Introduction to Complexity Science

1
Introduction toComplexity Science
Cybernetics, Dynamics Complexity
2
Dynamical Systems Theory
One example of a formal general approach to
describing and analysing systems is dynamical
systems theory (DST).

• The next few slides introduce some basic DST
  ideas.
• It is not that important to understand each idea
  in detail.
• But some of these concepts figure in many, many
  different approaches to dealing with systems so
  theyre important.
• They should convey to you the power of systems
  theory to capture important general aspects of
  system behaviour.
• Although simple examples will be used to explain
  DST concepts, you should easily be able to relate
  them to the domain-specific examples of the last
  few lectures

3
A Simple Example
One of the simplest dynamical systems is a
pendulum.

• a weighted rod is secured at one end

When released from a height the pendulum swings
backwards and forwards, gradually slowing until
it comes to rest.
Changing, for example, the length or mass of the
rod may change the rate at which the pendulum
swings, but does not qualitatively change its
basic behaviour.
4
State Space
To apply DST, we must first define a systems
state space the full range of possible states
that it can assume.

• For the pendulum, this is a two-dimensional space

However, since ? is periodic, the state space
is cylindrical.
Populating the space with vectors representing
legal state changes produces a phase portrait.
Notice that, for deterministic systems such as
the pendulum, vectors never intersect.
5
Variables Parameters
?
?
a
?
i
?
Parameters affect system dynamics but, unlike
variables, they are considered to be essentially
fixed.

• However if we manipulate parameters, the systems
  dynamics will change as a result
• For instance, as we decrease air resistance,
  friction etc., the pendulum dynamics gradually
  changes

The phase portrait tightens until perfect
oscillation is achieved by the frictionless
system.
We will revisit the change from spirals to
concentric circles later, when we discuss
bifurcation.
6
Change Stability
We are often interested in what a system will
tend to do in the long term, once it has settled
down.

• In order to determine this, we imagine tracing
  every trajectory of change for an infinite amount
  of time.

• many will be transients that tend never to be
  repeated.

• some will originate or terminate at fixed points.

• some will tend towards or away from limit cycles
  repeating again and again.

• this system has two fixed points a limit cycle
  which is significant ?

7
Attractors Their Basins
Some limit sets (limit cycles, fixed points,
etc.) have many trajectories leading to them,
whereas others do not.

• The former are attractors states towards which a
  system tends.

The volume of the state space that evolves
towards an attractor is termed its basin of
attraction.
Limit sets with no basin of attraction are termed
repellers.
Basins are divided by seperatrices.
We can reasonably expect that attractors with
large basins of attraction will often account for
the long-term behaviour of a system.
8
Coupling

• Sometimes we are interested in isolated systems,
  but often we are concerned with systems that
  interact in some way.

Coupling occurs when the parameters of one system
are the variables of another e.g., the girl and
her swing. For instance, the number of foxes
impacts on rabbit population dynamics.
Simultaneously, the number of rabbits affects the
dynamics of the fox population.
The populations oscillate out of phase Factors
affecting one population (pollution, habitat
change, etc.) will also impact on the other
population.
(This is an example of a classic Lotka-Volterra
model.)
9
Bifurcations (Branches)

• We have seen that changes to the parameters of a
  system will tend to have an effect on its overall
  dynamics.

• Sometimes these effects are just quantitative
• an attractors position changes, a limit cycle
  grows, etc.
• but sometimes they are more radical
  bifurcations.
• an attractor disappears, a fixed point becomes a
  cycle
• E.g., adding a rabbit toxin to the Lotka-Volterra
  model

• a small amount alters the kind of cycling
  exhibited by the system.
• but above some threshold, a catastrophe occurs
  the cycling is replaced by a fixed point
  extinction.

10
Chaos

• So far the systems we have considered have been
  very well behaved. Slight changes to system
  variables rarely have lead to significantly
  different system behaviours.

However, even the girl/swing system (a kind of
driven pendulum) is capable of exhibiting chaotic
behaviour
A tiny perturbation is capable of generating
exponential divergence in the systems trajectory
of change, yet the systems dynamics may still
exhibit strong patterns. This is known as the
butterfly effect a butterfly flapping its
wings in Cairo may cause a rainstorm in
Dudley. Often chaos in simple systems is avoided
by keeping the system within a safe part of the
parameter space. E.g., the Reynolds number for
fluid flow.
11
Adaptation

• The dynamics of some systems is special they
  appear to strive for some kind of local
  optimality or perfection.

• Imagine a simple 2-player game
• every round each player shouts heads or tails
  simultaneously
• when their shouts match they score 3 pts for
  heads, 2 for tails
• otherwise they score nothing
• players tend to copy their successful neighbours

Over time, the population will
gradually converge on one of the two
stable solutions
game-theoretic equilibria.
12
Feedback

• Feedback is a subtle notion relating to systems
  that mutually influence each other, i.e., coupled
  systems.

• In such cases, a system can monitor its own
  actions using information from the system that it
  is coupled to.
• e.g., hand-eye co-ordination, audience-singer
  interaction
• Negative feedback can reduce error, and maintain
  stability.
• e.g., Watt governor, white stick, criminal
  prosecution
• Positive feedback amplifies deviation, breaking
  symmetry
• e.g., copycat crime, fads fashion, racial
  segregation
• Some systems are capable of both economy,
  society, etc.
• Accurately characterising feedback is critically
  important to understanding almost any interesting
  system.

13
Homeostasis

• Homeostasis is the process whereby feedback is
  used to maintain stability or constancy in the
  face of disturbance.

• e.g., body temperature, blood sugar, cell
  metabolism are kept within critical limits
  through homeostatic processes
• a central bank setting interest rates to control
  inflation
• cf. the geopolitical balance of power the
  family unit

Ashbys law of requisite variety (perhaps the
most widely accepted cybernetic maxim) concerns
constraints on successful homeostasis only
variety can destroy variety the more kinds of
perturbation impinge on a system, the more
complex an effective homeostatic device must be
14
Alternative Approaches

• Here, weve taken a look at a mathematical
  approach.

• The explicitness and rigour of this type of
  analysis and modelling is attractive in a vague
  and complex domain.
• However, these methods easily become intractable.
• More philosophical /discursive approaches have
  influenced the thinking and writing of management
  social scientists.
• e.g., soft systems methodology, autopoiesis,
  etc.
• DST language is often used to describe target
  systems without the mathematical analysis
  associated with it.
• It is important to bear in mind that different
  researchers have different aims in this regard.