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 n Describe a framework for categorizing the different approaches researchers have taken to understanding the world n Make some general observations about the prospects and limitations of these methods n Share some of my personal views about the future of humanity

Models n Either explicitly or implicitly, most people are trying to understand the world by making models. n A model is a simplified description of a complicated process (ideally amenable to mathematical analysis). n “All models are wrong, but some are useful.” – George Box n The usefulness of a model may not relate to how realistic it is.

Agents n Person n Society n Industry n Organism n Neuron n Atom n…n… 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) xy

Linearity means the response is proportional to the stimulus: Linearity What linearity is not: xy = kx A chain of causality x1x1 x2x2 y = k 1 x 1 +k 2 x 2

Why Linear Models? n Simple – a good starting point n Most things are linear if x (and hence y) are sufficiently small n 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)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 Dynamics Only four things can happen in a linear system, no matter how complicated: Negative feedback: Exponential decay Decaying oscillation Positive feedback: Exponential growth Growing oscillation

Nonlinearities x y y = kx (Linear) y = -kx (Linear) diminishing returns economy of scale hormesis What doesn’t kill you strengthens you. cf: homeopathy (common) (uncommon)

Nonlinear Dynamics Nonlinear 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 Dynamics An 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 12345… … 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 Networks x′′′= – ax′′+ x′ 2 – x Sprott, PLA 228, 271 (1997) x′′′= – ax′′ – x′ + |x| – 1 Linz & Sprott, PLA 259, 240 (1999) NL N L LL x′′ x′ x |x| – 1 x′ 2

Matrix Representation 123 1LNL 2L00 30L LLN 2L00 30L LL0 2NLN 3NNL Sprott (1997) Linz & Sprott (1999) Lorenz (1963)

Lorenz System x′= σ(y – x) y′= – xz + rx – y z′= xy – bz Lorenz, 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 System A network of many nonlinearly-interacting agents

Reasons for Optimism 1. 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 n Nature is complicated n Things will change n “Prediction is very hard, especially when it's about the future.” –Yogi Berra n There will always be problems n Our every action changes the world

References n lectures/lessons.ppt (this talk) lectures/lessons.ppt n Complexity/sprott13.htm (condensed written version) Complexity/sprott13.htm n sa/ (my chaos textbook) sa/ n (contact me)