m1002: enrichment revised - uwa...published in 1952 | nobel prize for physiology and medicine in...

M1002: Enrichment

revised

Michael Small

School of Mathematics and StatisticsThe University of Western Australia

September 25, 2015

Michael Small (UWA) M1002: Enrichment revised September 25, 2015 1 / 31

Definition: Dynamical System

Dynamical:

characterized by constant change, activity, or progress

relating to forces producing motion

System:

a set of connected things or parts forming a complex whole

a set or principles or procedures according to which something isdone

In mathematics:

a set of difference or differential equations for which time t is anindependent variable

a mathematical model of (part of almost all of) reality


A model: Example 1

An example of a (sequence of) models:

The Hodgkin-Huxley model of action potential initiation andpropagation

Uses the squid giant axon as a model for a broad range of neurons

Experimental observation of dynamical behavior of this cell topropose a mathematical model of electrophysiological dynamics

Published in 1952 — Nobel Prize for Physiology and Medicine in1963 (Hodgkin, A., and Huxley, A. (1952): J. Physiol.117:500–544)


The Hodgkin-Huxley model

Let GNa, GK, and GL denote the sodium, potassium and leakagecurrent across a cell boundary. The cell potential V − Veq is theelectrical potential deviation from rest, Cm is the cell membranecapacitance and αm,n,h and βm,n,h are functions of V , peculiar to thesystem. The time course of the voltage V (t) is given by the followingsystem of differential equations

CMdV

dt= −gNam

3h(V − VNa)− gKn4(V − VK)− gL(V − VL)

dm

dt= αm(V )(1−m)− βm(V )m

dn

dt= αn(V )(1− n)− βn(V )n

dh

dt= αh(V )(1− h)− βh(V )h


... and ...

αm = 0.125− V

exp(25−V10

)− 1

βm = 4 exp

(−V18

)αh = 0.07 exp

(−V20

)βh =

1

exp(30−V10

)+ 1

αn = 0.0110− V

exp(10−V10

)− 1

βn = 0.125 exp

(−V80

)but, this model is a little complicated. So, ...


The FitzHugh-Nagumo model

.. reduce the previous model to the model of minimum complexity —and still capable of doing something interesting:

dv

dt= f(v)− w + Ia

dw

dt= bv − γw

f(v) = v(a− v)(v − 1)

where 0 < a < 1, 0 < b, 0 < γ. The variable v behaves “like” themembrane potential (the voltage V in the previous model), and wtakes the place of the internal variables m, n and h.


Eg. 2: A model to beat roulette

Henri Poincare asked (Science and Method, 1896 (French), 1914(English)) what is the probability of a roulette wheel landing onblack, and, what is the effect of small initial perturbations to thewheels motion

Until the ball starts bouncing, Newtonian mechanics and highschool calculus provide a good model

We show (M. Small and C.K. Tse (2012), Chaos) that this modelis good enough to make money — but it doesn’t account for theirregular bouncing motion


Definition: Model

A model is an abstraction of reality that is

simple enough to be useful, and

complex enough to be interesting.


Definition: Chaos

“Qualities” of Chaos (paraphrasing James Yorke — who was first touse the term Chaos in this context):

apparently irregular, but non-random — described bydeterministic equations

sensitive dependence on initial conditions

bounded, deterministic and aperiodic

deterministic, but not predictable

topological mixing — the images of open sets eventually intersect

periodic orbits are dense

may give rise to invariant sets with complex (fractal) boundaries— strange attractors


Eg. 1: Bernoulli map

Simplest example of chaos (and specifically sensitive dependence oninitial conditions:

xn+1 = f(xn) =

{2xn 0 ≤ xn < 0.5

2xn − 1 0.5 ≤ xn ≤ 1

Hence:

This map is bounded by (and onto) [0, 1]

f ′(x) = 2 a.e. and so local separation increases as 2n.

No x0 ∈ R \Q is periodic =⇒ chaos a.e.

Every x0 ∈ Q is periodic =⇒ periodic orbits are dense

Note, that this means that while almost every random initial conditionleads to chaos, every initial condition chosen by a computer is periodic(with period 1).


Eg. 2: Ikeda map

Define the discrete time map of a complex variable

zn+1 = A+Bzn exp[i(|z2n|+ C

)]where A, B, and C are real constants and zn ∈ C models the (complex)electric field in an optical ring cavity laser. Now forget the physics andstick with some numbers (zn = xn + iyn):

xn+1 = 1 + µ(xn cos θn − yn sin θn)

yn+1 = µ(xn sin θn + yn cos θn)

θn = 0.4− 0.6

1 + x2n + y2n

Exhibits sensitive dependence on initial conditions, mixing and astrange attractor...


Eg. 2: Ikeda map iterates

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

xn

y n

(xn, yn)Michael Small (UWA) M1002: Enrichment revised September 25, 2015 12 / 31


0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

xn

y n

(xn+5, yn+5)Michael Small (UWA) M1002: Enrichment revised September 25, 2015 13 / 31


0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

xn

y n



0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

xn

y n

(xn+35, yn + 35)Michael Small (UWA) M1002: Enrichment revised September 25, 2015 19 / 31


0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

xn

y n


Eg. 3: Double PendulumLet m1 and m2 be two masses at location (x1, y1) and (x2, y2)constrained by rigid arms of length `1 and `2. Hence

`21 = x21 + y21

`22 = (x2 − x1)2 + (y2 − y1)2

The force-balance equations (F = ma) under the effect of gravity ggives:

m1x1 =x1λ1`1− (x2 − x1)λ2

`2

m1y1 =y1λ1`1− (y2 − y1)λ2

`2−m1g

m2x2 =(x2 − x1)λ2

`2

m2y2 =(y2 − y1)λ2

`2−m2g


Eg. 4: Weather forecasting

Lorenz proposed a three dimension set of ordinary differentialequations to model convection forced via a temperature gradient:

dx

dt= −σx+ σy

dy

dt= −xz + rx− y

dz

dt= xy − bz

where (it turned out), for σ = 10, r = 28, and b = 83 , the system is

bounded and aperiodic. The variable x represents intensity ofconvective motion, y temperature gradient, and z is the degree ofnonlinearity in the temperature profile.



1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

−10

0

10

x(t)

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−20

0

20

y(t)

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

10203040

z(t)


Example

Line, square and cube (d = 1, 2 and 3).

Example

Middle-thirds Cantor dust (d = log 2log 3 ≈ 0.6309 . . .).

Example

Koch curve (d = log 4log 3 ≈ 1.2619 . . .).


Definition

Let the correlation sum θ(ε,N) be defined as

θ(ε,N) =1

N(N − 1)

∑i,j 6=i

H(‖vi − vj‖ < ε) (1)

(as before, H(·) is 1 if the condition in parenthesis is true, and zerootherwise) and then define the correlation dimension d2 as

d2 = limε→0

limN→∞

log θ(ε,N)

log ε(2)

Here the inner limit is just wishful thinking to invoke the impliedcorrelation integral which the correlation sum approximates —

θ → Θ(ε,N) :=

∫ ∫H(‖x− y‖ < ε)dµ(x)dµ(y)


Grassberger-Procaccia AlgorithmGrassberger-Procaccia (1983)

Direct implementation of (1) to compute (2) near the limit ε→ 0constitutes the so-called Grassberger-Procaccia algorithm. Theproblem with this algorithm is that it assumes that a scaling regionexists and then makes a best guess at what it should be. Schreiber andKantz (1997) discuss various palliative measures to overcome this.Fundamentally different estimators of d2 for systems which includenoise and may have scale dependent structure have been propose since.We note, in passing, that other dimensions have also been described —most common of these is the box-counting dimension d0, the Hausdorffdimension, and the information dimension d1. These deviate from thecorrelation dimension in the choice of metric used to measure closenessin (1).In what follows we describe pragmatic alternatives to the correlationdimension that are better suited to application to finite data sets.


This scheme for estimating dimension has (at least) the followingproblems

In the G-P implementation, the scaling region is first assumed toexist — then foundScaling in the correlation integral is bounded at large scales by thesize of the attractor (the curve flattens)At small scales, even without noise (which is another problem),quantisation screws things upAt small scale the distribution of inter point distances is biasedbecause the points (coming from a trajectory) are not independent(ala the triangle inequality)There are less points for small scales, and many points at largescales, this introduces a statistical correlation as the sample sizefrom which the correlation integral is estimated varies with scaleAs a result of all of the above, the scaling region is always over afinite range of length scales — this represents deterministicstructural properties of the attractor rather than the asymptoticEstimates of variance (error) in the estimate of correlationdimension are not forthcoming


Books

Reference

S.H. Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley,1994.

D.K. Arrowsmith and C.M. Place, Dynamical Systems, Chapmanand Hall, 1992.

K.T. Alligood, T. Sauer and J.A. Yorke, Chaos An introduction todynamical Systems, Springer, 1996.

Popular accounts

L.A. Smith, Chaos : A Very Short Introduction, Oxford UniversityPress, 2007.

J. Gleick, Chaos: Making a new Science, Penguin, 1988.

I. Stewart, Does God Play Dice?: The Mathematics of Chaos,Blackwell Publishers, 1990.


m1002: enrichment revised - uwa...published in 1952 | nobel prize for physiology and medicine in...

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