Dynamical systems : examples of complex behaviour

203 Pages·2005·11.749 MB·English

by  Jürgen Jost

Universitext
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Jürgen Jost
Dynamical Systems
Examples of Complex Behaviour
With50Figures
123
JürgenJost
MaxPlanckInstitute
forMathematicsintheSciences
Inselstr.22
04103Leipzig
Germany
e-mail:[email protected]
MathematicsSubjectClassification(2000):37-XX,34Cxx,34Dxx
LibraryofCongressControlNumber:2005925883
ISBN-10 3-540-22908-6 SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-22908-7 SpringerBerlinHeidelbergNewYork
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Preface
Our aim is to introduce, explain, and discuss the fundamental problems,
ideas,concepts,results,andmethodsofthetheoryofdynamicalsystemsand
to show how they can be used in specific examples. We do not intend to give
a comprehensive overview of the present state of research in the theory of
dynamical systems, nor a detailed historical account of its development. We
try to explain the important results, often neglecting technical refinements
and, usually, we do not provide proofs.1
One of the basic questions in studying dynamical systems, i.e. systems that
evolve in time, is the construction of invariants that allow us to classify
qualitativetypesofdynamicalevolution,todistinguishbetweenqualitatively
differentdynamics,andtostudytransitionsbetweendifferenttypes.Itisalso
important to find out when a certain dynamic behavior is stable under small
perturbations, as well as to understand the various scenarios of instability.
Finally, an essential aspect of a dynamic evolution is the transformation of
some given initial state into some final or asymptotic state as time proceeds.
Thetemporalevolutionofadynamicalsystemmaybecontinuousordiscrete,
butitturnsoutthatmanyoftheconceptstobeintroducedareusefulineither
case.
We first introduce some general notions and exemplify them for systems of
ordinary differential equations. We classify some simple types of dynamical
behavior, like fixed points, and discuss the stability issue. We introduce the
notion of typical or generic behavior and study bifurcations, i.e. transitions
between different types of behavior. Attractors represent important asymp-
toticdynamicalinvariants.Anotheraspectisthedistinctionbetweendynam-
ically contracting and expanding directions and those that are neither. The
latter constitute the so-called center manifold and encode the dynamically
nontrivial part of the evolution.
1 All proofs can be readily found in the references provided in the bibliography.
Myconscienceasamathematiciandoesnotallowmetosuggestthatyoustudy
dynamicalsystemswithoutseeingtheproofsofthedifficultresults.Therefore,I
hope that you will consult at least some of these references.
VI Preface
The theory of Conley allows a detailed investigation of qualitative features
of dynamical systems in terms of discrete algebraic invariants. The theory is
presented in detail.
Kolmogorov introduced the fundamental asymptotic invariant for a dynam-
ical system, the entropy. The topological entropy is an important tool for
analyzing so-called chaotic behavior, and the method of symbolic dynamics
transforms a continuous scenario into a discrete one.
Themetricaspectsofentropyallowustodiscusstheissueofcomplexityand
the absence or presence of intrinsic scales of a dynamical process. The mea-
sure theoretic entropy establishes a fundamental connection with Shannon’s
conceptofinformation.Lyapunovexponentsofadynamicalsystemareeasier
to compute than the entropy, but can sometimes provide an alternative to
the latter for analyzing the relation between expansion and contraction of a
dynamical process. For that aspect, a rather complete theory exists under a
certain assumption of structural stability, called hyperbolicity.
Wealsodiscusscellular automata and themoregeneral Boolean networks as
examples of discrete dynamical systems.
Of course, this short survey cannot treat the field of dynamical systems ex-
haustively. The most important omission is perhaps the theory of Hamil-
tonian and integrable dynamical systems and its profound connections with
symplecticgeometry forwhich astandard presentation isavailable in several
recent textbooks.
While any individual mathematician who develops a new concept or demon-
strates an important result is rightly proud of her or his achievement, in
general we mathematicians are inclined to consider important mathematical
theories and results to be the common property of all mathematicians, if not
of all of mankind. Perhapsfor that reason,wearenotalways very diligent in
tracing the history of individual contributions, and this may serve as a faint
excuse for not always carefully searching and listing all individual references
in the present survey.
The present book emerged from series of lectures given at Leipzig Univer-
sity and the Santa Fe Institute for the Sciences of Complexity to rather
diverse audiences. I thank them all for their interest, their inspiring ques-
tions, and their constructive criticism. I am grateful to Antje Vandenberg
and Pengcheng Zhao for technical help.
Leipzig, July 2004 Ju¨rgen Jost
Table of Contents
1 Introduction.............................................. 1
2 Stability of dynamical systems, bifurcations, and generic
properties ................................................ 7
2.1 Some general notions ................................... 7
2.2 Autonomous systems of ODEs ........................... 9
2.3 Examples: Bifurcation depending on a parameter λ∈R ..... 15
2.4 Chaos in differential and difference equations. The concept
of an attractor ......................................... 22
2.5 Interaction, or the interplay between concentration or
reaction and diffusion ................................... 36
2.6 Discrete and continuous systems. The Poincar´e return map . 41
2.7 Stability and bifurcations; generic properties ............... 42
2.8 The Hopf bifurcation ................................... 46
2.9 Lotka-Volterra equations ................................ 48
2.10 Stable, unstable, and center manifolds..................... 52
3 Discrete invariants of dynamical systems ................. 61
3.1 The topology of graphs.................................. 61
3.2 Floer homology ........................................ 62
3.3 Conley theory: examples and results ...................... 70
3.4 Cohomological Conley index ............................. 77
3.5 Homotopical invariants.................................. 80
3.6 Continuation properties of the Conley index ............... 92
3.7 The discrete Conley index ............................... 93
4 Entropy and topological aspects of dynamical systems .... 99
4.1 The entropy of a process as an asymptotic quantity ........ 99
4.2 Positive entropy and chaos............................... 103
4.3 Symbolic dynamics ..................................... 108
5 Entropy and metric aspects of dynamical systems ........ 111
5.1 The metric approach to topological entropy................ 111
5.2 Complexity and intrinsic scales........................... 114
VIII Table of Contents
6 Entropy and measure theoretic aspects of dynamical
systems................................................... 119
6.1 Probability spaces and measure preserving maps............ 119
6.2 Ergodicity ............................................ 121
6.3 Entropy and information ................................ 126
6.4 Invariant measures ..................................... 138
6.5 Stochastic processes .................................... 142
6.6 Stochastic bifurcations .................................. 149
7 Smooth dynamical systems ............................... 153
7.1 Lyapunov exponents ................................... 153
7.2 Hyperbolicity .......................................... 156
7.3 Information loss........................................ 165
8 Cellular automata and Boolean networks as examples of
discrete dynamical systems ............................... 169
8.1 Cellular automata ...................................... 169
8.2 Boolean networks....................................... 175
References.................................................... 181
Index......................................................... 185
1 Introduction
A dynamical system is a system that evolves in time through the iterated
application of an underlying dynamical rule. That transition rule describes
the change of the actual state in terms of itself and possibly also previous
states. The dependence of the state transitions on the states of the system
itselfmeansthatthedynamicsisrecursive.Inparticular,adynamicalsystem
isnotasimpleinput-outputtransformation,buttheactualstatesdependon
thesystem’sownhistory.Infact,aninputneednotevenbegiventothesys-
tem continuously, but rather it may be entirely sufficient if the input is only
given as an initial state and the system is then allowed to evolve according
only to its internal dynamical rule. This will represent the typical paradigm
of a dynamical system for us.
The transition rules for dynamical system will typically depend on certain
parameters. Investigating the qualitative nature of this dependence consti-
tutes an important aspect of the theory of dynamical systems.
The application of the transition rule can happen either at discrete time
steps, with the time parameter denoted by t taking values in the (positive)
integers Z(N), or infinitesimally with continuous underlying time taking val-
ues in R or R+ as in differential equations. If time is continuous, we assume
that the transition rules lead to an evolution that is continuous w.r.t. some
appropriatetopology.Thequalitativedynamicalbehaviorofthesystemmay,
however,changeduetophasetransitionsorbifurcations.Or,fromadifferent
perspective, there may be a transient and an asymptotic dynamical regime.
If time is discrete, we need to select a class of permissible state transitions
that preserve the identity of the system.
The underlying rule may be rather simple, but its iterated application may
still create an asymptotic behavior as time goes to infinity that is not so
easy to predict and analyze from the dynamical rule itself. In fact, in many,
and perhaps typical, cases, there is no simpler way to obtain or predict the
final result than to let the dynamical system run itself. Thus, in hindsight,
the attitude of Laplace seems rather naive that, given the complete initial
conditions, future states of the world could be computed. The point is that
the dynamical evolution may be so complicated that, for that prediction, a
computer that is essentially as powerful as the world itself would be needed,
and this computation would then take about as long as the corresponding