- Slides: 21
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 • Experiments n Person n Society n Organism n Neuron n Atom n … • Observations • Reductionism Inputs Outputs n Industry (stimulus) (response) Cause Effect Facts versus Theory
Nonstationarity Keep all inputs constant x y = f(x) y Why? • Transient (memory) • Inputs not kept sufficiently constant • Unidentified inputs • Noise or measurement errors • Internal dynamics
Linearity means the response is proportional to theisstimulus: What linearity not: x y = kx A chain of causality x 1 x 2 y = k 1 x 1+k 2 x 2
Why Linear Models? n n n 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 And The it feedback can be indirect can be either through positive other (reinforcing) agents (a loop orofnegative causality): (inhibiting). y(t) Time-varying Cause dynamics Effect can occur even in linear systems because of the inevitable time delay around the loop.
Linear Dynamics Only four things can happen in a linear system, no feedback: matter how complicated: Negative • Exponential decay • Decaying oscillation Positive feedback: • Exponential growth • Growing oscillation (Can alsothe have homeostasis Actually, above behaviorsand aresteady rarely seen oscillations, but thesegrowth) occur with zero nature (especially unlimited because probability is not linear. - they are “non-generic”. )
Nonlinearities y diminishing returns (common) y = kxeconomy of (Linear) scale (uncommon) x hormesis What doesn’t y= -kx kill you strengthens (Linear) you. cf: homeopathy
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
Networks An Of alternative necessity, most is to scientists characterize are the studying general a behaviors small partofoflarge a much nonlinear larger networks network. This as was done can lead for the to erroneous nonlinear conclusions. dynamics of simple systems.
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 1 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 Networks x′′′= – ax′′+ x′ 2 – x Sprott, PLA 228, 271 (1997) L x′ N x′′ x′ 2 x′′′= – ax′′ – x′ + |x| – 1 Linz & Sprott, PLA 259, 240 (1999) |x| – 1 L x′′ N L x′
Matrix Representation Sprott (1997) Linz & Sprott (1999) Lorenz (1963) 1 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
Lorenz System x′= σ(y – x) y′= – xz + rx – y z′= xy – bz Lorenz, JAS 20, 130 (1963) L N y x N z
Complex System A network of many nonlinearly-interacting agents • 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
Reasons for Optimism 1. Negative feedback is common 2. Most nonlinearities are beneficial 3. 4. 5. Complex systems self-organize to optimize their fitness Chaotic systems are sensitive to small changes 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 n http: //sprott. physics. wisc. edu/ lectures/lessons. ppt (this talk) http: //sprott. physics. wisc. edu/Chaos. Complexity/sprott 13. htm (condensed written version) http: //sprott. physics. wisc. edu/chaost sa/ (my chaos textbook) [email protected] wisc. edu (contact me)