthismonograph.Inthiswork,phase-planeanalysis,
combined with results from the theory of topological horseshoes and linked twist
maps are used to prove the existence of chaotic dynamics.
A small comment on the Linked Twist Maps (for short, LTMs): This is a
geometric configuration characterized by the alternation of two planar homeo-
morphisms(ordiffeomorphisms)whichtwisttwocircularannuliinsteadofasingle
map that expands the arcs along a domain homeomorphic to a rectangle, inter-
secting in two disjoint generalized rectangles A and B: Both the maps act in their
domainsothatatwisteffectisproduced.Thepresenceofchaos-likedynamicsfora
vertically driven planar pendulum, a pendulum of variable length, and other, more
general, related equations is proved.
This monograph emphasizes some mathematical aspects of the theory of
dynamicalsystems.Acertainlevelofmathematical sophisticationwouldbeuseful
throughoutthetext.Duringthelastfewyears,severalgoodtextbooksonnonlinear
dynamics have appeared for graduate students in applied mathematics. A majority
ofbooksarewithatheoreticalapproach.Severalpracticalissuesremainunclearfor
application of the theory to particular research problems. This book is oriented
towardsadvancedundergraduateorgraduatestudentsinmathematicsdoingapplied
research. It is also useful to professional researchers in physics, biology, engi-
neering, and economics who use dynamical systems as modeling tools in their
studies. Hence a moderate mathematical background in geometry, linear algebra,
analysis, differential equations, and dynamical systems is required. Wherever
necessary simple mathematical tools are used.
2 Burns, K., Weiss, H.: A geometric criterion for positive topological entropy. Comm. Math.
Phys.172,95–118(1995).
Preface ix
The book intends to provide the student (or researcher) with a solid basis in
dynamical systems theory. It would be also useful to researchers in Chaotic
Dynamical Systems as it offers a different route to Chaos. Graduate students could
profit from it.
The author is grateful to Professor Fabio Zanolin for his invaluable inputs and
encouragement.
Hyderabad, March 2014 Lakshmi Burra
Notations
We introduce some basic notations that will be used throughout this work.
N, Z; Q; R and C represent the sets of natural, integer, rational, real and
complex numbers, respectively. In particular N is the set of nonnegative integers.
ThesetsofnonnegativeandpositiverealnumberswillbeindicatedbyRþandRþ.
0
Accordingly, the first quadrant and the open first quadrant will be denoted by
ðRþÞ2 and ðRþÞ2; respectively.
0
WedenotebyIdtheidentitymapandbyId theidentityonthespaceW;when
W
we need to specify the domain. Given a function f; we indicate by domðfÞ its
domainandbyImðfÞitsrange.Byfj wemeantherestriction off toasubsetM
M
of its domain.
For a subset S of the topological space W, we denote by S, Int(S) and oS the
closure,theinteriorandtheboundaryofS;respectively.IfS(cid:2)W;weindicateby
WnSthecomplementofSinW:InthecasethatA; B (cid:2)W;wedenotebyAnBthe
relative complement of B in A: By jAj we mean the cardinality of the set A:
Acontinuousmapf :W !W iscalledaself-mapofthespaceW:Theiterates
of f are defined recursively with the convention f0 ¼Id ; f1 ¼f and
W
fn ¼f (cid:3)fn(cid:4)1; 8n(cid:5) 2. We say that w 2W is a fixed point for f if fðwÞ¼w:
We say that w 2 W is a periodic point for f if there exists an integer l(cid:5)1 such
that flðwÞ¼w:
The N-dimensional Euclidean space RN is endowed with the usual scalar
product h:; :i; norm k:k and distance distð:; :Þ. In the case of R; the norm k:;k
will be replaced with the absolute valuej:j. If X is a metric space different from
RN;wedenotebyd the distancedefined onit.Thedistance betweenM ; M (cid:2)
X 1 2
X is indicated by d ðM ;M Þ:¼inffd ðx; yÞ:x2 M ; y2 M g.
X 1 2 X 1 2
InanormedspaceðX;k:k Þ;wedenotebyBðx ; rÞandB½x ; r(cid:6)theopenand
X 0 0
closed balls centered in x 2X with radius r[0; i.e., Bðx ;rÞ:¼fx2 X :k
0 0
x(cid:4)x k \rg and B½x ; r(cid:6):¼fx2X :kx(cid:4)x k (cid:7)rg: For M (cid:2)X; we set
0 X 0 0 X
BðM;rÞ:¼fx2 X :9w2 M with kx(cid:4)wk \rg. The set B½M; r(cid:6) is defined
X
accordingly.
xi
xii Notations
Given a compact interval ½a;b(cid:6) (cid:2) R; we set L1ð½a;b(cid:6)Þ:¼ff :½a;b(cid:6)!R: fg
R
isLebesguemeasurableand bjfðtÞjdt\þ1;wheretheintegralistheLebesgue
a
integral.