Last time � Introduction � Fractals � NetLogo � Assignment 1 � Assignment 2 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 1 Outline for today � Nonlinear dynamic systems � The Logistic map � Strange attractors � The Hénon attractor � The Lorenz attractor � Producer-consumer dynamics � Equation-based modeling � Individual-based modeling 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 2 Dynamic Systems � A loose definition: � Anything that has motion � Questions � What is it that changes over time? � What rules governs how a dynamical system changes over time? 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 3 1
Dynamic Systems � Motions in the system depend on how the state of the system changes over time � If the system is deterministic, a set of rules governs how a system changes from one state to another � These rules exist whether we know of them or not 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 4 Type of Motions � Fixed point behavior � Limit cycle or periodic motion � Quasiperiodic motion � Similar to periodic motion, but never quite repeat itself � Chaos � Very common � Predictable in the short term 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 5 The Logistic Map � (The Quadratic map, The Feigenbaum map) � A simple population growth model 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 6 2
The Logistic Map 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 7 The Logistic Map 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 8 The Logistic Map 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 9 3
Bifurcation � Bifurcation: When a system goes from fix point to 2 limit cycle or from n-limit cycle and on 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 10 Bifurcation � When r increases the system will pass through bifurcation after bifurcation at an increasing pace 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 11 Bifurcation � Bifurcation rate = Feigenbaum constant � d ∞ = 4.669202… � Valid for one- dimensional maps that have a single bump � One can get an accurate estimate for a ∞ 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 12 4
Prediction of Chaotic Systems � Truly stochastic processes can only be characterized statistically � Chaotic processes can be predicted in short term � Long-term prediction of chaotic processes becomes more difficult the further one look � Limited precision in computers � Measurement errors � Irrational numbers � Repeating in computer simulations 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 13 Characteristics of Chaos � Deterministic � Sensitive � Sensitive to initial conditions � Easier to control � Ergodic � Embedded � There are an infinite number of limit cycles embedded within the chaotic attractor 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 14 Strange Attractors � Multidimensional systems � The Hénon attractor � The Lorenz attractor � The Mackey-Glass system 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 15 5
The Hénon Attractor � Multidimensional system 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 16 The Hénon Attractor – State Space � Strange attractor – fractal properties 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 17 The Hénon Attractor - Bifurcation � Period-7 limit cycle in the middle of chaos 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 18 6
The Lorenz Attractor � 1962, Edward Lorenz studied a simplified model for convection flow using differential equations 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 19 The Lorenz Attractor - Exemple 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 20 The Mackey-Glass System � Consists of a single delay-differential equation � Chaotic systems with nearly infinite complexity may often collapse into a lower-dimensional attractor 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 21 7
Producer-Consumer Dynamics � Two techniques for modeling populations � Model each species as a simple function of other species – Equation-based modeling � Model each individual and simulate all individuals simultaneously – Individual-based modeling � Both method give similar results � Complexity out of simplicity � Simplicity out of complexity 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 22 Producer-Consumer Interactions � All adaptive systems must possess a form of fluidity � Internal changes as a response to external changes in the environment � Stability in an adaptive system is easy when the state of an environment is mostly independent of the state of an individual � Much harder if the states are recursively dependent on each other 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 23 Predator-Prey Systems � The simplest predator-prey system � Two species, one eating the other � Very idealized � Alfred J. Lotka and Vito Volterra independently noticed the cyclic nature of population dynamics and set out to describe the phenomenon mathematically 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 24 8
Predator-Prey Systems 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 25 Generalised Lotka-Volterra Systems a) α = 0,75 b) α = 1,2 c) α = 1,32 d) α = 1,387 e) α = 1,5 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 26 Individual-Based Ecology � Lotka-Volterra consider all individuals in a species to be similar � Individual-based � IBM � IBS � ABM � ABS 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 27 9
Ecosystem model � A grid, width x height � Each point can have one creature or be empty � Three types of creatures � Plant, grows in empty points � Herbivore, eats plants, moves around � Carnivore, eats herbivores, moves around � In this case implemented with cellular automaton, other implementations possible 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 28 Ecosystem model – Flow of Resources 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 29 Ecosystem model - Algorithm � Table 12.1 � Not completely deterministic � Random update order for animals � Random choices when multiple options � Complexity � Individual-based vs. Lotka-Volterra (Equation- based) 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 30 10
Ecosystem model 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 31 Ecosystem model - Attractor 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 32 Chaotic Systems � Chaotic systems with nearly infinite complexity may often collapse into a lower- dimensional attractor 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 33 11
Summary � Nonlinear dynamic systems � The Logistic map � Strange attractors � The Hénon attractor � The Lorenz attractor � Producer-consumer dynamics � Equation-based modeling � Individual-based modeling 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 34 Next time � Cellular Automata 7/11 - 06 Emergent Systems, Jonny Pettersson, UmU 35 12
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