15-382 C OLLECTIVE I NTELLIGENCE – S19 L ECTURE 3: D YNAMICAL S YSTEMS 2 T EACHER : G IANNI A. D I C ARO
G ENERAL 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: * ⊆ "×! → ! Φ 0, $ = $ 1 (the initial condition) Ø Φ & 2 , 3 & 4 , $ = Φ(& 2 + & 4 , $) , (property of states) Ø for & 4 , & 4 +& 2 ∈ 6($) , & 2 ∈ 6(Φ(& 4 $)) , 6 $ = {& ∈ " ∶ (&, $) ∈ *} The evolution function Φ provides the system state (the value ) at time & Ø for any initial state $ 1 : ; = {Φ &, $ ∶ & ∈ 6 $ } orbit (flow lines) of the system through $ , Ø starting in $ , the set of visited states as a function of time: $(&) 2
T YPES OF D YNAMICAL 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
T YPES OF D YNAMICAL 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
F LOWS VS . I TERATED M APS Laminar (streamline) flow: No cross-currents or swirls, individual trajectories flow on parallel lines, do not intersect Turbulent flow: Individual trajectories can intersect 5
C ONTINUOUS -T IME D YNAMICAL 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
̇ ̇ C ONTINUOUS -T IME D YNAMICAL 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 2& 1 3 ", & = 2 1 2 1 1 2" 1 3 ", & 0 1 7
D ISCRETE -T IME D YNAMICAL 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
F ROM 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
̇ ̇ C ONTINUOUS - TIME DS: V ECTOR F IELDS AND O RBITS Uncoupled system § . is a vector field in ℝ 0 : a function associating a vector to 1 -dim point 2 " = 2" = % & (", )) § Solution: " 3 4 56 , ) 3 4 786 ) = −3) = % - (", )) . = (2", −3)) Orbits / Possible trajectories Direction and speed of solution Flow : Φ(:, " : 3 ) for any (", )) 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
̇ ̇ V ECTOR F IELDS , O RBITS , F IXED P OINTS (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
̇ E XAMPLE : L INEAR 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 ! ln ! = 6 7- ln ! − ln ! * = $/ = $/ ! * ! * ! = ! * 6 7- 12
L INEAR MODEL FOR POPULATION GROWTH Phase portrait , = *# #(!) = # & ' () Solution orbits / Flow (in !) : scalar (linear) vector field (a) * > 0: Exponential growth (b) * < 0: Exponential decrease 13
L OGISTIC 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: § ' . τ = )8 (dimensionless time) !0 !1 = 2 1 − 2 2 . = - 2 = ' - ∈ [0,1] (dimensionless population) 14
L OGISTIC MODEL FOR POPULATION GROWTH The logistic equation, even if not linear, can be also integrated by separation § of variables: !" !" !" !# = % 1 − % , " )*" = +τ, " )*" = ∫ +τ ∫ . +% +% ln % − ln 1 − % = τ + 2 % + . 1 − % = . +τ ln 1 − % 1 1 % = 1 + 53 * # % − 1 = 3 * # *4 = − τ − 2 % 1 The integration constant 5 depends on %(τ) = the initial condition % 8 1 + 53 * # < = ; − 9 8 ; 9(:) = 9 8 1 + <3 *=> 15
L OGISTIC MODEL FOR POPULATION GROWTH Equilibrium points: 1 !(τ) = , ! = ! 1 − ! = 0 à ! = 1, ! = 0 1 + () * + ! 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 ) L OGISTIC MODEL : D OES 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) = 6A @=/.B C DEFG //.B = 23.1 1 + <' =>) < = : − ! & : J = 0.031 ! # = ! & : = 189.4 @ !(1900) = 6A @=/.B C DIFFG //.B = 76 189.4 (the baby boom is ! # = à ! 1950 = 144.7 1 + 34.74' =&.&B6) not accounted!) 17
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