1. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 1 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?

2. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 2 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

3. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 3 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.

4. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 4 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 only depends on number in previous year. 2003 2004 2005 2006 … 234 261 305 318 …

5. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 5 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.)

6. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 6 Linear Growth/Decay A slightly less boring iterative map is defined by: s(t) = g s(t-1), g > 0 (some positive constant) It is clear 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. Example 1: g > 1, e.g. g = 1.1 means that every year the fish population grows by 10%. Reasonable to assume that the more fish produce more offspring. Example 2: In contrast, if g < 1, e.g. g = 0.9, then the fish population shrinks by 10% each year, ultimately going to zero. 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.

7. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 7 Geometric Interpretation see Bar-Yam’s book for pretty pictures

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

9. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 9 Geometric Interpretation see Bar-Yam’s book for pretty pictures

10. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 10 bifurcation diagram of logistic map fixed point

11. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 11 bifurcation diagram of logistic map fixed point two-cycle four-cycle chaotic regime