L ECTURE 10: D ISCRETE -T IME D YNAMICAL S YSTEMS 1 I NSTRUCTOR : G - PowerPoint PPT Presentation

15-382 C OLLECTIVE I NTELLIGENCE – S18 L ECTURE 10: D ISCRETE -T IME D YNAMICAL S YSTEMS 1 I NSTRUCTOR : G IANNI A. D I C ARO

(M ORE ) G ENERAL DEFINITION OF DYNAMICAL SYSTEMS A dynamical system is a 3-tuple 𝑈,𝑇, Φ : 𝑇 is a set of all possible states of the dynamical system (the state space) § 𝑈 is the set of values the time (evolution) parameter can take § Φ is the evolution function of the dynamical system, that associates to § each 𝒚 ∈ 𝑇 a unique image in 𝑇 depending on the time parameter 𝑢, (not all pairs (𝑢, 𝒚) are feasible, that requires the subset 𝐸) Φ:𝐸 ⊆ 𝑈×𝑇 → 𝑇 Ø Φ 0, 𝒚 = 𝒚 (the initial condition) Ø Φ 𝑢 2 ,Φ 𝑢 3 ,𝒚 = Φ(𝑢 2 + 𝑢 3 ,𝒚) , (property of states) for 𝑢 3 , 𝑢 3 + 𝑢 2 ∈ 𝐽(𝑦) , 𝑢 2 ∈ 𝐽(Φ(𝑢 3 𝑦)) , 𝐽 𝑦 = {𝑢 ∈ 𝑈 ∶ (𝑢, 𝒚) ∈ 𝐸} Ø The evolution function Φ provides the system state (the value ) at time 𝑢 for any initial state 𝒚 Ø 𝛿 ; = {Φ 𝑢,𝒚 ∶ 𝑢 ∈ 𝐽 𝒚 } orbit of the system through 𝒚 , starting in 𝒚 , the set of visited states as a function of time 2

T YPES OF D YNAMICAL SYSTEMS § Informally: A dynamical system defines a deterministic rule that allows to know the current state as a function of past states Given an initial condition 𝒚 < = 𝒚(0) ∈ 𝑇 , a deterministic trajectory § 𝒚 𝑢 , 𝑢 ∈ 𝐽(𝒚 < ) is produced by 𝑈, 𝑇,Φ § States can be “anything” mathematically well-behaved that represent situations of interest Continuous time dynamical systems (Flows): 𝑈 open interval of ℝ , § Φ continous and differentiable function à Differential equations 𝑦̇ = 𝑔(𝑦 𝑢 − 𝜐 ) Delay models § Ordinary Differential Equations E 𝑦̇ = 𝑔 𝑦 𝑢 + ∫ 𝑔 𝑦 𝜐 𝑒𝜐 § Integro-Differential E F Equations , accounting for history I H I H 3 IE H 𝜒 𝑦, 𝑢 = I; H 𝜒 𝑦,𝑢 § Partial Differential Equations , G H accounting for space and time Discrete-time dynamical systems (Maps): 𝑈 interval of ℤ , Φ a function § 𝑦 K = 𝑔(𝑦 KL3 ,𝑦 KL2 ,…, 𝑦 KLN ) , iterated updating 3

F ROM LOCAL RULES TO GLOBAL BEHAVIORS ? 𝑒𝒚 𝑒𝑢 = 𝑔(𝒚,𝑢) Flows Maps 𝑦 K = 𝑔(𝑦 KL3 ,𝑦 KL2 ,…, 𝑦 KLN ) ∆𝑢 = 1 , when ∆𝑢 → 0 à 𝑒𝑢 à Differential eq. § For an infinitesimal time, only the instantaneous variation, the velocity , makes sense à The next state is expressed implicitly, and all the instantaneous variations, local in time, must be integrated in order to obtain the global behavior 𝒚(𝑢) § Also in maps, the time-local iteration rule is a local description that can give rise to extremely complex global behaviors § à How do we integrate the local description into global behaviors? § à How do we predict global behaviors from the local descriptions? 4

M APS § Where maps can arise from? § Inherently discrete-time processes: looking at populations in terms of generations, epidemics in terms of weeks, economy in terms of quarters or years, traffic models per hour, growth per days, … § Discretization of differential equations : Euler method: 𝒚̇ = 𝒈(𝒚) à 𝒚 KR3 = 𝒚 K + ℎ𝒈(𝒚 K ) , …Runge-Kutta,… § § Discretization of algebraic equations: Newton’s method for solving 𝑔 𝑦 = 0 à Expand in Taylor series § near 𝑦 K :𝑔 𝑦 = 𝑔 𝑦 K + 𝑦 − 𝑦 K 𝑔 T 𝑦 K + ⋯ taking the usual linear approximation: 𝑔 𝑦 ≈ 𝑔 𝑦 K + 𝑦 − 𝑦 K 𝑔 T 𝑦 K , equating to 0: 𝑦 KR3 = 𝑦 K − 𝑔 𝑦 K /𝑔 T 𝑦 K § Let’s focus on one-dimensional maps § Even in one dimension, iterated maps can produce incredibly complex behaviors, including deterministic chaos! § Later on, we will consider multi-dimensional maps defined over a lattice (spatial grid) à Cellular Automata à even more complex behaviors 5

M APS : SAME TERMINOLOGY AS IN FLOWS 𝑔 𝑦 = 2𝑦 , 𝑔 is a map § The orbit of 𝑦 under the map 𝑔 is the set of points: § ,… } = {𝑦, 𝑔 𝑦 , 𝑔 2 𝑦 ,𝑔 Y 𝑦 ,… } {𝑦,𝑔 𝑦 ,𝑔 𝑔 𝑦 , 𝑔 𝑔 𝑔 𝑦 corresponding to the iterated application of the map § The initial point provides the initial conditions A point 𝑦 ∗ , such that 𝑔 𝑦 ∗ = 𝑦 ∗ is a fixed point, the orbits remain § in 𝑦 ∗ for all future iterations Cobweb plots for individual orbits 6

F IXED POINTS § Starts at 1.6, converges to 1 § Starts at 1.8, converges to -1 Stability of a fixed point 𝑦 ∗ ? Fixed points correspond to intersection between graph 𝑔(𝑦) and 𝑦 § General map: 𝑦 KR3 = 𝑔(𝑦 K ) , 𝑔(𝑦 ∗ ) = 𝑦 ∗ § Let’s consider a near orbit, 𝑦 K = 𝑦 ∗ + 𝜁 K : is the orbit attracted or § repelled from 𝑦 ∗ ? If it’s attracted we can say that 𝑦 ∗ is stable Does the perturbation 𝜁 K grow or decay with 𝑜 ? § By the definition, 𝑦 ∗ + 𝜁 KR3 = 𝑦 KR3 = 𝑔(𝑦 ∗ + 𝜁 K ) , and using the Taylor § series expansion about 𝑦 ∗ : 𝑦 ∗ + 𝜁 KR3 = 𝑦 KR3 = 𝑔 𝑦 ∗ + 𝜁 K = 𝑔 𝑦 ∗ + 𝑔 T 𝑦 ∗ 𝜁 K + 𝑃(𝜁 K2 ) Given that 𝑔(𝑦 ∗ ) = 𝑦 ∗ à 𝜁 KR3 = 𝑔 T 𝑦 ∗ 𝜁 K + 𝑃(𝜁 K2 ) § § If we take the linear approximation à Linearized map: 𝜁 KR3 = 𝑔 T 𝑦 ∗ 𝜁 K § Eigenvalue / multiplier: 𝜇 = 𝑔 T 𝑦 ∗ § 7

F IXED POINTS Linearized map: 𝜁 KR3 = 𝑔 T 𝑦 ∗ 𝜁 K § Solution of linearized map: 𝜁 3 = 𝜇𝜁 < , 𝜁 2 = 𝜇𝜁 3 = 𝜇 2 𝜁 < ….. 𝜁 K = 𝜇 K 𝜁 < § If 𝜇 = |𝑔 T 𝑦 ∗ | < 1 à 𝜁 K → 0, for 𝑜 → ∞, and 𝑦 ∗ is linearly stable § If 𝜇 = |𝑔 T 𝑦 ∗ | > 1 à 𝜁 K → ∞, for 𝑜 → ∞, and 𝑦 ∗ is unstable § The linear stability holds also for the general map § The marginal case 𝜇 = |𝑔 T 𝑦 ∗ | = 1 doesn’t allow to draw conclusions. § In this case the quadratic term 𝑃(𝜁 K2 ) determines the stability If 𝜇 =0, then the fixed point is said superstable § 𝑦 KR3 = sin𝑦 K § 𝑦 ∗ = 0 is a fixed point § 𝜇 = 𝑔 T 𝑦 ∗ = 1 , marginal case § § Cobweb à It’s stable! Is it global? For all orbits 𝑦 K → 0 ? § For any 𝑦 < , 𝑦 3 ∈ [−1,1] since |sin𝑦 3 | < 1 § 1 § à From cobweb we can say it’s global 8

A NOTHER EXAMPLE , LIMITING BEHAVIOR 𝑦 KR3 = cos 𝑦 K § K→l 𝑦 K ? … by iterating the map (e.g., use calculator!), 𝑦 K → 0.739.. lim § Solution of trascendental equation 𝑦 = cos𝑦 § The fixed point 0.739… has 𝜇 < 0 à Damped oscillations § For 0 < 𝜇 < 1 convergence to a stable fixed point is monotonic § 9

L OGISTIC MAP 𝑦 KR3 = 𝑠𝑦 K (1 − 𝑦 K ) § 𝑦 K is a dimensionless measure of the population in the 𝑜 th generation § and 𝑠 is the intrinsic growth rate (with capacity being limited to 1) Let’s restrict 0 ≤ 𝑠 ≤ 4 § à The map maps 0,1 → 0,1 Let’s fix 𝑠 and study the evolution § § Trivially, for small growth rates, 𝑠 < 1, the population always goes 1 𝑦 K extinct, as 𝑦 K → 0 2 For 1 < 𝑠 < 3 , population grows § and eventually reaches a non-zero steady state Watch out: this a time series! 10

A PATH TO THE CHAOS … 11

“R EGULAR ” BEHAVIOR , PERIODIC ATTRACTORS 12

“R EGULAR ” BEHAVIOR , PERIODIC ATTRACTORS 13

T RANSITION TO CHAOTIC BEHAVIOR 14

C HAOS : S ENSITIVITY TO INITIAL CONDITIONS 15

P ERIODS IN THE LOGISTIC MAP § Oscillating about the previous steady state, alternating between small and large populations Period-4 cycle § § Period-2 cycle: Oscillation repeats every two iterations, periodic orbit Period-doubling to cycles appears by increasing 𝑠 § § They correspond to bifurcations i n phase diagram § Successive bifurcations come faster and faster! Limiting value 𝑠 K → 𝑠 l = 3.569946 … § § Geometric convergence, in the limit the distance between successive values shrink to a constant: 16

C HAOS … § 𝑠 > 𝑠 l ? § For many values of 𝑠, the sequence never settles down to a fixed point or a periodic orbit § Aperiodic, bounded behavior! 17

O RBIT DIAGRAM What happens for larger 𝑠 ? Sure, more chaos …. Even more interesting things! § Orbit diagram: system’s attractors as a function of 𝑠 § § Construction: Choose a value of 𝑠 § Select a random initial condition 𝑦 < and generate the orbit: lets iterate for ~ 300 cycles to let § the system settle down, then plot the next ~ 300 points from the map iterations Move to an adjacent value of 𝑠 and repeat, sweeping the 𝑠 interval § At 𝑠 ≈ 𝑠 l = 3.57 the map § becomes chaotic and the attractor changes from a finite to an infinite set of points For 𝑠 > 3.57 , mixture of order § and chaos, with periodic windows interspersed between clouds of (chaotic) dots 18

O RBIT DIAGRAM 19

C HAOS AND ORDER 20

C HAOS AND ORDER , S ELF - SIMILARITY The large window at 𝑠 ≈ 3.83 § contains a stable period-3 orbit § Looking at the period-3 window even closer: a copy of the orbit diagram reappears in miniature! à Self-similarity 21