Fundamentals of Dynamical Systems / Discrete-Time …

1 Dr. Dylan Fundamentals of Dynamical Systems / D iscrete- time Models Dynamical Systems theory Considers how Systems autonomouslychange along time Ranges from Newtonian mechanics tomodern nonlinear dynamics theories Probes underlying Dynamical mechanisms,not just static properties of observations Provides a suite of tools useful forstudying complex systemsWhat is a Dynamical system ? A system whose state is uniquelyspecified by a finite set of variablesand whose behavior is uniquelydetermined by predetermined "rules" Simple population growth Simple pendulum swinging Motion of celestial bodies Behavior of two rational agents in anegotiation gameMathematical formulations of Dynamical Systems Discrete-Time model :xt = F(xt-1, t) Continuous- time model : (differential equations) dx/dt = F(x, t) xt: State variable(s) of the system at time t F: Some function that determines the rule that the system s behavior will obey (difference/recurrence equations.)

2 Iterative maps) Discrete-Time Models Discrete-Time model Easy to understand, develop andsimulate Doesn t require an expression for the rateof change (derivative) Can model abrupt changes and/or chaoticdynamics using fewer variables Directly translatable to simulation in acomputer Experimentally, we often have samples ofsystem states at specific points of timeDifference equation and time series Difference equationxt = F(xt-1, t) produces series of values of variable x starting with initial condition x0: { x0, x1, x2, x3, .. } time series A prediction made by the above model (to be compared to experimental data)Linear vs. nonlinear Linear: Right hand side is just a first-orderpolynomial of variablesxt = a xt-1 + b xt-2 + c xt-3.

3 Nonlinear: Anything elsext = a xt-1 + b xt-22 + c xt- 1 xt-3 .. 1st -order vs. higher-order 1st-order: Right hand side refers only to theimmediate pastxt = a xt-1 ( 1 xt-1 ) Higher-order: Anything elsext = a xt-1 + b xt-2 + c xt-3 .. (Note: this is different from the order of terms in polynomials) Autonomous vs. non-autonomous Autonomous: Right hand side includes only statevariables (x) and not t itselfxt = a xt-1 xt-2 + b xt-32 Non-autonomous: Right hand side includes terms thatexplicitly depend on the value of txt = a xt-1 xt-2 + b xt-32 + sin(t) Things that you should know (1) Non-autonomous, higher-orderequations can always be convertedinto autonomous, 1st-order equations xt-2 yt-1, yt = xt-1 t yt, yt = yt-1 + 1, y0 = 0 Autonomous 1st-order equations cancover dynamics of any non-autonomoushigher-order equations too!

4 Things that you should know (2) Linear equations are analytically solvable show either equilibrium, exponentialgrowth/decay, periodic oscillation (with>1 variables), or their combination Nonlinear equations may show more complex behaviors do not have analytical solutions in generalSimulating Discrete-Time Models Simulating Discrete-Time models Simulation of a Discrete-Time modelcan be implemented by iteratingupdating of the system s states Every iteration represents one discretetime stepExercise Implement simulators of the followingmodels and produce time series for t= 1~10xt = 2 xt-1 + 1, x0 = 1 xt = xt-12 + 1, x0 = 1 Exercise Simulate the following set ofequations and see what happens if thecoefficients are variedxt = xt-1 + 1 yt-1yt = xt-1 + 1 yt-1x0 = 1.

5 Y0 = 1 Building Your Own model Equation Mathematical modeling tips Grab an existing model and tweak it Implement each assumption one by one Find where to change, replace it by afunction, and design the function Adopt the simplest form Check the model with extreme valuesExample: Saturation of growth Simple exponential growth model :xt = a xt-1 Problem: How can one implement thesaturation of growth in this model ? Think about a new nonlinear model :xt = f(xt-1) xt-1 Coefficient replaced by a function of xModeling saturation of growth xt = f(xt-1) xt-1 f(x) should approach 1 (no netgrowth) when x goes to a carrying capacity of the environment, say K f(x) should approach the originalgrowth rate a when x is very small ( , with no saturation effect) What should f(x) be?

6 F(x) x K 1 a 0 f(x) = x + aa 1 K A new model of growth xt = f(xt-1) xt-1 = ( (a 1) xt-1 / K + a ) xt-1 Using r = a 1:xt = ( r xt-1 / K + r + 1 ) xt-1 = xt-1 + r xt-1 ( 1 xt-1 / K ) Net growthExample: Logistic growth model N: Population r: Population growth rate K: Carrying capacity Discrete-Time version:Nt = Nt-1 + r Nt-1 ( 1 Nt-1/K ) Continuous- time version:dN/dt = r N ( 1 N/K ) Nonlinear terms Problem: Develop a nonlinear model ofa simple ecosystem made of predatorand prey populationsModeling with multiple variables Think about how variables behave in isolation Naturally grows to carrying capacity if isolated + Naturally decays if isolated - Rabbit Population : xFox Population : yInitial assumptions Rabbits will grow based on the logisticgrowth model , with carrying capacity= 1 for simplicity Foxes will decay exponentiallyRabbit: xt = xt-1 + a xt-1 (1 xt-1) Fox.

7 Yt = b yt-1(0<a, 0<b<1) Naturally grows to carrying capacity if isolated Naturally decays if isolated Think about how variables interact with each other + - Positive influence Foxes growth rate increases with increasing rabbits + Negative influence Rabbits survival rate decreases with increasing foxes - Rabbit Population : x Fox Population : yRevised model Introduced coefficient (1 c yt-1) (0<c) tothe first term of x Negative influence of foxes on rabbits survivalrate Replaced b with (b + d xt-1) (0<d) Positive influence of rabbits on foxes growthrateRabbit: xt = (1 c yt-1) xt-1+ a xt-1 (1 - xt-1)Fox: yt = (b + d xt-1) yt-1 (0<a, 0<b<1, 0<c, 0<d) FYI: Lotka-Volterra model This model can be rewritten as.

8 Xt xt-1 = xt-1 (1-xt-1) xt-1 yt-1 yt yt-1 = - yt-1 + xt-1 yt-1 Known as the Lotka-Volterra equations (ofdiscrete- time version with carrying capacity) Models predator-prey dynamics in a generalform One of the most famous nonlinear Systems withmultiple variablesAnalysis of Discrete-Time Models Equilibrium point A state of the system at which statewill not change over time fixed point, steady state Can be calculated by solvingxt = xt-1 Exercise Calculate equilibrium points in thefollowing modelsNt = Nt-1 + r Nt-1 ( 1 Nt-1/K ) xt = 2xt-1 xt-12 xt = xt-1 xt-22 + 1 Phase Space Visualization Geometrical approach Developed in the late 19 Cby J.

9 Henri Poincare Visualizes the behavior ofdynamical Systems astrajectories in a phase space Produces a lot of intuitive insights ongeometrical structure of dynamicsthat would be hard to infer usingpurely algebraic methodsPhase space (state space) A theoretical space in which everystate of a Dynamical system ismapped to a spatial locationPhase space (state space) Created by orthogonalizing stat evariables of the system Its dimensionality equals # ofvariables needed to specify thesystem state ( degrees offreedom) Temporal change of the system statescan be drawn in it as a trajectoryAttractor and basin of attraction Attractor:A state (or a set of states) fromwhich no outgoing edges or flowsrunning in phase space Static attractors (equilibrium points) Dynamic attractors ( limit cycles) Basin of attraction.

10 A set of states which will eventuallyend up in a given attractor a simple vertical spring oscillator State can be specified by two realvariables (location x, velocity v)v x Trajectory (orbit) Dynamics of continuous models can be depicted as flow in a continuous phase spacePhase space of continuous-state models Cobweb plot A visual tool to study the behavior of1-D iterative maps Take xt-1 and xt for two axes Draw the map of interest (xt=F(xt-1))and the xt=xt-1 reference line They will intersect at equilibrium points Trace how time series develop froman initial value by jumping betweenthese two curvesCobweb Plot Cobweb PlotRescaling Variables Rescaling variables Dynamics of a system won t changequalitatively by linear rescaling ofvariables ( , x x ) You can set arbitrary rescalingfactors for variables to simplify themodel equations If you have k variables, you mayeliminate k parametersExercise Simplify the logistic growth model byrescaling x x xt = xt-1 + r xt-1 (1 - xt-1/K)