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 gener- ally 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 dy- namics 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: obser- vation of period doubling etc. [Hilborn figs 1.2-1.8] important features: • diode: reverse recovery time (micro-seconds); use appropriate frequency of sig- nal 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, “but- terfly 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 equa- tions 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!1 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!1 k=1 − ln(n) ' 0:57721, golden mean p 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.