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Lessons Learned from 20 Years of Chaos and Complexity. J. C. Sprott Department of Physics University of Wisconsin - Madison Presented to the Society for Chaos Theory in Psychology and Life Sciences in Milwaukee, Wisconsin on August 1, 2014. Goals. - PowerPoint PPT Presentation
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Lessons Learned from 20 Years of Chaos and Complexity
J. C. SprottDepartment of Physics
University of Wisconsin - Madison
Presented to the
Society for Chaos Theory in Psychology and Life Sciences
in Milwaukee, Wisconsin
on August 1, 2014
Goals Describe a framework for
categorizing the different approaches researchers have taken to understanding the world
Make some general observations about the prospects and limitations of these methods
Share some of my personal views about the future of humanity
Models Either explicitly or implicitly, most people
are trying to understand the world by making models.
A model is a simplified description of a complicated process (ideally amenable to mathematical analysis).
“All models are wrong, but some are useful.” – George Box
The usefulness of a model may not relate to how realistic it is.
Agents Person Society Industry Organism Neuron Atom …
Inputs(stimulus)
Outputs(response)
Cause Effect
• Experiments
• Observations
• Reductionism
Facts versus Theory
Nonstationarity
Keep all inputs constant
Why?• Transient (memory)• Inputs not kept sufficiently constant• Unidentified inputs• Noise or measurement errors• Internal dynamics
y = f(x)
x y
Linearity means the response is proportional to the stimulus:
Linearity
What linearity is not:
x y = kx
A chain of causalityx1
x2
y = k1x1+k2x2
Why Linear Models? Simple – a good starting point
Most things are linear if x (and hence y) are sufficiently small
Linear systems can be solved exactly and unambiguously for any number of agents
Feedback
Time-varying dynamics can occur even in linear systems because of the inevitable time delay around the loop.
The feedback can be either positive (reinforcing) or negative (inhibiting).
y(t)
And it can be indirect through other agents (a loop of causality):
Cause Effect
Actually, the above behaviors are rarely seen (especially unlimited growth) because nature is not linear.
(Can also have homeostasis and steady oscillations, but these occur with zero probability - they are “non-generic”.)
Linear DynamicsOnly four things can happen in a linearsystem, no matter how complicated:Negative feedback:• Exponential decay
• Decaying oscillation
Positive feedback:• Exponential growth
• Growing oscillation
Nonlinearities
x
yy = kx
(Linear)
y = -kx (Linear)
diminishing returns
economy of scale
hormesis
What doesn’t kill you strengthens you.
cf: homeopathy
(common)
(uncommon)
Nonlinear DynamicsNonlinear agents with feedback loops
• All four linear behaviors
• Multiple stable equilibria
• Stable periodic cycles
• Quasiperiodicity
• Bifurcations (“tipping points”)
• Hysteresis (memory)
• Coexisting (hidden) attractors
• Chaos
• Hyperchaos
Of necessity, most scientists are studying a small part of a much larger network. This can lead to erroneous conclusions.
An alternative is to characterize the general behaviors of large nonlinear networks as was done for the nonlinear dynamics of simple systems.
Networks
Network DynamicsAn important distinction is dynamics ON the
network versus dynamics OF the network (and the two are usually concurrent and coupled).
Network Architectures• Random networks
• Sparse networks
• Near-neighbor networks
• Small-world networks
• Scale-free networks
1 2 3 4 5 …
1
2
3
4
5
…
• Cellular automata (discrete in s, t, v)
• Coupled map lattices (discrete in s, t)
• Systems of ODEs (discrete in s)
• Systems of PDEs (continuous in s, t, v)
Minimal Chaotic Networksx′′′= – ax′′+ x′ 2 – xSprott, PLA 228, 271 (1997)
x′′′= – ax′′ – x′ + |x| – 1Linz & Sprott, PLA 259, 240 (1999)
NL
N
L
L L
x′′
x′′
x′
x′
x
|x| – 1
x′ 2
Matrix Representation1 2 3
1 L N L
2 L 0 0
3 0 L 0
1 2 3
1 L L N
2 L 0 0
3 0 L 0
1 2 3
1 L L 0
2 N L N
3 N N L
Sprott(1997)
Linz & Sprott(1999)
Lorenz(1963)
Lorenz Systemx′= σ(y – x)y′= – xz + rx – yz′= xy – bzLorenz, JAS 20, 130 (1963)
N
L
N
x
y z
• Complex ≠ complicated• Not real and imaginary parts• Not very well defined• Contains many interacting parts• Interactions are nonlinear• Contains feedback loops (+ and -)• Cause and effect are intermingled• Driven out of equilibrium• Evolves in time (not static)• Usually chaotic (perhaps weakly)• Can self-organize, adapt, learn
Complex SystemA network of many nonlinearly-interacting agents
Reasons for Optimism1. Negative feedback is common
2. Most nonlinearities are beneficial
3. Complex systems self-organize to optimize their fitness
4. Chaotic systems are sensitive to small changes
5. Our knowledge and technology will continue to advance
Summary Nature is complicated
Things will change
“Prediction is very hard, especially
when it's about the future.” –Yogi
Berra
There will always be problems
Our every action changes the world
References http://sprott.physics.wisc.edu
/ lectures/lessons.ppt (this talk)
http://sprott.physics.wisc.edu/Chaos-Complexity/sprott13.htm (condensed written version)
http://sprott.physics.wisc.edu/chaostsa/ (my chaos textbook)
[email protected] (contact me)