SWEN90004
Modelling Complex Software Systems
Lecture Cx.02 Dynamical Systems and Chaos
Artem Polyvyanyy, Nic Geard
artem.polyvyanyy@unimelb.edu.au;
nicholas.geard@unimelb.edu.au
Semester 1, 2021
1 / 31
Overview
In this lecture, you will learn about:
using mathematics to model a system’s behaviour
two mathematical models of population growth (exponential and logistic)
chaotic behaviour in (discrete) systems
2 / 31
Outline
Dynamical systems
Population growth
Chaos
3 / 31
Recap: what is a dynamical system?
a set of possible states (the state space)
time t (which we may treat as discrete or continuous)
a rule that determines the state at the present time (xt) in
terms of states at earlier times (xt−1, xt−2, . . .)
For now, we will require this rule to be deterministic, so that the history of the system (it’s past state) uniquely determines the present state.
also, an initial condition x0 : the state that the system is in when t = 0
4 / 31
Functions and iteration
A function takes a number as an input, does something to it, and produces a number as an output; eg,
f (x) = 3x
The function f takes x as an input, multiplies it by 3 and returns
the resulting value as an output
Iteration is simply using the output of the previous application of a function as the input for the next application:
x0,f(x0),f(f(x0)),f(f(f(x0))),… x0,x1,x2,x3,…,xt
What happens to the state of a system as time progresses?
5 / 31
Outline
Dynamical systems
Population growth
Chaos
6 / 31
Red foxes in Australia
Red foxes were introduced to Australia by settlers from England in the 1830s
They are now listed among the world’s 100 worst invasive species
http://en.wikipedia.org/wiki/Red_fox http://en.wikipedia.org/wiki/Australia
7 / 31
Red foxes in Australia
Red foxes are an apex predator: they predate on other species (like the bilbies and numbats), but no other species predate on them
FigureBilby FigureNumbat
http://www.murweh.qld.gov.au/bilby-night-talk-and-tour http://www.abc.net.au/news/2011-08-23/numbat-numbers-in-decline-feature/2851582
8 / 31
Red foxes in Australia
A female red fox reproduces once per year, producing around 4 kits (baby foxes)
To keep things simple, let’s make the following assumptions:
the initial population contained 2 female red foxes a female red fox reproduces in its first year of life a female red fox only reproduces once in it’s life half of newborn kits are female
Therefore, the number of female red foxes will double each year: xt+1 = 2xt
where xt is the number of female red foxes alive in year t
9 / 31
Red foxes in Australia
Figurefunction plot Figuretime series plot
What is the general formula for xt?
xt = x02t
The exponential model is often stated as xt = x0rt, where r is a parameter that governs how steeply the curve rises
10 / 31
But…
This model for the growth of the red fox population in Australia predicts that there should be approximately 2 × 1055 red foxes by 2016!
What is missing?
Before we move on to a different model of population growth: in the
model above, we used r = 2. Think about what happens when:
r=1 r<1
11 / 31
Refining our model
Clearly the unlimited growth predicted by the exponential model is unrealistic: the red foxes will eventually run out of food or space
Next, we will consider a model with the additional (more realistic) assumptions that:
when there are few red foxes, there will be plenty of food, and their numbers will grow rapidly
when there are many red foxes, there won’t be enough food to go around, and their numbers will grow more slowly
12 / 31
The logistic model of population growth
Pt+1 = rPt where Pt is the population size at time t
Let us say that A is the population size at which the red foxes eat all available food, leading to starvation and zero red foxes next year
Pt Pt+1=rPt 1−A
When P is close to A, there will be fewer red foxes in the next year When P is close to 0, there will be more red foxes in the next year Dividing through by A and letting x = P/A, we can rewrite this as
xt+1 =rxt(1−xt) This is known as the logistic map
13 / 31
The logistic map
The logistic map (based on a the continuous equation introduced in 1838 by Pierre François Verhulst) displays a diverse range of behaviours, depending on the value of the parameter r
FigureThe logistic map for r = 1 xt+1 =rxt(1−xt)
rxt corresponds to positive feedback
(1 − xt ) corresponds to negative feedback
14 / 31
The logistic map
r = 0.9
15 / 31
The logistic map
r = 1.5
16 / 31
The logistic map
r = 1.5
17 / 31
The logistic map
r = 3.2
18 / 31
The logistic map
r = 3.52
19 / 31
The logistic map
r = 3.56
20 / 31
The logistic map
r = 3.84
21 / 31
The logistic map
r = 4.0
22 / 31
Outline
Dynamical systems
Population growth
Chaos
23 / 31
A definition of chaos
A system is chaotic if it displays all of the following four properties:
The dynamical update rule is deterministic
The system behaviour is aperiodic
The system behaviour is bounded
The system displays sensitivity to initial conditions
24 / 31
The butterfly effect
Coined by Edward Lorenz while working on a mathematical model of weather: he observed that his model’s behaviour could produce wildly different outputs with only very small modifications to input data:
Two states differing by imperceptible amounts may eventually evolve into two considerably different states . . . If, then, there is any error whatever in observing the present state—and in any real system such errors seem inevitable—an acceptable predic- tion of an instantaneous state in the distant future may well be impossible . . . In view of the inevitable inaccuracy and in- completeness of weather observations, precise very-long-range forecasting would seem to be nonexistent.
Lorenz (1963) Deterministic nonperiodic flow, Journal of Atmospheric Sciences 20: 130–141
25 / 31
The butterfly effect
26 / 31
Sensitive dependence on initial conditions
FigureTwo trajectories beginning at x = 0.05 (red) and x = 0.0501 (black)
27 / 31
Bifurcation diagrams
FigurePoints on attractors of the logistic map for various values of r
Feldman (2012) Chaos and Fractals: An Elementary Introduction, Oxford University Press
28 / 31
Bifuraction diagrams
FigureThe bifurcation diagram for the logistic map, r in [2.4, 4.0]
http://en.wikipedia.org/wiki/Bifurcation_diagram
29 / 31
Summary
To model a dynamical system, we define:
our system variables and the states they can take the function that computes the next state
This will usually involve simplifying assumptions about the real world
important to record these
When exploring the behaviour of our model, we need to
evaluate it against what we know about the real system This may result in us refining our model
Dynamical systems have several different types of characteristic behaviours
fixed points
limit cycles
aperiodic orbits
The dynamic behaviour that a system exhibits can depend critically on the values of system parameters
30 / 31
Further reading
(both books available online through UniMelb library) For a gentle introduction:
David P. Feldman, Chaos and Fractals: An Elementary Introduction, Section I on dynamical systems and Section II (parts 9-11) on chaos
For a more mathematically rigorous treatment:
Nino Boccara, Modeling Complex Systems, chapters 4 and 5
Video lectures – Introduction to Dynamical Systems and Chaos (Santa Fe Institute):
http://www.complexityexplorer.org/courses/22-introduction- to-dynamical-systems-and-chaos-winter-2015/
31 / 31