15-382 COLLECTIVE INTELLIGENCE – S19

LECTURE 3: DYNAMICAL SYSTEMS 2

TEACHER: GIANNI A. DI CARO GENERAL DEFINITION OF DYNAMICAL SYSTEMS

A dynamical system is a 3-tuple ", !, Φ : § ! 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: $(&) 2 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)

3 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)

4 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 5 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)

Vector field on ℝ- Flow orbits $ = (20, −34)

6 CONTINUOUS-TIME DYNAMICAL SYSTEMS

§ Ordinary Differential Equations (system of ODEs)

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

,- § Partial Differential Equations, accounting for space and time 1 21 21 3 ", & = 3 ", & 01 2&1 2"1

7 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 = ; 8 FROM LOCAL RULES TO GLOBAL BEHAVIORS?

'( = )((, ") Flows Maps -. = )(-./0, -./1, … , -./3) '" ∆" = 1, when ∆" → 0 à '" 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 ((") § 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? 9 CONTINUOUS-TIME DS: VECTOR FIELDS AND ORBITS

Uncoupled system § . is a vector field in ℝ0: a function "̇ = 2" = %& (", )) associating a vector to 1-dim point 2 56 786 )̇ = −3) = %- (", )) § Solution: "34 , )34

. = (2", −3)) Orbits / Possible trajectories Direction and speed of solution Φ(:, " : ) for any Flow: 3 (", )) Vector field Phase portrait Rate of change, velocity § 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) 10 VECTOR FIELDS, ORBITS, FIXED POINTS

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

(Rescaled) vector field Closed (periodic) orbit Equilibrium point

Direction of increasing time

§ -∗ is an equilibrium (fixed) point of the ODE if / -∗ = 0 § ↔ Once in "∗, the system remains there: -∗ = - 2; -∗ , 2 ≥ 0 11 EXAMPLE: LINEAR MODEL FOR POPULATION GROWTH

!̇ = $! § Linear model of population growth (Malthus model, 1798) § Works well for small populations !(0) = !* § ! = size of population, $ = growth rate

§ This linear equation can easily be integrated by separation of variables: , +, +, +, - = $!, = $./, 0 = $ ∫ ./ +- , , -1 ,1 ! ! 7- ln ! − ln !* = $/ ln = $/ = 6 !* !* 7- ! = !*6

12 LINEAR MODEL FOR POPULATION GROWTH

Phase portrait , = *# () Solution orbits / Flow (in !): #(!) = #&' scalar (linear) vector field

(a) * > 0: Exponential growth (b) * < 0: Exponential decrease

13 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: ) = intrinsic rate of increase [1/t] !" " = )' 1 − - = max carrying capacity [# individuals] !# , '. = '(0)

§ à Non-dimensional equation with no parameters: !0 ' τ = )8 (dimensionless time) = 2 1 − 2 2 = . ' !1 . 2 = ∈ [0,1] - - (dimensionless population) 14 LOGISTIC MODEL FOR POPULATION GROWTH

§ The logistic equation, even if not linear, can be also integrated by separation of variables: !" !" !" = % 1 − % , = +τ, ∫ = ∫ +τ !# " )*" " )*"

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

1 The integration constant 5 depends on %(τ) = 1 + 53*# the initial condition %8

; ; − 98 9(:) = < = 1 + <3*=> 98 15 LOGISTIC MODEL FOR POPULATION GROWTH

1 Equilibrium points: !(τ) = 1 + ()*+ , ! = ! 1 − ! = 0 à ! =1, ! = 0 !

Phase ! = 1 ! portrait ! = 0 ! =1

! = 0

Asymptotic divergence

Flow, different ( values (=1

Flow function /(0, !2) is not defined for all values of 0 16 (BASIC) LOGISTIC MODEL: DOES IT WORK?

§ Population of the US in 1800: 5.3 millions à Predict population in 1900 and 1950 § Population of the US in 1850: 23.1 millions 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:

@ !(1850) = = 23.1 6A @=/.B CDEFG//.B : : − !& J = 0.031 ! # = =>) < = 1 + <' ! @ & !(1900) = = 76 : = 189.4 6A @=/.B CDIFFG//.B 189.4 ! # = (the baby boom is 1 + 34.74'=&.&B6) à ! 1950 = 144.7 not accounted!) 17 LOGISTIC MODEL VS. EXPONENTIAL GROWTH

● real population values in the US ▬ Logistic model predictions

Exponential explosion Logistic asymptote ● real population values in the US ▬ Logistic model predictions

Little difference for small populations

Both linear and logistic model work well 18