1. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 1 Motivation: natural processes unfold over time: swinging of a pendulum decay of radioactive material a chemical reaction growth of a plant formation of a Tornado galloping of a horse reaching for a cup of tea action potential traveling down an axon remembering an event Dynamical Systems, Iterative Maps, and Chaos A universal mathematical language for describing processes unfolding in time is dynamical systems theory. An important part of this is the study of differential equations and iterative maps.

2. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 2 Iterative Maps and Chaos Goals: what is deterministic chaos? how does it relate to randomness? what is an iterative map? what are the behaviors of a linear iterative map? graphical analysis of iterative maps? what is the quadratic map (logistic map)? what is a bifurcation (intuitive)? how does the quadratic map exhibit chaos?

3. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 3 Chaos: Dictionary Definition Main Entry: cha·os Function: noun Etymology: Latin, from Greek -- more at GUM Date: 15th century 1 obsolete : CHASM, ABYSS 2 a often capitalized : a state of things in which chance is supreme;GUMCHASMABYSS especially : the confused unorganized state of primordial matter before the creation of distinct forms -- compare COSMOSCOSMOS b : the inherent unpredictability in the behavior of a natural system (as the atmosphere, boiling water, or the beating heart) 3 a : a state of utter confusion b : a confused mass or mixture - cha·ot·ic adjective - cha·ot·i·cal·ly adverb source: Webster’s Dictionary

4. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 4 Deterministic Chaos Determinism: Dynamical systems that adhere to deterministic laws, no randomness: with perfect knowledge of initial state of the system, system behavior is perfectly predictable. But… Sensitivity to initial conditions: The slightest uncertainty about initial state leads to very big uncertainty after some time. With such initial uncertainties, the system’s behavior can only be predicted accurately for a short amount of time into the future. Note: In all physical systems there is always uncertainty about the initial system state (Heisenberg uncertainty principle in physics). Illustration: Butterfly effect: the flapping of the wings of a butterfly at the Amazon can determine the occurrence of a later hurricane thousands of miles away.

5. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 5 The fish pond example Every spring at a fixed day we go and count the number of fish in the pond: Question: can we create a mathematical model that allows us to predict the number of fish at some point in the future (up to a certain accuracy). Assume: number of fish depends deterministically on number in previous year. 2003 2004 2005 2006 … 234 261 305 318 …

6. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 6 Iterative Maps State of the system: s(t) s : state (real number, for generality), t : time (discrete: 0,1,2,…) Iterative Map: s(t) = f( s(t-1) ) In words: state at time t is obtained by applying a function f to the state at the previous time step t-1. Simplest Example: f(s) = s, i.e. f is the identity function. It follows: s(t) = s(t-1) Interpretation: the state always stays the same (This would correspond to the same number of fish every year.)

7. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 7 Linear Growth/Decay A slightly less boring iterative map is defined by: s(t) = g s(t-1), g > 0 (some positive constant) Reasonable to assume that the more fish produce more offspring. It follows that s(t) = g t s 0 = exp(ln(g)t) s 0, if s 0 is the initial state of the system at time t=0: exponential law. In this model the fish population will either grow to infinity or approach zero. It does not capture the (observed) phenomenon that the fish population grows up to some limit --- the maximum of fish that can be supported by the pond. Example 2: g < 1, e.g. g = 0.9 : fish population shrinks by 10% each year, ultimately going to zero. Example 1: g > 1, e.g. g = 1.1 :every year fish population grows by 10%.

8. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 8 Y. Bar-Yam

9. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 9 The Quadratic Map Linear growth: s(t) = g s(t-1) Let’s add a term that leads to competition among fish and limits overall growth: s(t) = a s(t-1) – a s(t-1) 2 = a s(t-1) ( 1 – s(t-1) ) Interpretation: fish from previous year: s(t-1) production of new fish due to breeding: (a-1) s(t-1) removal of fish due to overpopulation: -a s(t-1) 2 From now on assume: 0 ≤ s(t) ≤ 1, 0 ≤ a ≤ 4.

10. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 10 Y. Bar-Yam

11. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 11 Y. Bar-Yam

12. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 12 Y. Bar-Yam

13. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 13 From fixed point to two cycle a = 2.8: system settles on a fixed point a = 3.2: system settles on a period two oscillations, i.e. the same two numbers keep repeating over and over again The transition between fixed point and period two oscillation behavior occurs exactly at a = 3.0. This is a simple example of a bifurcation. The system behavior qualitatively changes if a continuous parameter (in this case a) changes its value.

14. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 14 Chaotic Behavior Further increases in a lead to transitions to four cycles, eight cycles and so forth: the transitions are called period doublings. For a bigger than a* = 3.569946… the system becomes non-periodic. The sequence goes on and on without repeating itself.

15. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 15 Bifurcation diagram of logistic map fixed point a s(n)

16. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 16 Bifurcation diagram of logistic map fixed point two-cycle four-cycle chaotic regime “period doublings”

17. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 17 Period doublings and Feigenbaum Number For increasing a, period doublings occur faster and faster: period 2: a 1 = 3 period 4: a 2 = 3.449… period 8: a 3 = 3.54409… period 16: a 4 = 3.5644… … The sequence of values a, where period doublings occur obeys a simple law: δ is the famous Feigenbaum number

18. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 18 Sensitivity to initial condition If system is in chaotic regime, e.g. a = 3.7, system is sensitive to initial conditions. Two nearby starting points will rapidly move apart.