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 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)

sprott@physics.wisc.edu (contact me)