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

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

Apr 03, 2024 344 likes β€’602 views

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 ,, : is a set of all

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LECTURE 10: DISCRETE-TIME DYNAMICAL SYSTEMS 1

INSTRUCTOR: GIANNI A. DI CARO

15-382 COLLECTIVE INTELLIGENCE – S18

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(MORE) GENERAL DEFINITION OF DYNAMICAL SYSTEMS

A dynamical system is a 3-tuple π‘ˆ,𝑇, Ξ¦ : Β§ 𝑇 is a set of all possible states of the dynamical system (the state space) Β§ π‘ˆ is the set of values the time (evolution) parameter can take Β§ Ξ¦ is the evolution function of the dynamical system, that associates to each π’š ∈ 𝑇 a unique image in 𝑇 depending on the time parameter 𝑒, (not all pairs (𝑒, π’š) are feasible, that requires the subset 𝐸) Ξ¦:𝐸 βŠ† π‘ˆΓ—π‘‡ β†’ 𝑇 Ø Ξ¦ 0, π’š = π’š (the initial condition) Ø Ξ¦ 𝑒2,Ξ¦ 𝑒3,π’š = Ξ¦(𝑒2 + 𝑒3,π’š), (property of states) for 𝑒3,𝑒3 + 𝑒2 ∈ 𝐽(𝑦), 𝑒2 ∈ 𝐽(Ξ¦(𝑒3𝑦)), 𝐽 𝑦 = {𝑒 ∈ π‘ˆ ∢ (𝑒, π’š) ∈ 𝐸} Ø The evolution function Ξ¦ provides the system state (the value) at time 𝑒 for any initial state π’š Ø 𝛿; = {Ξ¦ 𝑒,π’š ∢ 𝑒 ∈ 𝐽 π’š } orbit of the system through π’š, starting in π’š , the set of visited states as a function of time

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TYPES OF DYNAMICAL SYSTEMS

Β§ Informally: A dynamical system defines a deterministic rule that allows to know the current state as a function of past states Β§ Given an initial condition π’š< = π’š(0) ∈ 𝑇, a deterministic trajectory π’š 𝑒 , 𝑒 ∈ 𝐽(π’š<) is produced by π‘ˆ, 𝑇,Ξ¦ Β§ States can be β€œanything” mathematically well-behaved that represent situations of interest Β§ Continuous time dynamical systems (Flows): π‘ˆ open interval of ℝ, Ξ¦ continous and differentiable function Γ  Differential equations Β§ Discrete-time dynamical systems (Maps): π‘ˆ interval of β„€, Ξ¦ a function

Β§ 𝑦̇ = 𝑔(𝑦 𝑒 βˆ’ 𝜐 ) Delay models Β§ 𝑦̇ = 𝑔 𝑦 𝑒 + ∫ 𝑔 𝑦 𝜐 π‘’πœ

E EF

Integro-Differential Equations, accounting for history Β§

3 GH IH IEH πœ’ 𝑦, 𝑒 = IH I;H πœ’ 𝑦,𝑒

Partial Differential Equations, accounting for space and time Ordinary Differential Equations

𝑦K = 𝑔(𝑦KL3,𝑦KL2,…, 𝑦KLN), iterated updating

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FROM LOCAL RULES TO GLOBAL BEHAVIORS?

Flows Maps βˆ†π‘’ = 1, when βˆ†π‘’ β†’ 0 Γ  𝑒𝑒 Γ  Differential eq. π‘’π’š 𝑒𝑒 = 𝑔(π’š,𝑒) 𝑦K = 𝑔(𝑦KL3,𝑦KL2,…, 𝑦KLN) Β§ For an infinitesimal time, only the instantaneous variation, the velocity, makes sense Γ  The next state is expressed implicitly, and all the instantaneous variations, local in time, must be integrated in order to

  • btain the global behavior π’š(𝑒)

Β§ Also in maps, the time-local iteration rule is a local description that can give rise to extremely complex global behaviors Β§ Γ  How do we integrate the local description into global behaviors? Β§ Γ  How do we predict global behaviors from the local descriptions?

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MAPS

Β§ Where maps can arise from? Β§ Inherently discrete-time processes: looking at populations in terms of generations, epidemics in terms of weeks, economy in terms of quarters

  • r years, traffic models per hour, growth per days, …

Β§ Discretization of differential equations: Β§ Euler method: π’šΜ‡ = π’ˆ(π’š) Γ  π’šKR3 = π’šK + β„Žπ’ˆ(π’šK), …Runge-Kutta,… Β§ Discretization of algebraic equations: Β§ Newton’s method for solving 𝑔 𝑦 = 0 Γ  Expand in Taylor series near 𝑦K:𝑔 𝑦 = 𝑔 𝑦K + 𝑦 βˆ’ 𝑦K 𝑔T 𝑦K + β‹― taking the usual linear approximation:𝑔 𝑦 β‰ˆ 𝑔 𝑦K + 𝑦 βˆ’ 𝑦K 𝑔T 𝑦K , equating to 0: 𝑦KR3 = 𝑦K βˆ’ 𝑔 𝑦K /𝑔T 𝑦K Β§ Let’s focus on one-dimensional maps Β§ Even in one dimension, iterated maps can produce incredibly complex behaviors, including deterministic chaos! Β§ Later on, we will consider multi-dimensional maps defined over a lattice (spatial grid) Γ  Cellular Automata Γ  even more complex behaviors

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MAPS: SAME TERMINOLOGY AS IN FLOWS

Β§ 𝑔 𝑦 = 2𝑦, 𝑔 is a map Β§ The orbit of 𝑦 under the map 𝑔 is the set of points: {𝑦,𝑔 𝑦 ,𝑔 𝑔 𝑦 , 𝑔 𝑔 𝑔 𝑦 ,… } = {𝑦, 𝑔 𝑦 , 𝑔2 𝑦 ,𝑔Y 𝑦 ,… } corresponding to the iterated application of the map Β§ The initial point provides the initial conditions Β§ A point π‘¦βˆ—, such that 𝑔 π‘¦βˆ— = π‘¦βˆ— is a fixed point, the orbits remain in π‘¦βˆ— for all future iterations Cobweb plots for individual

  • rbits
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FIXED POINTS

Β§ Starts at 1.6, converges to 1 Β§ Starts at 1.8, converges to -1 Stability of a fixed point π‘¦βˆ—? Β§ Fixed points correspond to intersection between graph 𝑔(𝑦) and 𝑦 Β§ General map: 𝑦KR3 = 𝑔(𝑦K), 𝑔(π‘¦βˆ—) = π‘¦βˆ— Β§ Let’s consider a near orbit, 𝑦K = π‘¦βˆ— + 𝜁K: is the orbit attracted or repelled from π‘¦βˆ—? If it’s attracted we can say that π‘¦βˆ— is stable Β§ Does the perturbation 𝜁K grow or decay with π‘œ? Β§ By the definition, π‘¦βˆ— + 𝜁KR3 = 𝑦KR3 = 𝑔(π‘¦βˆ— + 𝜁K), and using the Taylor series expansion about π‘¦βˆ—: π‘¦βˆ— + 𝜁KR3 = 𝑦KR3 = 𝑔 π‘¦βˆ— + 𝜁K = 𝑔 π‘¦βˆ— + 𝑔T π‘¦βˆ— 𝜁K + 𝑃(𝜁K2) Β§ Given that 𝑔(π‘¦βˆ—) = π‘¦βˆ— Γ  𝜁KR3 = 𝑔T π‘¦βˆ— 𝜁K + 𝑃(𝜁K2) Β§ If we take the linear approximation Γ  Β§ Linearized map: 𝜁KR3 = 𝑔T π‘¦βˆ— 𝜁K Β§ Eigenvalue / multiplier: πœ‡ = 𝑔T π‘¦βˆ—

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FIXED POINTS

Β§ Linearized map: 𝜁KR3 = 𝑔T π‘¦βˆ— 𝜁K Β§ Solution of linearized map: 𝜁3 = πœ‡πœ<, 𝜁2 = πœ‡πœ3 = πœ‡2𝜁< ….. 𝜁K = πœ‡K𝜁< Β§ If πœ‡ = |𝑔T π‘¦βˆ— | < 1 Γ  𝜁K β†’ 0, for π‘œ β†’ ∞, and π‘¦βˆ— is linearly stable Β§ If πœ‡ = |𝑔T π‘¦βˆ— | > 1 Γ  𝜁K β†’ ∞, for π‘œ β†’ ∞, and π‘¦βˆ— is unstable Β§ The linear stability holds also for the general map Β§ The marginal case πœ‡ = |𝑔T π‘¦βˆ— | = 1 doesn’t allow to draw conclusions. In this case the quadratic term 𝑃(𝜁K2) determines the stability Β§ If πœ‡ =0, then the fixed point is said superstable Β§ 𝑦KR3 = sin𝑦K Β§ π‘¦βˆ— = 0 is a fixed point Β§ πœ‡ = 𝑔T π‘¦βˆ— = 1, marginal case Β§ Cobweb Γ  It’s stable! Β§ Is it global? For all orbits 𝑦K β†’ 0? Β§ For any 𝑦<, 𝑦3∈ [βˆ’1,1] since |sin𝑦3| < 1 Β§ Γ  From cobweb we can say it’s global

1

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ANOTHER EXAMPLE, LIMITING BEHAVIOR

Β§ 𝑦KR3 = cos 𝑦K Β§ lim

Kβ†’l𝑦K? … by iterating the map (e.g., use calculator!), 𝑦K β†’ 0.739..

Β§ Solution of trascendental equation 𝑦 = cos𝑦 Β§ The fixed point 0.739… has πœ‡ < 0 Γ  Damped oscillations Β§ For 0 < πœ‡ < 1 convergence to a stable fixed point is monotonic

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LOGISTIC MAP

Β§ 𝑦KR3 = 𝑠𝑦K(1 βˆ’ 𝑦K) Β§ 𝑦K is a dimensionless measure of the population in the π‘œth generation and 𝑠 is the intrinsic growth rate (with capacity being limited to 1)

1 2

𝑦K Β§ Let’s restrict 0 ≀ 𝑠 ≀ 4 Γ  The map maps 0,1 β†’ 0,1 Β§ Let’s fix 𝑠 and study the evolution Β§ Trivially, for small growth rates, 𝑠 < 1, the population always goes extinct, as 𝑦K β†’ 0 Β§ For 1 < 𝑠 < 3, population grows and eventually reaches a non-zero steady state Watch out: this a time series!

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A PATH TO THE CHAOS …

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β€œREGULAR” BEHAVIOR, PERIODIC ATTRACTORS

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β€œREGULAR” BEHAVIOR, PERIODIC ATTRACTORS

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TRANSITION TO CHAOTIC BEHAVIOR

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CHAOS: SENSITIVITY TO INITIAL CONDITIONS

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PERIODS IN THE LOGISTIC MAP

Β§ Oscillating about the previous steady state, alternating between small and large populations Β§ Period-2 cycle: Oscillation repeats every two iterations, periodic orbit Β§ Period-4 cycle Β§ Period-doubling to cycles appears by increasing 𝑠 Β§ They correspond to bifurcations in phase diagram Β§ Successive bifurcations come faster and faster! Β§ Limiting value 𝑠

K β†’ 𝑠 l = 3.569946 …

Β§ Geometric convergence, in the limit the distance between successive values shrink to a constant:

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CHAOS …

Β§ 𝑠 > 𝑠

l ?

Β§ For many values of 𝑠, the sequence never settles down to a fixed point

  • r a periodic orbit

Β§ Aperiodic, bounded behavior!

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ORBIT DIAGRAM

Β§ What happens for larger 𝑠? Sure, more chaos…. Even more interesting things! Β§ Orbit diagram: system’s attractors as a function of 𝑠

Β§ Construction: Β§ Choose a value of 𝑠 Β§ 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 Β§ Move to an adjacent value of 𝑠 and repeat, sweeping the 𝑠 interval

Β§ At 𝑠 β‰ˆ 𝑠

l = 3.57 the map

becomes chaotic and the attractor changes from a finite to an infinite set of points Β§ For 𝑠 > 3.57, mixture of order and chaos, with periodic windows interspersed between clouds of (chaotic) dots

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ORBIT DIAGRAM

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CHAOS AND ORDER

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CHAOS AND ORDER, SELF-SIMILARITY

Β§ The large window at 𝑠 β‰ˆ 3.83 contains a stable period-3 orbit Β§ Looking at the period-3 window even closer: a copy of the orbit diagram reappears in miniature! Γ  Self-similarity

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LOGISTIC MAP, ANALYSIS

Β§ 𝑦KR3 = 𝑠𝑦K(1 βˆ’ 𝑦K), 0 ≀ 𝑠 ≀ 4, 0 ≀ 𝑦 ≀ 1, Fixed points and stability? Β§ Fixed points, are roots of: π‘¦βˆ— = 𝑔(π‘¦βˆ—) = π‘ π‘¦βˆ—(1 βˆ’ π‘¦βˆ—) Γ  π‘¦βˆ— = 0, π‘¦βˆ— = 1 βˆ’

3 y

Β§ Since 𝑦 β‰₯ 0, π‘¦βˆ— is in the range of allowable values only if 𝑠 β‰₯ 1 Β§ Stability depends on multiplier πœ‡ = 𝑔T π‘¦βˆ— = 𝑠 βˆ’ 2π‘ π‘¦βˆ— Β§ π‘¦βˆ— = 0: 𝑔T π‘¦βˆ— = 0 = 𝑠 Γ  Origin is stable for 𝑠 < 1, unstable for 𝑠 > 1 Β§ π‘¦βˆ— = 1 βˆ’

3 y : 𝑔T 1 βˆ’ 3 y

= 2 βˆ’ 𝑠 Γ  1 βˆ’

3 y is stable for 1 < 𝑠 < 3,

unstable for 𝑠 > 3

Β§ For 𝑠 = 1, a second fixed point appears, while the origin loses its stability Β§

  • Γ  Transcritical bifurcation at 𝑠 = 1

Β§ When the slope of the parabola at π‘¦βˆ— = 0 becomes too steep, the origin loses its stability (it happens at 𝑠 = 3) Β§ Γ  Flip bifurcation at 𝑠 = 3, that are (usually) associated with period doubling and in this case a 2-period cycle is spawn

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ANALYSIS: SPAWNING OF TWO-CYCLE

Β§ The logistic map has a two-cycle for all 𝑠 > 3 Β§ Period-2 cycle: there are two states π‘ž and π‘Ÿ, such that : Β§ 𝑔 π‘ž = π‘Ÿ, 𝑔 π‘Ÿ = π‘ž, or equivalently, Β§ 𝑔 𝑔 π‘ž = π‘ž Β§ Γ  π‘ž (and π‘Ÿ) fixed points of second-iterate map, 𝑔2(𝑦) ≑ 𝑔 𝑔 𝑦 Β§ 𝑔2(𝑦) is a quartic polynomial, that for 𝑠 > 3 looks like: Β§ π‘ž, π‘Ÿ corresponds to where the graph of 𝑔2(𝑦) intersects the diagonal: 𝑔2 𝑦 = 𝑦 Β§ … π‘ž,π‘Ÿ =

yR3 (yLY)(yR3) 2y

, real for 𝑠 > 3 Β§ Γ  A two-cycle exists for all 𝑠 > 3 Β§ At 𝑠 = 3, the two-cycle bifurcates continuously from π‘¦βˆ—

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FLIP BIFURCATIONS AND PERIOD DOUBLING

Β§ If tangent slope, 𝑔T π‘¦βˆ— β‰ˆ βˆ’1 and the graph of the function is concave near π‘¦βˆ—, the cobweb tends to produce a small, stable 2-cycle around the fixed point Β§ The critical slope β‰ˆ βˆ’1 corresponds to a flip bifurcation that gives rise two a 2-cycle Β§ How can we determine that the 2-cycle is stable or not? Β§ π‘ž, π‘Ÿ are the solutions of 𝑔2 𝑦 = 𝑦 Γ  The 2-cycle determined by π‘ž, π‘Ÿ is stable iff π‘ž,π‘Ÿ stable both stable fixed points of the 𝑔2 map Β§ Doing the usual analysis … for both π‘ž,π‘Ÿ β†’ πœ‡ = 4 βˆ’ 2𝑠 βˆ’ 𝑠2 Β§ Γ  The 2-cycle is stable iff 4 βˆ’ 2𝑠 βˆ’ 𝑠2 < 1 Γ  𝑠 < 1 + 6

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A FIRST BIFURCATION DIAGRAM …

Β§ The dashed lines indicate fixed points that are instable Β§ The first bifurcation is a flip one, that creates a new equilibrium, losing the stability of the original one Β§ Each further pitchfork bifurcation is a supercritical one, with two new stable equilibrium points appearing and the original equilibrium losing its stability.