Complexity in Systems
Complexity: Ch. 2
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• Merely means systems that evolve
with time
• not intrinsically interesting in our
context
• What is interesting are certain nonlinear dynamical systems
• So lets work our way up to those
Complexity in Systems
Dynamical Systems
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• Dynamical systems
• Nonlinear dynamical systems
• Nonlinear chaotic dynamical
systems
• Order in nonlinear chaotic
dynamical systems
Complexity in Systems
What’s coming in this chapter
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Complexity in Systems
Dynamical systems
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• A triumph of reductionism
• Led to Laplace’s description of a
predictable, clockwork universe
• Famous 19th century complacency
that physics was essentially
complete except for some
details
Complexity in Systems
Newtonian Mechanics
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• Quantum mechanics and
Heisenberg’s Principle
• not sure this is relevant
• Discovery of presumably wellposed problems that are
pathologically dependent on
initial conditions
Complexity in Systems
Worms in the apple of certainty
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The state of a dynamical system, such
as the Solar System, depends on
• the position and velocity of all of its
components at some particular time,
called initial conditions, and
• the application of its equations of
motion (i.e., Newton’s laws, updated).
Complexity in Systems
On initial conditions
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The logistic map is the name of a class
of dynamical systems that play a large
role in complexity theory.
Think of it as the
logistic model of population growth
Complexity in Systems
The logistic map
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“Logistic” may come from the
French word meaning “to house”
but that’s only a guess.
“Map” just means function.
Complexity in Systems
The logistic map
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Here it is:
𝑥𝑛+1 = 𝑅𝑥𝑛 1 − 𝑥𝑛
𝒙𝒏 is in [0, 1] and is the population at
time n relative to the maximum
possible population
R is in [0, 4] is a measure of the
replacement rate
Complexity in Systems
The logistic population model
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Complexity in Systems
Show us pictures!
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Complexity in Systems
Same R, different starting value
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Complexity in Systems
Larger R
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Complexity in Systems
Slightly larger R
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Complexity in Systems
R = 4: SDIC and chaos
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Complexity in Systems
Bifurcation map
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Complexity in Systems
First bifurcation
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Complexity in Systems
Bifurcation map
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Complexity in Systems
Onset of chaos (period doubling)
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Complexity in Systems
Mitchell Feigenbaum
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Complexity in Systems
Feigenbaum’s Constant
4.6692016
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Complexity in Systems
Example: logistic map
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Complexity in Systems
δ = 4.669 201
609 102 990 671
853 203 821 578
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• Simple, deterministic systems can
generate apparent random behavior
• Long term prediction for such
systems may be impossible in
principle
• Such systems may show surprising
regularities: period-doubling and
Feigenbaum’s Constant
Complexity in Systems
Takeaway
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