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 orbitsVector field on ℝ-$ = (20, −34)

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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

101

212&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 = ;

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FROM LOCAL RULES TO GLOBAL BEHAVIORS?

FlowsMaps

∆" = 1, when ∆" → 0à '"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?

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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))Directionand speedof solution

for any

(", ))

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VECTOR FIELDS, ORBITS, FIXED POINTS

"̇ = $ = %& (", $)$̇ = −" − $+ = %, (", $)

Closed (periodic) orbitEquilibrium 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

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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:

+,+- = $!, +,

, = $./, 0,1

,+,, = $ ∫-1

- ./

ln ! − ln !* = $/

! = !*67-

ln !!*= $/

!!*= 67-

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LINEAR MODEL FOR POPULATION GROWTH

Phase portrait

Solution orbits / Flow (in !): #(!) = #&'()

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

, = *#scalar (linear) vector field

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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

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LOGISTIC MODEL FOR POPULATION GROWTH

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

!"!# = % 1 − % , !"

" )*" = +τ, ∫ !"" )*" = ∫+τ

.+%% + . +%1 − % = .+τ ln % − ln 1 − % = τ+ 2

ln 1 − %% = −τ− 2 1% − 1 = 3*#*4 1

% = 1 + 53*#

%(τ) = 11 + 53*#

The integration constant 5 depends on the initial condition %8

9(:) = ;1 + <3*=>

< = ; − 9898

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LOGISTIC MODEL FOR POPULATION GROWTH

!(τ) = 11 + ()*+

(=1

, ! = ! 1 − ! = 0à ! =1, ! = 0Equilibrium points:

Flow, different ( values

!!

! =1

! = 0

Phase portrait

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

Asymptotic divergence

! = 0

! = 1

!

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(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 1950Answer: 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 àWe 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!)

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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