L ECTURE 3: D YNAMICAL S YSTEMS 2 T EACHER : G IANNI A. D I C ARO G - - PowerPoint PPT Presentation

LECTURE 3: DYNAMICAL SYSTEMS 2

TEACHER: GIANNI A. DI CARO

15-382 COLLECTIVE INTELLIGENCE – S19

2

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 :

3

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 Φ give raise to different classes of dynamical systems (and resulting properties and trajectories)

4

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)

5

FLOWS VS. ITERATED MAPS

Laminar (streamline) flow: No cross-currents or swirls, individual trajectories flow on parallel lines, do not intersect Turbulent flow: Individual trajectories can intersect

6

CONTINUOUS-TIME DYNAMICAL SYSTEMS

§ Continuous-time dynamical systems (Flows): ! open interval of ℝ, Φ continous and differentiable function § If the flow Φ is generated by a vector field $ on % ⊆ ℝ', then the orbits ((*) of the flow are the images of the integral curves of the vector field § Vector field on a ,-dim space %: assignment of a ,-dim vector to each point of the space, the vector defines a direction and a velocity in the point (that the field would exert on a point-like particle in the point) Flow orbits Vector field on ℝ-

$ = (20, −34)

7

CONTINUOUS-TIME DYNAMICAL SYSTEMS

§ Delay models, past state is determining present state ̇ " = $(" & − ( ) § Integro-Differential Equations, accounting for history ̇ " = $ " & + +

,- ,

$ " ( .( § Partial Differential Equations, accounting for space and time 1 01 21 2&1 3 ", & = 21 2"1 3 ", & § Ordinary Differential Equations (system of ODEs)

8

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 § Note that integro-differential equations are in principle order-∞ Markov, since infinite states from the past affect current state § Another, well-known example: Fibonacci recurrence equation §

• ( = -(01 + -(03

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

9

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 ((") § 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?

10

CONTINUOUS-TIME DS: VECTOR FIELDS AND ORBITS

̇ " = 2" = %

& (", ))

̇ ) = −3) = %

• (", ))

§ . is a vector field in ℝ0: a function associating a vector to 1-dim point 2 § Solution: "3456, )34786 Vector field Rate of change, velocity Phase portrait § Autonomous system à no explicit dependence from time in ., all information about the solution is represented

§ A fundamental theorem guarantees (under differentiability and continuity assumptions) that two orbits corresponding to two different initial solutions never intersect with each other (laminar flow)

Orbits / Possible trajectories Flow: Φ(:, " :3 ) Uncoupled system

. = (2", −3))

Direction and speed

• f solution

for any (", ))

11

VECTOR FIELDS, ORBITS, FIXED POINTS

̇ " = $ = %

& (", $)

̇ $ = −" − $+ = %

, (", $)

Closed (periodic) orbit Equilibrium point §

• ∗ is an equilibrium (fixed) point of the ODE if / -∗ = 0

§ ↔ Once in "∗, the system remains there: -∗ = - 2; -∗ , 2 ≥ 0 Direction of increasing time

/ = ($, −" − $+) E.g., /(1,1) = (1, −2)

(1,1) (1, −2) (Rescaled) vector field

12

EXAMPLE: LINEAR MODEL FOR POPULATION GROWTH

§ Linear model of population growth (Malthus model, 1798) § Works well for small populations § ! = size of population, $ = growth rate

̇ ! = $! !(0) = !*

§ This linear equation can easily be integrated by separation of variables:

+, +- = $!, +, , = $./, ,1 ,+, , = $ ∫

• 1
• ./

ln ! − ln !* = $/ ! = !*67- ln ! !* = $/ ! !* = 67-

13

LINEAR MODEL FOR POPULATION GROWTH

Phase portrait Solution orbits / Flow (in !):

#(!) = #&'()

(a) * > 0: Exponential growth (b) * < 0: Exponential decrease , = *# scalar (linear) vector field

14

LOGISTIC MODEL FOR POPULATION GROWTH

§ General form for population growth:

!" !# = %(')

§ What is a good model that captures essential aspects? ü Every living organism must have at least one parent of like kind ü In a finite space, due to the limiting effect of the environment, there is an upper limit to the number of organisms that can occupy that space: resources competition constraint § à Logistic model (1838), non-linear:

!" !# = )' 1 − " , ) = intrinsic rate of increase [1/t]

• = max carrying capacity [# individuals]

'. = '(0)

§ à Non-dimensional equation with no parameters:

!0 !1 = 2 1 − 2

• 2. =

'.

• (dimensionless time)

2 = '

• ∈ [0,1] (dimensionless population)

τ = )8

15

LOGISTIC MODEL FOR POPULATION GROWTH

§ The logistic equation, even if not linear, can be also integrated by separation

• f variables:

!" !# = % 1 − % , !" " )*" = +τ,

∫

!" " )*" = ∫ +τ

. +% % + . +% 1 − % = . +τ

ln % − ln 1 − % = τ + 2 ln 1 − % % = −τ − 2 1 % − 1 = 3*#*4 1 % = 1 + 53*# %(τ) = 1 1 + 53*#

The integration constant 5 depends on the initial condition %8

9(:) = ; 1 + <3*=> < = ; − 98 98

16

LOGISTIC MODEL FOR POPULATION GROWTH

!(τ) = 1 1 + ()*+

(=1

, ! = ! 1 − ! = 0 à ! =1, ! = 0 Equilibrium points: Flow, different ( values

! ! ! =1 ! = 0

Phase portrait Flow function /(0, !2) is not defined for all values of 0 Asymptotic divergence

! = 0 ! = 1 !

17

(BASIC) LOGISTIC MODEL: DOES IT WORK?

§ Population of the US in 1800: 5.3 millions § Population of the US in 1850: 23.1 millions à Predict population in 1900 and 1950 Answer: 76 (1900), 150.7 (1950) § Let’s look first at what the linear (i.e., exponential growth) model would predict: !(#) = !&'() à We need first to derive an estimate for growth parameter *: !(1850) = !(1800)'() à 23.1 = 5.3'/&( à * = 0.29 ! 1900 = ! 1800 '&.3456&& = 100.7 ! 1950 = ! 1800 '&.3456/& = 438.8 § The non-linear (i.e., logistic growth) model in the dimensional form has two parameters à We need more information: let’s assume we know the 1900 answer: ! # = : 1 + <'=>) < = : − !& !& !(1850) =

@ 6A @=/.B CDEFG//.B = 23.1

!(1900) =

@ 6A @=/.B CDIFFG//.B = 76

J = 0.031 : = 189.4

! # = 189.4 1 + 34.74'=&.&B6)

à ! 1950 = 144.7 (the baby boom is not accounted!)

18

LOGISTIC MODEL VS. EXPONENTIAL GROWTH

• real population values in the US

▬ Logistic model predictions

• real population values in the US

▬ Logistic model predictions Little difference for small populations Both linear and logistic model work well Logistic asymptote Exponential explosion