Nonlinear Dynamics in Complex Systems: Theory and Applications for the Life-, Neuro- and Natural Sciences

235 Pages·2012·10.174 MB·English

by  Armin Fuchs

Armin Fuchs
Nonlinear Dynamics
in Complex Systems
Theory and Applications for the Life-,
Neuro- and Natural Sciences
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ArminFuchs
CenterforComplexSystems&BrainSciences
FloridaAtlanticUniversity
BocaRaton,FL
USA
ISBN978-3-642-33551-8 e-ISBN978-3-642-33552-5
DOI10.1007/978-3-642-33552-5
SpringerHeidelbergNewYorkDordrechtLondon
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To my parents
Foreword
As attested to by the numerous anddiverse volumes in Springer Complexity
themathematicalandcomputationaltoolsofnonlineardynamicsareprovid-
ing major breakthroughs in our understanding of complex systems in many
fieldsofinquiry.Thisishardlysurprising,sincetheuniverseitselfisnotstatic
nor, for that matter, is the brain we use to understand it. What it means,
however,is that in orderto understandourselvesandthe worldwe live in–in
all its manifestations–itis necessaryto transcend,if noterasethe traditional
boundaries between the disciplines.
Now imagine a situation, rare25 yearsago,where graduate students from
very different disciplines are thrust together by the shared belief that some-
how the systems and problems confronting science and technology are com-
plex,nonlinearanddynamical,subjecttovariability,interactions,emergence
andchangeonmanydifferentlevelsofdescription.Thesearejustwords.But
how do you fill them with content? How do you go about the challenge of
understanding complex systems? Given that the ”stuff” of these systems is
typically very different, and the range of scientific problems so broad and
apparentlydiffuse, whatconcepts,methods andtoolsareyougoingto useto
understand how these systems behave? And how on earth is it going to be
possible for young people from vastly different backgrounds to communicate
with each other? And in what language?
These were some of the problems confronting us when we started The
Center for Complex Systems and Brain Sciences at Florida Atlantic Univer-
sity (FAU) many years ago. A Center, by the way, that would not just have
a research mission or be a convenient way for people from different disci-
plines to get together and perhaps collaborate. Rather, a Center that would
also have a pedagogical mission–to create a new breed of interdisciplinary
scientist, one who is theoretically sophisticated and computer literate, com-
fortable with models and modeling–comfortablealso with handling data and
thus equally adeptinexperimental laboratoryandnaturalobservationalset-
tings. Such was the goal of a NIH-supported National Training Program in
Complex Systems and Brain Sciences at FAU; such is what we thought, at
VIII Foreword
the very least, a PhD degree in Complex Systems and BrainSciences should
consist of, regardless of specific implementation or field of inquiry.
But how to bring this about? One of the key solutions is to be found in
Armin Fuchs’s book: it is based on the idea that students of complex sys-
tems, many with little or no mathematical background to speak of, should
become intimate with the mathematical concepts and techniques of nonlin-
ear dynamical systems. Fuchs himself comes originallyfroma backgroundin
theoretical physics, especially synergetics, Hermann Haken’s theory of how
patterns are formed in nature’s nonequilibrium systems that are open to en-
ergyand matter exchangeswith the environment.Fuchs himself is anexpert
in the analysis and modeling of brain activity, pattern recognition by com-
puters and the coordinated movements of human beings, having performed
fundamental research in all these areas. As revealed in this book, with the
authoritative voice of a genuine practitioner, Fuchs is a master teacher of
how to handle complex dynamical systems. The reason is that he has de-
voted himself over a period of many years to communicating the theory of
dynamical systems to students some of whom possess only a minimum, even
a self-expressed fear of mathematics. What I find beautiful in this book is
its clarity, the clear definition of terms, every step explainedsimply and sys-
tematically. When an equation is introduced we know where it comes from
andhowitmaybe applied. We knowwhatmeasuresto calculateandhowto
calculate them. And all along, superb, clean computer graphics that enable
visualization of solutions all the while enhancing intuition. Both the art and
the science of modeling are portrayed in this book. The power of dynamical
systemsisseennotjustasadescriptivetool.Fuchsshows,forexampleinthe
case of the HKB model of movement coordination, that it can be predictive
too, promoting the interplay of theory and experiment. This book is a mas-
terful treatment one might even say a gift, to the interdisciplinary scientist
ofthefuture.Itfillsthegapbetweenhighlymathematicaltreatmentslimited
to experts and the non-technical picturesque treatments that populate the
shelvesofbookstoresandamazondotcom.What’smoreisthatthisbook–its
contents laid out systematically in a manner that can be grasped by the cu-
riousandtheeager–hasbeenshowntowork.Ithasbeentriedandtestedon
first yeargraduatestudents (numbering near the hundreds)in the PhD Pro-
gramin Complex Systems and BrainSciences many of whom have put what
they have learnedinto practice. Now, for the first time, the book is available
to everyone. For that we can all be grateful to its author, an extraordinary
scientist and teacher, Armin Fuchs.
Boca Raton, Florida J.A. Scott Kelso
September, 2012
Preface
Duringthelastdecadesthetheoryofnonlineardynamicalsystemshasfound
its way into fields such as biology, psychology, the neuro- and other life- and
natural sciences, which have been viewed as ‘soft’ or qualitative branches of
science.Withthisongoingdevelopmentmanyresearchersandstudentsdonot
have a strong enough background in mathematics to apply the quantitative
methods and powerful tools provided by dynamical systems theory to their
own research or even follow the literature to a satisfactory depth. I hope
this text that provides the basics and foundations of nonlinear dynamics
will be helpful for those who need an introduction to the topics or want to
deepen their understanding of the analysis and modeling procedures applied
to complex systems. A variety of books introducing dynamical systems exist
already, but most of them are either aimed at readers who have a deeper
understandingofadvancedmethodsofcalculusanddifferentialequations,or
they are non-technical introductions where the flashy topics like fractals or
deterministic chaos are overemphasized.
Most scientists and graduate students have at least some elementary
knowledgeofcalculusandlinearalgebra,whichisallthatisneededtounder-
standandtosomeextendevenapplydynamicalsystemstheory.Itistheaim
of this book to introduce mathematical tools, show how these tools work in
certain applications and provide some of the necessary mathematical back-
ground to those who are not using formulas and equations on a daily basis.
With this said, the present book is full of equations and may scare off many
readers who just briefly skim through it, but I hope that closer inspection
revealsthatthemanyequationsactuallymakethingseasierbyshowingevery
step in all the calculations.
It is one of my main intentions here to not only present the problems
andsolutions but to provideeachstepalongthe waytogether with the basic
rulesthathavebeenapplied.Mostpeoplewhodonotusemathematicaltools
regularlyunderestimatetheirownknowledgeandgiveupifthegapsfromone
step to the next are too big and too many, and they assume that there must
be some sortof higher math magic at work. I hope, by explicitly showing all
X Preface
details, to convince the readerthat this impressionis not only false but that
themathematicsneededtofollowthroughallthesestepsisquitelimitedand
known by most people doing research in science.
This book consists of three parts: Part I is an introduction to the theory
ofdynamicalsystems, whichmakesintense use ofequationsanddiagramsto
create a profound exploration of the properties and laws of complex nonlin-
ear systems. Phase space plots, potential functions and bifurcation diagrams
illuminate the complexity inherent in the dynamics of nonlinear systems. In
addition to an extensive discussion of the continuous case described by ordi-
narydifferentialequations,wealsohavealookatdiscretemapsanditerative
systems in space.
PartIIshowshownonlineardynamicalsystemstheoryisappliedtothelife-
and neurosciences. We discuss in detail some of the most successful quanti-
tativemodelsincluding theHaken-Kelso-Bunzmodel ofmovementcoordina-
tion and some of its extensions. The Haken-Zwanzig model serves as a gate
to self-organization and synergetics, and we examine the famous neuronal
models (i.e. Hodgkin-Huxley, Fitzhugh-Nagumo and Hindmarsh-Rose) and
their dynamical properties as important applications in theoretical biology.
PartIII gives a brief overviewof some ofthe mathematical tools we make
useofextensivelyinPartsIandII.Itisintendedasarefresherandreference
source on mathematical topics that most scientists and graduate students
should be familiar with to a certain degree but may have put some rust on
overtime. Finally, in the appendix I give a detailed derivationof the Haken-
Kelso-Bunzmodel fromthe levelofcouplednonlinearoscillatorsandpresent
a brief introduction to numerical procedures together with some example
code in Matlab.
I am most thankful to Senior Editor Dr. Thomas Ditzinger at Springer
Verlag for valuable advice, guidance, patience and availability throughout
the writing ofthis book. My deepestgratitude goestomy student Stephanie
Lewkowitz for her thorough proof-reading and the many corrections to my
language to make it proper English.
Boca Raton, Florida Armin Fuchs
July 2012
Contents
Part I: Nonlinear Dynamical Systems
1 Introduction............................................. 3
1.1 Complex Systems...................................... 3
1.2 Differential Equations – The Basics ...................... 6
1.3 First Steps ........................................... 7
1.4 Terminology – Dynamical Systems Jargon ................ 9
1.5 A First Encounter with Phase Space ..................... 11
1.6 Limitations of Linear Systems........................... 12
2 One-dimensional Systems................................ 15
2.1 Linear Stability Analysis ............................... 16
2.1.1 A Quantitative Measure of Stability ............... 17
2.1.2 Bistability, Multistability and Qualitative Change ... 18
2.1.3 Exotic Cases: When Linear Stability Analysis
Breaks Down ................................... 19
2.2 Potential Functions .................................... 21
2.3 Bifurcation Types ..................................... 26
2.3.1 Saddle-Node Bifurcation ......................... 27
2.3.2 Transcritical Bifurcation ......................... 27
2.3.3 Supercritical Pitchfork Bifurcation................. 28
2.3.4 Subcritical Pitchfork Bifurcation .................. 29
2.3.5 Systems with Hysteresis.......................... 30
2.4 Periodic One-Dimensional Systems ...................... 32
2.5 Problems for Chapter 2 ................................ 34
3 Two-Dimensional Systems ............................... 37
3.1 The Harmonic Oscillator ............................... 37
3.1.1 Second Order Systems – A First Look.............. 38
3.1.2 Second Order Systems – A Second Look............ 39
3.1.3 Damped Harmonic Oscillator ..................... 40
XII Contents
3.2 Classification of Two-Dimensional Linear Systems ......... 42
3.3 Nonlinear Systems: Linear Stability Analysis in Two
Dimensions ........................................... 46
3.4 Limit Cycles .......................................... 52
3.5 Hopf Bifurcation ...................................... 53
3.6 Potential Functions in Two-Dimensional Systems .......... 60
3.7 Generalization of Potentials: Lyapunov Functions.......... 64
3.8 Nonlinear Oscillators................................... 66
3.8.1 Van-der-Pol Oscillator: N(x,x˙)=x2x˙ ............. 67
3.8.2 Rayleigh Oscillator: N(x,x˙)=x˙3 ................. 68
3.8.3 Hybrid Oscillator: N(x,x˙)={x2x˙,x˙3}............. 69
3.9 Poincar´e-BendixonTheorem ............................ 71
3.10 Problems for Chapter 3 ................................ 71
4 Higher-Dimensional Systems and Chaos ................. 73
4.1 Periodicity and Quasi-periodicity ........................ 73
4.2 Lorenz System ........................................ 74
4.3 R¨ossler System........................................ 80
4.4 Fractal Dimension ..................................... 82
4.5 Lyapunov Exponents................................... 84
4.6 Time Series Analysis................................... 87
5 Discrete Maps and Iterations in Space................... 89
5.1 Logistic Map.......................................... 89
5.1.1 Geometric Approach............................. 92
5.2 Lyapunov Exponents................................... 93
5.3 H´enon Map........................................... 95
5.4 Feigenbaum Constants ................................. 97
5.5 Using Discrete Maps to Understand Continuous Systems ... 98
5.6 Iterated Function Systems .............................. 100
5.7 Mandelbrot Set ....................................... 103
6 Stochastic Systems ...................................... 105
6.1 Brownian Motion...................................... 106
6.2 Features of Noise ...................................... 108
6.2.1 How to Describe Kicks ........................... 108
6.2.2 Averages ....................................... 110
6.2.3 Distributions ................................... 111
6.2.4 Properties of the Gaussian Distribution ............ 112
6.2.5 Correlations .................................... 113
6.2.6 Colors of Noise.................................. 120
6.3 Langevin and Fokker-PlanckEquation ................... 121
6.4 Reconstruction of Drift and Diffusion Term ............... 123
6.5 Multiplicative Noise ................................... 125
6.6 Mean First PassageTime............................... 126
Contents XIII
6.7 Fingerprints of Stochasticity ............................ 127
6.8 Problems for Chapter 6 ................................ 128
Part II: Model Systems
7 Haken-Kelso-Bunz (HKB) Model........................ 133
7.1 Basic Law of Coordination: Relative Phase ............... 133
7.2 Stability: Perturbations and Fluctuations................. 137
7.3 Oscillator Level ....................................... 139
7.4 Symmetry Breaking Through the Components ............ 141
8 Self-organization and Synergetics ........................ 147
8.1 Haken-Zwanzig System................................. 148
8.2 Lorenz Revisited ...................................... 150
8.3 Formalism of Synergetics ............................... 155
9 Neuronal Models ........................................ 159
9.1 Hodgkin-Huxley Model................................. 160
9.2 Fitzhugh-Nagumo Model ............................... 166
9.3 Hindmarsh-Rose Model ................................ 171
Part III: Mathematical Basics
10 Mathematical Basics..................................... 181
10.1 Complex Numbers..................................... 181
10.2 Linear Systems of Equations ............................ 185
10.3 Eigenvalues and Eigenvectors ........................... 187
10.4 Taylor Series.......................................... 193
10.5 Delta and Sigmoid Functions............................ 195
A The Coupled HKB System .............................. 199
A.1 The Oscillators........................................ 199
A.2 The Coupling ......................................... 200
A.3 The Coupled System................................... 202
B Numerical Procedures and Computer Simulations ....... 205
B.1 Integration Schemes for Ordinary Differential Equations.... 205
C Solutions ................................................ 213
C.1 Chapter 2 ............................................ 213
C.2 Chapter 3 ............................................ 218
C.3 Chapter 6 ............................................ 223
References................................................... 229
Index........................................................ 231