PY2T20: CHAOS and COMPLEXITY (12 Lectures) Stefan Hutzler

Home , Feigenbaum constants, Helge von Koch, Mitchell Feigenbaum

PY2T20: CHAOS AND COMPLEXITY (12 lectures) Stefan Hutzler

lecture notes at: http://www.tcd.ie/Physics/Foams/Lecture Notes/PY2T20ChaosAndComplexity

February 7, 2014 Contents

1 A glossary 1

2 Examples of non-linear and chaotic behaviour 3 2.1 Population dynamics ...... 3 2.2 Non-linear electrical circuit ...... 6 2.3 Lorenz model of atmospheric convection ...... 7 2.4 Summary of observations ...... 7

3 Universal properties and self-similarity 8 3.1 Feigenbaum constants ...... 8 3.2 Measuring chaos ...... 9 3.3 Universality of chaos ...... 10

4 Determinism 12

5 Dynamics in phase space: Motion of the pendulum 13 5.1 Equation of motion for damped driven pendulum ...... 13 5.2 Phase space ...... 15 5.3 Damped driven pendulum: period doubling and chaos ...... 17 5.4 Properties of trajectories ...... 17

6 Some theory of chaotic dynamics 19 6.1 Long-term behaviour of dissipative systems ...... 19 6.2 Stability of ﬁxed points ...... 20 6.2.1 one dimension ...... 20 6.2.2 two dimensions ...... 21 6.2.3 three dimensions ...... 23 6.3 Analysis of limit cycles ...... 23

i 6.4 Examples for damped driven pendulum ...... 23 6.5 Quasi-periodicity ...... 24 6.6 Different routes to chaos ...... 24

7 Iterated maps 26 7.1 Motivation ...... 26 7.2 Bernoulli shift ...... 26

8 Fractals 29 8.1 A mathematical monster: the Koch curve ...... 29 8.2 Fractal dimensions ...... 30 8.3 Examples of fractals ...... 30

9 Strange attractors 32 9.1 Deﬁnition ...... 32 9.2 Baker’s transformation ...... 33 9.3 Stretching and folding for the logistic map ...... 33

10 Advanced topics 35 10.1 Hamiltonian systems (motivation) ...... 35 10.2 Quantum chaos (motivation) ...... 36

11 Some examples of complex behavior 38

12 Cellular Automata 39 12.1 Concept ...... 39 12.2 Example in one dimension ...... 39 12.3 A two-dimensional CA: The game of life ...... 40

13 Power-laws and self-organised criticality 41 13.1 Power-laws ...... 41 13.2 A model for Sandpile dynamics ...... 42 13.3 Earthquake model ...... 42

14 Pedestrian Dynamics 44 14.1 Observations ...... 44 14.2 Model of pedestrian motion ...... 45 14.3 Results of computer simulations ...... 46

ii 14.4 Optimisation of pedestrian facilities ...... 47 14.5 Design of new pedestrian facilities ...... 47

A References 48

iii Chapter 1

A glossary

φιλos [philos]: friendly, loving; σoφια [sophia] knowledge ⇒ philosophy: love of wisdom, later: study of “reality” and human nature

Aristoteles (384-322 BC) treatise on movement of natural/material bodies: φνσικα [physica] derived from adjective for “natural”, things

early names for physics: “natural science”, “natural philosophy”; nature : natural motion

TCD School of Physics: Erasmus Smith’s Chair of Natural and Experimental Phi- losophy (1724), currently vacant

χαos [chaos]:

• disordered formless matter supposed to have existed before the ordered universe (as in Greek and Babylonian mythology, Old Testament “without form, void”)

• complete disorder, utter confusion

• (Math.) Stochastic behaviour occurring in a deterministic system (Royal Society 1986)

stochastic (stochastikos): “skillful in aiming” at a target ’stokhos’, using laws of chance for personal beneﬁt determinism: every event is result of antecedent causes (⇒ predictability)

1 in the context of this course: chaos ≡ deterministic chaos: “lawless behaviour governed entirely by law” (Ian Stewart)

complexity: Greek [plexus]: braided, [com-]: together ⇒ “braided together”

simple: once folded

complex system: organised system, at “edge of chaos”, this may lead to pattern formation some comments on (non-)linearity

• many equations in physics are linear ⇒ the sum of two solutions of an equation is again a solution (superposition)

• this simpliﬁes the mathematics, but might not necessarily describe the actual physics (see treatment of pendulum in chapter 5: sin x ' x gives qualitatively wrong results for large amplitudes)

• now: very often non-linearity is at the centre of scientiﬁc problems; one generally needs computers for solutions

• for non-linear systems small changes in a parameter may lead to sudden dramatic changes in both qualitative and quantitative behaviour of system, ie. change from periodicity to aperiodicity

Stanislaw Marcin Ulam (Manhattan project): “Calling the subject non-linear dynamics is like calling zoology ‘non-elephant studies’ ”

2 Chapter 2

Examples of non-linear and chaotic behaviour

2.1 Population dynamics

The model assumptions: population of species (green-ﬂies) whose individuals are born and die in the same year (“breed in summer and leave eggs that hatch in spring”)

discrete times ⇒ difference equations

simplest model (geometric growth):

Pn+1 = aPn (2.1)

Pn: population in generation n, a: growth constant, if a > 1: population explosion, a < 1: extinction more realistic growth model:

Pn+1 = Pn(a − bPn) (2.2) where b is a constant. This models the fact that population is bounded by a ﬁnite carrying capacity of its environment, i.e. overcrowding, diseases, lack of food etc.

let’s introduce the rescaled variable xn: a P = x (2.3) n b n

3 thus

xn+1 = axn(1 − xn) ≡ fa(x) (2.4) This is the logistic map. (P.F. Verhulst, 1845; french “logis” means “house, ac- commodation”) Note that it is a fully deterministic equation.

comments:

• xn > 1 ⇒ xn+1 < 0 (unphysical) ⇒ restrict xn[0, 1] and choose 0 ≤ a ≤ 4

• it is a map: range [0,1] is mapped into itself

• it is an iterative function: population in N years is found by iterating/repeating the calculation of eqn. 2.4.

• the sequence of x values is called the trajectory or orbit of the map

∗ ∗ • ﬁxed points are deﬁned by xa = fa(xa)

∗ ∗ here: there are two ﬁxed points, xa = 0 (trivial) and xa = 1 − 1/a, the latter ﬁxed point is stable only for 1 < a < 3.

Discussion of the logistic map study map either by direct computation/iteration or graphically

[ﬁgure: Hilborn 1.12 + applet http://brain.cc.kogakuin.ac.jp/ kanamaru/Chaos/e/Logits/];

ﬁxed points are the intersections of the parabola fa(x) with y = x

example: a=0.6 ﬁgure: evolution of a population here: 0 is the attractor, interval [0:1] is the basin of attraction; effect: population dies out for any size of initial population

attractor: set of points to which trajectories tend as number of iterations goes to inﬁnity

example a=2: ∗ ∗ ∗ 2 ﬁxed points, x1 = 0, x2 = 1 − 1/a = 0.5, here x2 = 0.5 is an attractor

4 example a = 3.1 [Hilborn, table 1.1]

x0 = 0.250 : observation of period doubling (bifurcation = splitting into two parts): population is high in one year and low in the following year two-point attractor, basis of attraction ]0,1[, except x∗ = 1/a = 1./3.1, which is a ﬁxed point

four-point attractor at a = 3.44948

8-point attractor at ...

at a ≥ 3.5699: trajectory values do no longer repeat: chaotic behaviour

[Table 1.2 (Hilborn)] shows the evolution of similar starting values x0 in the chaotic regime. What’s going on?

The above observations may be summarised in a bifurcation diagram where for a given value of the parameter a one plots the attracting points of the map. [figure: Hilborn fig. 1.14, Gould-Tobochnik fig 6.2] (pitch-fork bifurcation applet at http://brain.cc.kogakuin.ac.jp/ kanamaru/Chaos/e/BifArea/) questions: relevance to actual biological system? complicated assessment; see figure of evolution of blowflies [Kendall 2002] relevance to physical systems?

5 2.2 Non-linear electrical circuit

[Hilborn ﬁg. 1.1] components:

• ac signal generator (voltage V0)

• diode: valve

• inductor (coil): produces electrical potential difference proportional to the rate of change of current

• bias dc generator (voltage Vdc)

measure diode voltage as function of time; control parameter: driving voltage V0 of the periodic signal

initially diode voltage has same frequency as signal generator; increase V0: observation of period doubling etc. [Hilborn ﬁgs 1.2-1.8] important features:

• diode: reverse recovery time (micro-seconds); use appropriate frequency of signal generator, i.e. kHz

• closing time depends on strength of current

• energy stored in capacitance of diode (100pF)

• inductor: extra degree of freedom; picks out special frequencies for oscillations in circuit (resonance)

6 2.3 Lorenz model of atmospheric convection

Edward Lorenz, meteorologist (model published in 1963)

historical importance: this highlights the importance of initial conditions, “butterfly effect”: a butterfly flapping its wings could change the course of the weather... model atmosphere treated as fluid layer, heated at bottom, cooled at top. Navier-Stokes equations for motion of the fluid + thermal diffusion ... =⇒ 3 dimensionless variables X,Y,Z Lorenz equations, final form:

X˙ = p(Y − X) Y˙ = −XZ + rX − Y Z˙ = XY − bZ (2.5)

[ﬁgures: Hilborn 1.19, 1.22]

2.4 Summary of observations

• sudden changes in qualitative behaviour as control parameters are slowly varied

• chaos is not random noise: divergence of nearby trajectories

• well deﬁned changes from regular motion via period doubling to chaos

7 Chapter 3

Universal properties and self-similarity

3.1 Feigenbaum constants idea of Mitchell Feigenbaum 1978: let’s look again at the period doubling scenario from chapter 2.1:

deﬁne a1: value of control parameter at change from period 1 to period 2 behaviour

a2: value of parameter at change from period 2 to period 4

a3 etc. an−an−1 compute δn = an+1−an

Feigenbaum found that δn is approximately the same for all n; it can be shown that this holds for all maps which are parabolic near the maximum, as for example the map

xn+1 = bsin(πx)(b < 1)

Feigenbaum δ = limn→∞ δn = 4.6692016091...

δ belongs to a group of fundamental numbers such as π ' 3.1415927, e ' 2.7182818, Pn 1 Euler-Mascheroni constant γ = limn→∞ k=1 − ln(n) ' 0.57721, golden mean √ k φ = 1+ 5 ' 1.618034, ﬁne structure constant e2 ' 1 2 4π¯hc0 137.03599976

8 experimental data? comparison is limited due to small n < 7, nevertheless there is reasonable agreement (error < 20%) [Hilborn table 2.3, p.52] comment: use δn to predict period doubling note: other Feigenbaum constants are easier to compute: [ﬁgure: Gould/Tobochnic, Fig. 6.6] Feigenbaum α: d α = lim n = 2.5029078750... n→∞ dn+1

dn: distances between nearest attractor elements in bifurcation diagram where xmax is part of trajectory (f(xmax) is maximum of f(x))

self similarity of bifurcation diagram: “different pieces are smaller replicas of other pieces” expansion by δ in x-direction, expansion by α in y-direction (for upper branch)

self-similar objects: fractals, see chapter 8 self-similarity: no inherent (length) scale or size

3.2 Measuring chaos how can we explore the sensitivity to initial conditions? choose two different starting values, after every iteration compute the difference between trajectories |∆x0|, |∆x1| · · ·

[ﬁgure: Gould, ﬁg. 6.8]

one ﬁnds numerically: ln |∆xn| ∝ n

˜ |∆xn| = |∆x0| exp(λn) (3.1)

9 in the limit n → ∞, λ˜ is called the Lyapunov exponent λ (Alexander Michailow- itsch Lyapunov, 1857-1918) λ > 0 indicates exponential divergence of nearby trajectories, this is a signature of chaos

[ﬁgure: variation of λ with control parameter a of the logistic equation (Gould, p.149, ﬁg. 6.9)] actual computation of λ:

1 ∆xn λ˜ = ln (3.2) n ∆x0 expansion: ∆x ∆x ∆x ∆x n = n n−1 ··· 1 (3.3) ∆x0 ∆xn−1 ∆xn−2 ∆x0 =⇒ n−1 1 X ∆xi+1 λ˜ = ln (3.4)

n i=0 ∆xi

dxi+1 0 take limit ∆xi → 0 ⇒ dxi and note that = f (xi) as we have xi+1 = f(xi); dxi we obtain for the Lyapunov exponent:

n−1 1 X 0 λ(x0) = lim ln |f (xi)| (3.5) n→∞ n i=0

compute λ at every iteration (each xi is a starting point)

compute λ(x0) over a sample of starting points x0 to obtain average Lyapunov 1 P exponent λ = N i λi

3.3 Universality of chaos

We looked at systems whose physical details were vastly different:

• population dynamics

• electrical circuits

• Lorentz model of atmosphere

but found complex behaviour with common features:

10 • period doubling

• Feigenbaum numbers

• chaotic behaviour

In order to understand this: study systems at a different, higher level in an abstract phase space (“phase portraits”).

11 Chapter 4

Determinism

Galileo Galilei (1564 -1642) – “kinematics”: position ~r(t) d~r(t) ˙ velocity ~v(t) = dt = ~r(t) d~v(t) ˙ d2~r(t) ¨ acceleration ~a(t) = dt = ~v(t) = dt2 = ~r(t)

Isaac Newton (1642-1727) – “dynamics”:

2nd law: m~r¨ = F~ (~r, ~r,˙ t) (4.1)

where F~ is the net force acting on a particle with mass m note that higher derivatives of ~r(t) are not required

determinism: Given exact starting positions ~r(t = t0) and velocities ~v(t = t0) at ~ time t0, the equation F = m~a determines future positions and velocities for forever! (assuming the forces are known) (Laplace)

Asked by Napoleon where God comes into play, Laplace answered: “Je n’avais pas besoin de cette hypothese-l` a.”` Historically, Laplace’s exact determinism led to an approximate determinism, where an approximate knowledge of ~r0,~v0 is enough for determination of the future.

Note that this is NOT correct for many non-linear systems!

12 Chapter 5

Dynamics in phase space: Motion of the pendulum

5.1 Equation of motion for damped driven pendulum mass m attached to weight-less rigid rod of constant length L, θ measures angle to the vertical

(see sketch)

d ˙ velocity v of bob along the arc: v = dt (Lθ) = Lθ acceleration: a = Lθ¨ component of gravitational force mg along arc: mg sin θ in direction of decreasing angle θ thus Newton’s 2nd law F = ma gives:

d2θ mL = −mg sin θ (5.1) dt2 minus sign: displacement from equilibrium sets up a restoring force acting in op- posite direction of increasing displacement, back towards equilibrium harmonic approximation: sin θ ' θ for small θ 2 natural frequency: ω0 = g/L simple harmonic oscillator:

d2θ = −ω2θ (5.2) dt2 0

13 with solution

θ(t) = θ0 sin(ω0t + φ) (5.3)

θ0 is amplitude, φ is phase constant.

dθ Let’s include a damping force proportional to velocity, γL dt ; dimension of [γ] = mass/time and a driving force with amplitude A and frequency ω˜D; dimension: [A] = force thus: ¨ ˙ mLθ = −mg sin θ − γLθ + A cosω ˜Dt (5.4) To simplify a theoretical analysis this is often written in dimensionless form. In- q troduce dimensionless time variable τ = tω0 = t g/L.

d d dτ q d d • dt = dτ dt = g/l dτ = ω0 dτ

d2 2 d2 • dt2 = ω0 dτ 2 Eqn. 5.4 may then be rewritten as: d2θ γ dθ A 2 + sin θ + = cos ωDτ (5.5) dτ mω0 dτ mg

ω˜D with ωD = . ω0

deﬁne p = A and 1/q = γ we ﬁnally obtain: mg mω0 1 θ¨ + θ˙ + sin θ = p cos ω t (5.6) q D where we simply renamed τ again as t (so from now on t is dimensionless) 1 p, q , ωD are three control parameters

Eqn.5.6 needs to be solved numerically. This may be done by re-writing it as a set of three ﬁrst order differential equations and then applying some standard numerical procedure (e.g. Euler, Runge-Kutta). Introducing two new variables ω and Φ we obtain

dΦ = ω dt D dθ = ω dt dω 1 = − ω − sin θ + p cos Φ (5.7) dt q

14 Integration of these equations may in principle be done using the “simple Euler method”, based on ﬁrst two terms of Taylor expansion of a function. 1

dΦ

Φ(t0 + ∆t) = Φ(t0) + ∆t = Φ(t0) + ωD∆t dt t0 dθ

θ(t0 + ∆t) = θ(t0) + ∆t = θ(t0) + ω(t0)∆t dt t0 dω 1

ω(t0 + ∆t) = ω(t0) + ∆t = ω(t0) + (− ω0 − sin θ(t0) + p cos Φ(t0))∆t (5.8) dt t0 q

with starting values Φ0, θ0, ω0

5.2 Phase space

The pairs x(t) and x˙(t) (or θ(t) and θ˙(t)) specify the behaviour of a dynamical system completely (see chapter4). The system may thus be characterised at any instant of time by a point in a plot x˙ against x. [sketch: phase space, phase portrait]

Construction of phase portraits harmonic oscillator (1/q = 0, p = 0, ωD = 0) using harmonic approximation we obtain from eqn.(5.7) θ˙ = ω ω˙ = −θ (5.9) time does not feature explicitly in phase portraits; let’s eliminate t:

ω˙ dω dω θ = dt = = − (5.10) ˙ dθ θ dt dθ ω thus Z Z ωdω = − θdθ (5.11) ω2 + θ2 = C (5.12) where C is a constant of integration Note that this is the equation of a circle. [sketch] 1note that this simplest algorithm is also very crude and thus is not used in practice

15 Note

2 2 2 Using the dimensional form of the variables we obtain ω + ω0θ = C, which is the equation of an ellipse. C is proportional to the total energy of the pendulum,

C ∝ Etot = Ekin + Epot. damped harmonic oscillator: 1/q > 0, p = 0, ωD = 0 determine the ﬁxed point: dθ = ω := 0 ⇒ ω = 0 dt dω = −ω/q − θ = −θ := 0 ⇒ θ = 0 (5.13) dt The ﬁxed point is an attractor when trajectories approach the point (asymptotically). Here the basin of attraction is the whole plane.

dω = −1/q − θ/ω (5.14) dθ [sketches] non-linear pendulum: equation for phase portrait dω − sin θ = (5.15) dθ ω ω2 − cos θ = C (5.16) 2 ﬁxed points: dθ := 0 ⇒ ω = 0 dt dω = − sin θ := 0 ⇒ sin θ = 0 ⇒ θ = 0, ±π, ±2π, ··· (5.17) dt further analysis shows that θ = 0, ±2π ··· are stable, θ = ±π, ± 3π, ··· are unstable with respect to small perturbation (see also chapter 6.2). [sketches]

16 5.3 Damped driven pendulum: period doubling and chaos

[phase portraits see Baker/Gollub, ﬁgs 2.14, 2.15, 3.6 + applets]

• observation of limit cycles and period doubling (as for the logistic map)

• Bifurcation diagram [Baker/Gollub, ﬁg. 3.13] is complex; it includes chaotic regions and narrow periodic windows

• from the bifurcation diagram one obtains Feigenbaum δpendulum = 4 ± 1 (com-

pared to theoretical value δF eigenbaum ' 4.669)

• sensitivity to initial conditions in the chaotic regime [Figure: Baker/Gollub ﬁg. 3.2]; observations: a) area shrinks (dissipation) (b) stretching and contraction in different directions; in order for the trajectories to remain bounded without intersections, they must fold back on themselves; this results in a 3d layered attractor

5.4 Properties of trajectories

• orbits/trajectories cannot cross each other as this would violate the principle of determinism

• conservative systems (systems where energy is conserved) preserve area in phase space: points in given area of phase space at one time move in a way that at later time the area occupied by these points remains the same

[sketch] test for dissipation consider two-dimensional phase space with time evolution equations

x˙ 1 = f1(x1, x2)

x˙ 2 = f2(x1, x2) (5.18)

consider the following area in phase space: A = (X1C − X1B)(X2C − X2B)

17 [sketch]

dA = (X − X )[f (X ,X ) − f (X ,X )] dt 1C 1B 2 1B 2C 2 1B 2B +(X2C − X2B)[f1(X1C ,X2B) − f1(X1B,X2B)] (5.19) Taylor expansion: ∂f 1 f1(X1C ,X2B) = f1(X1B,X2B) + (X1C − X1B) + ··· (5.20) ∂X1 X1B ,X2B similarly for f2 this ﬁnally leads to 1 dA ∂f ∂f = 1 + 2 (5.21) A dt ∂X1 ∂X2 A similar argument for the evolution of a N-dimensional volume V gives: 1 dV i=N ∂f = X i ≡ div(f) = 5~ · f~ (5.22) V dt i=1 ∂Xi

Comments

• 5~ · f~ = 0: system is conservative (conserves energy); in hydrodynamics: ﬂow of an incompressible ﬂuid (Hamiltonian system)

• 5~ · f~ < 0: system is dissipative; trajectories collapse onto an attractor whose geometrical dimension is less than that of the phase-space

Examples

• undamped harmonic oscillator: θ¨ + θ = 0; using θ˙ = ω;ω ˙ = −θ, one obtains   ω f~ =   −θ

~ ~ ∂ω ∂(−θ) thus 5 · f = ∂θ + ∂ω = 0 (conservative system) • damped harmonic oscillator: θ¨ + 1/qθ˙ + θ = 0 using θ˙ = ω;ω ˙ = −θ − ω/q, one obtains   ω f~ =   −θ − ω/q

~ ~ ∂ω ∂(−θ−ω/q) thus 5 · f = ∂θ + ∂ω = −1/q (dissipative system)

18 Chapter 6

Some theory of chaotic dynamics

6.1 Long-term behaviour of dissipative systems

(the limit t → ∞) trajectory in phase space will evolve towards a ﬁnal point, curve, area etc. (the attractor) basin of attraction: set of initial conditions that lead to attractor will show later: attractor of chaotic systems is a fractal, i.e. it has a dimension which is non-integer one-dimensional state-space x˙ = f(x); ﬁxed points divide x-axis in a number of non-interacting regions

node attracts nearby trajectories: −→ · ←−

[sketch]: f(x0 + x) < 0 ⇒ x˙ < 0 =⇒ move back towards x0; f(x0 − x) > 0 ⇒

df(X) x˙ > 0 =⇒ move forward towards x0; thus dX is important X0

repellor repells nearby trajectories: ←− · −→ saddle, attracting on one side, repelling on the other: −→ · −→ or ←− · ←−

19 two-dimensional state-space

[sketch of node, repellor and saddle] ⊕ possibility of limit cycles, corresponding to oscillatory motion

Poincare-Bendixson´ theorem

GIVEN that: (a) long-term motion of a point in 2d-space is limited to a ﬁnite size region R and (b) any trajectory starting within R stays within R, THEN the following holds: any trajectory starting in R can only approach a ﬁxed point OR a limit cycle for t → ∞ (no possibility for chaos in two dimensions)

6.2 Stability of ﬁxed points

6.2.1 one dimension ˙ ˙ dynamical equation: X = f(X) with ﬁxed point: X|X=X0 = f(X0) = 0 three possibilities: node (sink), repellor (source), saddle point Taylor expansion of f(X) around ﬁxed point: df 1 d2f 1 d3f 2 3 f(X) = f(X0) + (X − X0) + (X − X0) 2 + (X − X0) 3 + ... dX X0 2 dX X0 6 dX X0 (6.1)

introduce new variable x = X − X0 (distance away from ﬁxed point) ˙ ˙ using f(X) = X =x ˙ + X0 =x ˙ and neglecting higher order derivatives, one obtains

x˙ = x (6.2) dX X0 solution: x(t) = x(0) exp λt (6.3)

df with λ = dX . X0 λ is called the characteristic value of the ﬁxed point X0

λ < 0: node: X approaches X0 exponentially

λ > 0: repellor: X is repelled from X0 exponentially

λ is also called Lyapunov exponent for region around ﬁxed point.

20 6.2.2 two dimensions dynamical equation: ˙ X~ = f~(X~ ) (6.4) ~ ~ with X = (X1,X2) and f = (f1, f2), thus,

˙ X1 = f1(X1,X2) ˙ X2 = f2(X1,X2) (6.5) ~ ﬁxed point: X = (X10,X20)

Taylor expansion:

∂f ∂f ˙ 1 1 X1 = (X1 − X10) + (X2 − X20) + ··· ∂X1 X10 ∂X2 X20 ∂f ∂f ˙ 2 2 X2 = (X1 − X10) + (X2 − X20) + ··· (6.6) ∂X1 X10 ∂X2 X20 new variables: ˙ x1 = X1 − X10 =⇒ x˙ 1 = X1 ˙ x2 = X2 − X20 =⇒ x˙ 2 = X2

linearisation (neglecting higher order terms):

∂f ∂f 1 1 x˙ 1 = x1 + x2 = f11x1 + f12x2 ∂X1 X10 ∂X2 X20 ∂f ∂f 2 2 x˙ 2 = x1 + x2 = f21x1 + f22x2 (6.7) ∂X1 X10 ∂X2 X20 write as matrix equation:       x˙ 1 f11 f12 x1   =     ≡ J~x (6.8) x˙ 2 f21 f22 x2 J is called Jacobi matrix.

Solution of eqn.(6.8):

~x = ξ~exp λt (6.9) where ξ~ is an eigenvector of the Jacobi matrix and λ is the corresponding eigenvalue.

21 Proof:

~x˙ = λξ~exp λt = λ~x (6.10) using eqn.(6.8) J~x = λ~x =⇒ Jξ~ = λξ (6.11) Note that this is the eigenvalue problem for the Jacobi matrix J, λ is the eigenvalue, ξ~ the eigenvector. Eigenvalues are the roots of the polynomial equation:

det(J − λI) = 0 (6.12) where I is the identity matrix. The eigenvectors are determined by eqn.(6.11). The eigenvalues determine the type of ﬁxed point, i.e.:

• λ1 > λ2 > 0: repellor

• λ1 < λ2 < 0: node

• λ2 < 0 < λ1: saddle

• λ1, λ2 complex; λ1,2 = c1 ± c2i: spiral point; c1 = 0: circle with centre at ﬁxed point

The behaviour may be summarised in a stability diagram:

[ﬁgure Boyce, ﬁg. 9.1.9] here p = trJ = f11 + f22; q = detJ = f11f22 − f21f12 and ∆ = p2 − 4q.

Example: Brusselator model of a particular chemical reaction, X(t) and Y (t) are concentrations of two different reacting chemicals, A and B are two positive control parameters for the reaction

˙ 2 X = A − (B + 1)X + X Y ≡ F1(X,Y ) ˙ 2 Y = BX − X Y ≡ F2(X,Y ) (6.13)

ﬁxed point are at X0 = A and Y0 = B/A common choice: A = 1 which leaves B as the control parameter √ 2 eigenvalues: λ1,2 = −1 + B/2 ± 1/2 B − 4B detailed analysis shows that for 0 < B < 2, the concentrations X and Y will q oscillate with frequency |B(B − 4)| until they spiral into the ﬁxed point

22 6.2.3 three dimensions same type of eigenvalue analysis, e.g. node at ﬁxed point if all eigenvalues negative; however, there are further possibilities, e.g. node in one plane and repellor in transverse plane

6.3 Analysis of limit cycles

Poincare´ section: reduction of n-dimensional problem to n-1 dimensions example: n=2 [sketch, Hilborn ﬁg 3.12]

1. draw line segment that cuts through the limit cycle

2. start trajectory close to limit cycle

3. monitor the intersections Pn of a trajectory with the Poincare´ line segment

attracting limit cycle: P1 −−−P2 −−−P3 −−−−P −−−P3 −−−P2 −−−P1

repelling limit cycle: P3−−−P2−−−P1−−−P −−−P1−−−P2−−−P3−−−

saddle cycle: P1 − − − P2 − − − P3 − − − P − − − P1 − − − P2 − − − P3 − −− or P3 − − − P2 − − − P1 − − − P − − − P3 − − − P2 − − − P1 − −−

In principle there exists a function F that relates Pn to Pn+1,

Pn+1 = F (Pn) (6.14) this is the Poincare´ map function (iterative relation)

6.4 Examples for damped driven pendulum

[Figures Baker/Gollub 2.14,2.15,2.19, 3.6]

Here we have a three-dimensional parameter space with axes Φ/ωD (time), ω and θ. Attractor is a one dimensional line in this space. Shown in the lecture are two- dimensional projections of the trajectories onto the (ω, θ) plane, revealing a limit cycle.

Poincare´ cuts/sections are taken parallel to the ω − θ plane at ﬁxed intervals of

Φ/ωD (slices perpendicular to the Φ/ωD axis). (stroboscopical view with period of

23 forcing)

Non-chaotic regime: Poincare´ cuts result in ﬁnite number of isolated points.

Chaotic regime: cuts shows a layered structure (with infinite number of points) with an infinite number of layers. The fine structure is similar to the gross structure: self-similarity. It can be described using fractal geometry. The attractors are called strange attractors. layered structure: stretching and folding (similar to the ”kneading of dough”)

6.5 Quasi-periodicity new type of motion, not possible in 1d or 2d

1. trajectories constrained to motion on surface of torus with two separate frequencies [ﬁg: Hilborn 4.8, 4.9]

2. Poincare´ plane is a cut through the torus. The obtained pattern of Poincare´ map points depends on the ratio of the two frequencies

3. rational fraction (as for two oscillators that are frequency locked): Poincare´ plane consists of a number of points

4. irrational fraction: Poincare´ plane is a curve; motion on the torus surface will never repeat itself, it is quasi-periodic; attractor is a two-dimensional surface

6.6 Different routes to chaos

1. period doubling: starting from some limit cycle (Poincare´ section: point) new periodic motion develops as control parameter is increased =⇒ period 2 limit cycle (2 Poincare´ points) increase control parameter =⇒ period 4 etc.

2. quasi-periodicity: System begins with a limit-cycle trajectory with frequency

f1. As a control parameter is changed, a spiral node ﬁxed point turns into a

spiral repellor and a stable limit cycle develops around it with a frequency f2

24 (Hopf bifurcation); if ratio f2/f1 irrational =⇒ quasi-periodic motion; further increase of control parameter might lead to chaos (Ruelle-Takens scenario)

3. intermittency and crisis: irregular occurring burst of chaotic behaviour inter- spersed with intervals of apparent regular behaviour, variation of control parameter causes chaotic bursts to become larger and more frequent

4. chaotic transients and “homoclinic orbits”: long term behaviour inﬂuenced by “interactions” of unstable ﬁxed points/cycles and attractors; parameter change: complicated transient behaviour, eventually lasting forever open questions which scenario for which dynamical system? additional scenarios?

25 Chapter 7

Iterated maps

7.1 Motivation

[ﬁgure: Hilborn 5.1]

• The intersections of phase space trajectories with Poincare´ planes at point (un, vn) may be viewed as mappings of the following kind:

un = Pu(un−1, vn−1)

vn = Pv(un−1, vn−1) (7.1)

• The idea is to study the properties of such maps in general, and then apply the gained knowledge to the corresponding differential equation.

• Often the consideration of one-dimensional maps (xn+1 = f(xn)) is sufﬁcient. Dissipative systems: collapse of a volume of initial conditions to a volume V=0, the Poincare´ points will collapse onto a line (of complicated shape).

• May be seen for example by plotting un as a function of un−1. Examples: the diode circuit of chapter 2.2 [Hilborn ﬁg. 5.3] and the damped driven pendulum, chapter 5.1 [Hilborn, ﬁg. 5.4], or indeed the logistic map of chapter 2.1. Data taken at successive time intervals will result in single valued

functional relationships un+1 = f(un)

7.2 Bernoulli shift

This simple map reproduces several signature features of chaos.

26   2xt 0 ≤ xt < 1/2 xt+1 =  2xt − 1 1/2 ≤ xt < 1

example: x0 = 1/7;

=⇒ x1 = 2/7, x2 = 4/7, x3 = 1/7, x4 = 2/7 ··· periodic! different way of specifying this map: σ(x) = 2 x mod 1, where a mod b means divide a by b and take the remainder, i.e. 11 mod 4 = 3 [sketch of σ(x)]

A different way of looking at σ(x) let’s write x0 in binary presentation (x0 < 1): ∞ X −ν −1 −2 x0 = aν2 = a12 + a22 + · · · ≡ (a1, a2, a3,...) (7.2) ν=1

x0 < 1/2 ⇒ a1 = 0; x0 > 1/2 ⇒ a1 = 1   2x0 for a1 = 0 −→ (a2, a3,...) σ(xo) =  2x0 − 1 for a1 = 1 −→ (a2, a3,...) The Bernoulli shift deletes the ﬁrst digit (of the binary representation) and shifts the remaining sequence to the left.

Properties of the Bernoulli map σ(x)

• the iterates of σ(x) depend sensitively on the the initial value x0: let x and x’ differ after the n-th digit =⇒ σn(x) and σn(x0) already differ in the ﬁrst digit

• σ(x) can produce sequences of numbers that might as well be produced by ﬂip- ping a coin with random output head H or tail T sequence TTHHTHHHT... ⇒ 110010001 · · · ⇒ 1 × 2−1 + 1 × 2−2 + 1 × 2−5 + 1 × 2−9 + ··· Thus σ(x) applied to a an irrational number (a non-repeating inﬁnite sequence in the above expansion) is just as random as the result from tossing a coin. “given a random sequence of 0 and 1 one cannot say whether this is the result of coin throwing or of computation using the Bernoulli shift”

27 • feature of the map: stretching and folding: interval [0,1[ is ﬁrst stretched into interval [0,2[ and then [1,2[ is folded back into [0,1[ [sketch]

• the map has a positive Lyapunov exponent: using λ˜ = 1 ln ∆xn n ∆xo n n ∆xn = f (xo + ∆x) − f (x0) n n λ = lim 1 ln ∆f (xo) = lim 1 ln df (xo) n→∞,∆x0→0 n ∆xo n→∞ n dxo Bernoulli shift: slope of σ(x) is 2 everywhere apart from x=1/2 where it is not 1 n deﬁned; λ = limn→∞ n ln 2 = ln 2 > 0; this is a feature of chaotic behaviour!

28 Chapter 8

Fractals

8.1 A mathematical monster: the Koch curve

(Niels Fabian Helge von Koch (1870 - 1924), Swedish mathematician) consider curve consisting of N(l) segments, each of length l; total lengthL is given by L = N(l)l consider the following so-called Koch-curve [sketch] step number of segments N(l) segment length l total length L 0 1 1 1 1 4 1/3 4/3 2 42 (1/3)2 16/9 ...... n 4n (1/3)n (4/3)n n 1 n length of Koch curve: L = liml→0 N(l)l = liml→0 4 3 −→ ∞ what’s going on?

Deﬁnition

Df If liml→0 N(l)l = ∞ and there exists a number Df with liml→0 N(l)l = B < ∞, then Df is called the fractal dimension of the curve.

Df liml→0 l = liml→0 B/N(l);

liml→0 Df ln l = liml→0(ln B − ln N(l)) ln B Df = liml→0 ln l − liml→0 ln N(l)/ln l ln N(l) =⇒ Df = lim (8.1) l→0 ln(1/l)

29 ln 4n ln 4 Koch curve: Df = liml→0 ln 3n = ln 3 = 1.2619 Note that here Df takes value between the dimension of a line (dim=1) and an area (dim=2).

8.2 Fractal dimensions

Box counting method construct boxes or d-spheres to cover the boundary; if N(l) varies as N(l) ∝ l−D for l → 0 then D is called the box counting fractal dimension (also: capacity dimension)

(the topological dimension of the Koch curve dtop = 1) example for computation of D for the Koch curve see lecture

Note: a variety of different deﬁnitions of fractal dimensions is in use; in mathematical liter- ature: Hausdorff dimension makes use of variable size boxes, dH < Dbox in practical computations box-counting is computer time intensive; better: corre- lation dimension which is computed directly from a trajectory on an attractor

8.3 Examples of fractals

Mathematics

Julia set [Gaston Julia, 1893-1978]: consider mapping

2 zn+1 = zn + c (8.2) where zn are complex variables, c is a complex constant [figure Hilborn, fig. 9.8] plot all initial positions (x axis = Re(z), y-axis = Im(z)) which result in trajectories that are bounded (don’t shoot off to infinity) these type of maps (e.g. Mandelbrot set [Benoit Mandelbrot, *1924]) are of relevance to fractal basin boundaries, as for example in the damped driven oscillator

30 Nature coast lines, trees, dendritic crystal growth [ﬁgures and applets see lecture]

31 Chapter 9

Strange attractors

9.1 Definition there is no generally accepted definition, however: A strange attractor has two defining properties.

1. It is an attractor. (bounded region in phase space to which all sufﬁciently close trajectories from basin are attracted asymptotically, trajectories visits every point on attractor in course of time)

2. It displays a sensitive dependence on initial conditions: points on the attractor that are initially close to each other become exponentially separated with time. This makes it “strange”. note: all strange attractors that have been found have fractal (non-integer) dimension example: chaotic system in 3D space

~x˙ = F~ (~x), divF~ < 0 (dissipative system) Typically flow contracts a volume element in one dimension, but stretches it in another; in order to remain confined in a bounded domain, folding is needed. [sketch] divF~ < 0 → dV/dt < 0, but the volume cannot become two-dimensional due to Poincare-Bendixson´ (no chaos in 2d) =⇒ resulting geometry has fractal dimension (flow generates a set of points whose dimension is less than three)

32 Example: Lorenz attractor

dV From Lorenz equations (2.5) ﬁnd dt = −(p + 1 + b)V < 0; (p > 0, b > 0). The Hausdorff dimensions was computed as 2.06.

9.2 Baker’s transformation dissipative (non-area preserving) map, reminiscent of a baker’s kneading of dough [see applets in lecture]

xn+1 = 2xnmod1   ayn 0 ≤ xn < 1/2 yn+1 =  1/2 + ayn 1/2 ≤ xn ≤ 1

a < 0.5: dissipative; a = 1/2: non-dissipative [see sketch]

Lyapunov exponents x-direction: λx = ln 2 (derivation see Bernoulli shift)

y-direction: λy = ln a n using λ = lim 1 ln df (xo) n→∞ n dxo 1 n slope is a everywhere apart from y=1/2 where it is not deﬁned; λ = limn→∞ n ln a = ln a < 0 fractal dimension: x-direction: one-dimensional −D y-direction: use self-similarity, deﬁnition liml→o N(l) ∝ l N(a) −D −2D D ln 1/2 ln 2 N(a2) = 1/2 = a /a = a =⇒ D = ln a = − ln 2/ ln a = | ln a| ln 2 D = Dx + Dy = 1 + | ln a| a=1/2: D = 1+1 =2 (non-dissipative case: dimension of an area)

9.3 Stretching and folding for the logistic map

dfa fa(x) = ax(1 − x) =⇒ dx = a − 2ax maximum at x = 1/2 set a=4

33 interval [0,1/2] is stretched into [0,1] interval [1/2,1] is mapped into [0,1] in reverse order, it is stretched and folded this process of stretching and folding leads to a loss of information about initial conditions (fa(x) is not invertible)

34 Chapter 10

Advanced topics

10.1 Hamiltonian systems (motivation) dynamical systems discussed so far were dissipative, characterised by collapse of a volume of initial conditions in phase space onto an attractor

what happens for conservative (Hamiltonian) systems?

chaos is possible, there are chaotic regions in phase space (but not attractive), densely interweaved with regular regions

no academic question: solar system is (very nearly) conservative (the study of its stability was Poincare’s´ motivation to look at dynamical systems)

3-body problem: how do 3 bodies move due to their gravitational interaction?

treat this as 2-body problem (resulting in oscillation in ellipses with frequency ω1 around centre of gravity) + periodic perturbation by 3rd body with frequency ω2

for rational ratio ω1 = p/q system can be in a stable resonance or exhibit chaos ω2 (unstable resonance), dependent of initial conditions

ω1 example: orbit of Saturn around the sun, ω1 is disturbed by Jupiter, ω2: = ω2

35 4331days = 0.4000923 ' 2/5. 10825days

KAM theorem (Kolmogorov 1954, Arnold 1963, Moser 1962):

ω1 needs to be “sufﬁciently irrational” in order to give motion that is stable under ω2 perturbation √ most stable motion: golden mean! Φ = ( 5 − 1)/2 ' 0.6180339 = 1/Φ − 1 perturbations primarily inﬂuence all motions with ω1 nearly rational ω2 astronomical consequences:

• distribution of asteroids between Mars and Jupiter (where most of the asteroids are) shows gaps (resonance with Jupiter −→ chaotic trajectory, asteroids are being swept away to distances without resonances [Stewart, ﬁg. 109]

• gaps in the rings of Saturn due to inﬂuence of Saturn’s moons[Schuster, p. 199]

• tumbling motion of Saturn moon Hyperion

10.2 Quantum chaos (motivation) quantum mechanics: extremely successful theory for molecular, atomic and sub-atomic systems some features:

Uncertainty Principle (NO trajectories!), wave function Ψ(t), probability interpretation of |Ψ(t)|2 general belief: predictions of quantum mechanics ought to agree with predictions of classical mechanics in some appropriate limit

Correspondence Principle: limit Planck’s constant h → 0

36 This seems to imply that there needs to be chaotic behaviour also in quantum mechanics (where else would it come from in classical mechanics?) But Schrodinger¨ equation is linear! non-linear extension of Schrodinger¨ equation?

Bialynicki-Birula 1976:

∂ "−h¯2 # ih¯ Ψ(~r, t) = ∇2 + V (~r, t) − b ln(|Ψ(~r, t)|2) Ψ(~r, t) (10.1) ∂t 2m with parameter b > 0.

prediction b ' 10−12eV

measurements: even smaller? b ≤ 3.3 × 10−15eV

question and motivation: is quantum mechanics the only linear theory which is not the limit of some non-linear theory?

37 Chapter 11

Some examples of complex behavior

complexity, complex systems: no generally accepted deﬁnition is as yet available, but it could be along the line of “collective behaviour that emerges out of the (nonlinear) interactions of many in- dividual units”

examples include [ﬁgures see lecture] :

• Formation of networks (trails of humans and animals, social networks)

• emergence of correlated motion e.g. in the dynamics of trafﬁc ﬂow or pedestrian motion

• power-law statistics for example in stock market ﬂuctuations

• Growth processes (bacteria, liquid crystals): fractal structures; mathematical model: diffusion-limited aggregation

• Pattern formation (e.g.“spots of the leopard”); model: reaction-diffusion equations

All of the above are modelled by physicists (and mathematicians,biologists, economists, sociologists...). Computers are often essential for modelling. They also have added to a new way of thinking and looking at problems, for example the “cellular automata” approach.

38 Chapter 12

Cellular Automata

12.1 Concept origins: Von Neumann, Ulam 1948 cellular automaton is a discrete dynamical system that can be simulated exactly, set up on a checkerboard, every cell changes at tick of an external clock according to rules based on the present conﬁguration (microstate) of cells in its neighbourhood

Characteristics of a Discrete Dynamical System:

• space is discrete; regular array of cells with a ﬁnite set of values

• time is discrete; update of values is sequential

• update rules; dependence on local neighbourhood only

• simultaneous updates; dependent of previous time-step

12.2 Example in one dimension two states for every cell: shown is the local neighbourhood of the central value, 23 = 8 possible conﬁgurations, order as decreasing (binary) numbers from left to right (Stephen Wolfram , Rev., Mod. Phys. 55, 601 (1983))

t: 111 110 101 100 011 010 001 000 t+1: 0 1 0 1 1 0 1 0 at time t+1 only the updated central sites are shown

39 this update rule is 01011010, interpretation as a binary number ⇒ rule 90 (= 26 + 24 + 23 + 21)

there are 28 = 256 possible update rules that one can construct

rules can be classiﬁed according to long term behaviour of these CA (see slides shown in lecture, calculations use periodic boundary conditions)

• homogeneous state, equilibrium solution

• simple periodic structures

• chaotic, aperiodic structures

• complex patterns with local structure