L ECTURE 13: D ISCRETE -T IME D YNAMICAL S YSTEMS 1 TEACHER : G IANNI - - PowerPoint PPT Presentation

LECTURE 13: DISCRETE-TIME DYNAMICAL SYSTEMS 1

TEACHER:

GIANNI A. DI CARO

15-382 COLLECTIVE INTELLIGENCE – S19

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RECAP: GENERAL DEFINITION OF DYNAMICAL SYSTEMS

§ ! is a set of all possible states of the dynamical system (the state space) § " is the set of values the time (evolution) parameter can take § Φ is the evolution function of the dynamical system, that associates to each $ ∈ ! a unique image in ! depending on the time parameter &, (not all pairs (&, $) are feasible, that requires introducing the subset *) Φ: * ⊆ "×! → ! Ø Φ 0, $ = $1 (the initial condition) Ø Φ &2, 3 &4, $ = Φ(&2 + &4, $), (property of states) for &4, &4+&2 ∈ 6($), &2 ∈ 6(Φ(&4$)), 6 $ = {& ∈ " ∶ (&, $) ∈ *} Ø The evolution function Φ provides the system state (the value) at time & for any initial state $1 Ø :; = {Φ &, $ ∶ & ∈ 6 $ } orbit (flow lines) of the system through $, starting in $ , the set of visited states as a function of time: $(&)

A dynamical system is a 3-tuple ", !, Φ :

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RECAP: TYPES OF DYNAMICAL 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 !" = !(0) ∈ (, a deterministic trajectory ! ) , ) ∈ + !" , is produced by ,, (, Φ § States can be “anything” mathematically well-behaved that represent situations of interest § The nature of the set . and of the function / give raise to different classes of dynamical systems (and resulting properties and trajectories)

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TYPES OF DYNAMICAL SYSTEMS

§ Continuous time dynamical systems (Flows): ! open interval of ℝ, Φ continous and differentiable function à Differential equations § Φ represents a flow, defining a smooth (differentiable) continuous curve § The notion of flow builds on and formalizes the idea of the motion of particles in a fluid: it can be viewed as the abstract representation of (continuous) motion of points over time. § Discrete-time dynamical systems (Maps): ! interval of ℤ, Φ a function § Φ, represents an iterated map, which is not a flow (a differentiable curve) anymore, since the trajectory is a discrete set of points § Trajectory is represented through linear interpolation and it can easily present large slope changes at the points (e.g., cuspids)

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DISCRETE-TIME DYNAMICAL SYSTEMS

§ Discrete-time dynamical systems (Maps): ! interval of ℤ, Φ a function § The iterated map Φ is generated by a set of recurrence equations $ on % ⊆ ℝ( (also referred to as difference equations) § The orbits )(+) are sets of discrete points resulting from the closed-form solution (not always achievable) of the recurrence equations § Example with one single recurrence equation:

• ( = /(-(01, -(03, … , -(05)

§ Order-6 Markov states: relevant state information includes all past 6 states § Well-known example: Fibonacci recurrence equation §

• ( = -(01 + -(03

§ Initial condition (that uniquely determines the orbit): -8 = 9, -1 = :

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FROM LOCAL RULES TO GLOBAL BEHAVIORS?

Flows Maps ∆" = 1, when ∆" → 0 à '" à Differential eq. '( '" = )((, ")

• . = )(-./0, -./1, … , -./3)

§ 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 ((") § In maps time step is finite and discrete, 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?

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MAPS

§ 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: ̇ " = $(") à "'() = "' + ℎ$("'), …Runge-Kutta,… § Discretization of algebraic equations: § Newton’s method for solving , - = 0 à Expand in Taylor series near

• ': , - = , -' + - − -' ,1 -' + ⋯ taking the usual linear

approximation:, - ≈ , -' + - − -' ,1 -' , equating to 0:

• '() = -' − , -' /,1 -'

§ 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

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MAPS: SAME TERMINOLOGY AS IN FLOWS

§ ! " = 2", ! is a map § The orbit of " under the map ! is the set of points: {", ! " , ! ! " , ! ! ! " , … } = {", ! " , !) " , !* " , … } corresponding to the iterated application of the map § The initial point provides the initial conditions § A point "∗, such that ! "∗ = "∗ is a fixed point, the orbits remain in "∗ for all future iterations Cobweb plots for individual orbits

!(")

! " = 2"(1 − ")

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FIXED, EQUILIBRIUM POINTS

§ Starts at 1.6, converges to 1 § Starts at 1.8, converges to -1 Stability of a fixed point !∗? § Fixed points correspond to intersection between graph #(%) and % § General map: !'() = +(!'), +(!∗) = !∗ § Let’s consider a near orbit, !' = !∗ + -': is the orbit attracted or repelled from !∗? If it’s attracted we can say that !∗ is stable § Does the perturbation -' grow or decay with /? § By the definition, !∗ + -'() = !'() = +(!∗ + -'), and using the Taylor series expansion about !∗: !∗ + -'() = !'() = + !∗ + -' = + !∗ + +0 !∗ -' + 1(-'2) § Given that +(!∗) = !∗ à -'() = +0 !∗ -' + 1(-'2) § If we take the linear approximation à Linearized map: -'() = +0 !∗ -' § Eigenvalue / multiplier: 3 = +0 !∗

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FIXED POINTS

§ Linearized map: !"#$ = &' (∗ !" § Solution of linearized map: !$ = *!+, !- = *!$ = *-!+ ….. !" = *"!+ § If * = |&' (∗ | < 1 à !" → 0, for 3 → ∞, and (∗ is linearly stable § If * = |&' (∗ | > 1 à !" → ∞, for 3 → ∞, and (∗ is unstable § The linear stability holds also for the general map § The marginal case * = |&' (∗ | = 1 doesn’t allow to draw conclusions. In this case the quadratic term 6(!"-) determines the stability § If * =0, then the fixed point is said superstable § ("#$ = sin (" § (∗ = 0 is a fixed point § * = &' (∗ = 1, marginal case § Cobweb à It’s stable! § Is it global? For all orbits (" → 0? § For any (+, ($∈ [−1,1] since |sin ($| < 1 § à From cobweb we can say it’s global

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ANOTHER EXAMPLE, LIMITING BEHAVIOR

§ !"#$ = cos !" § lim

"→-!"? … by iterating the map (e.g., use calculator!), !" → 0.739..

§ Solution of trascendental equation ! = cos ! § The fixed point 0.739… has . < 0 à Damped oscillations § For 0 < . < 1 convergence to a stable fixed point is monotonic

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LOGISTIC MAP

§ !"#$ = &!"(1 − !") § !" is a dimensionless measure of the population in the +-th generation and & is the intrinsic growth rate (with capacity being limited to 1)

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!" § Let’s restrict 0 ≤ & ≤ 4 à Then map maps 0,1 → 0,1 § Let’s fix & and study the evolution § Trivially, for small growth rates, & < 1, the population always goes extinct, as !" → 0 § For 1 < & < 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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REGULAR BEHAVIOR, PERIODIC ATTRACTORS

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REGULAR BEHAVIOR, PERIODIC ATTRACTORS

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TRANSITION TO CHAOTIC BEHAVIOR

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CHAOS: SENSITIVITY TO INITIAL CONDITIONS