15-382 collective intelligence –s19gdicaro/15382/slides/... · 4 typesofdynamicalsystems...
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LECTURE 3: DYNAMICAL SYSTEMS 2
TEACHER:GIANNI A. DI CARO
15-382 COLLECTIVE INTELLIGENCE – S19
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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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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)
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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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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
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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)
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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
§ 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