1 Dynamic Systems Motions in the system depend on how the state of - - PDF document

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7/11 - 06 1 Emergent Systems, Jonny Pettersson, UmU

Last time

Introduction Fractals NetLogo Assignment 1 Assignment 2

7/11 - 06 2 Emergent Systems, Jonny Pettersson, UmU

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

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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?

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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

• ne state to another

These rules exist whether we know of them or

not

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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 6 Emergent Systems, Jonny Pettersson, UmU

The Logistic Map

(The Quadratic map, The Feigenbaum map) A simple population growth model

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The Logistic Map

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The Logistic Map

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The Logistic Map

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Bifurcation

Bifurcation: When a system goes from fix point to

2 limit cycle or from n-limit cycle and on

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Bifurcation

When r increases the system will pass through

bifurcation after bifurcation at an increasing pace

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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∞

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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

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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

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Strange Attractors

Multidimensional systems The Hénon attractor The Lorenz attractor The Mackey-Glass system

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The Hénon Attractor

Multidimensional system

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The Hénon Attractor – State Space

Strange attractor – fractal properties

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The Hénon Attractor - Bifurcation

Period-7 limit cycle in the middle of chaos

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The Lorenz Attractor

1962, Edward Lorenz studied a simplified model for

convection flow using differential equations

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The Lorenz Attractor - Exemple

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The Mackey-Glass System

Consists of a single delay-differential equation Chaotic systems with nearly infinite complexity may

• ften collapse into a lower-dimensional attractor

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Producer-Consumer Dynamics

Two techniques for modeling populations

Model each species as a simple function of

• ther 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

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Producer-Consumer Interactions

All adaptive systems must possess a form

• f 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

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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

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Predator-Prey Systems

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Generalised Lotka-Volterra Systems

a) α = 0,75 b) α = 1,2 c) α = 1,32 d) α = 1,387 e) α = 1,5

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Individual-Based Ecology

Lotka-Volterra consider all individuals in a

species to be similar

Individual-based

IBM IBS ABM ABS

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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,

• ther implementations

possible

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Ecosystem model – Flow of Resources

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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)

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Ecosystem model

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Ecosystem model - Attractor

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Chaotic Systems

Chaotic systems with

nearly infinite complexity may often collapse into a lower- dimensional attractor

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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 35 Emergent Systems, Jonny Pettersson, UmU

Next time

Cellular Automata