Jochen Triesch, UC San Diego, triesch 1 Motivation: natural processes unfold over time: swinging of a pendulum decay of radioactive.

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 Dynamical Systems, Iterative Maps, and ChaosMaps, 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.

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

Iterative Maps and ChaosIterative 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?

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

Chaos: Dictionary DefinitionChaos: Dictionary Definition

Main Entry: cha·os  Function: nounEtymology: Latin, from Greek -- more at GUMDate: 15th century1 obsolete : CHASM, ABYSS2 a often capitalized : a state of things in which chance is supreme;especially : the confused unorganized state of primordial matter beforethe creation of distinct forms -- compare COSMOSb : 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<a chaos of television antennas>- cha·ot·ic adjective- cha·ot·i·cal·ly adverb

source: Webster’s Dictionary

Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 4

Deterministic ChaosDeterministic 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 initialstate leads to very big uncertainty after some time. With such initialuncertainties, the system’s behavior can only be predicted accuratelyfor a short amount of time into the future.

Note: In all physical systems there is always uncertainty about the initialsystem state (Heisenberg uncertainty principle in physics).

Illustration:Butterfly effect: the flapping of the wings of a butterfly at the Amazoncan determine the occurrence of a later hurricane thousands of milesaway.

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

The fish pond exampleThe 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 predictthe 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 …

Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 6

Iterative MapsIterative 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 stateat 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.)

Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 7

Linear Growth/DecayLinear 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) = gt s0 = exp(ln(g)t) s0,

if s0 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%.

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

Y. B

ar-Y

am

Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 9

The Quadratic MapThe 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.

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

Y. B

ar-Y

am

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

Y. B

ar-Y

am

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

Y. B

ar-Y

am

Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 13

From fixed point to two cycleFrom 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 occursexactly at a = 3.0. This is a simple example of a bifurcation. The system behaviorqualitatively changes if a continuous parameter (in this case a) changes its value.

Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 14

Chaotic BehaviorChaotic 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.

Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 15

Bifurcation diagram of logistic mapBifurcation diagram of logistic map

fixed point

a

s(n)

Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 16

Bifurcation diagram of logistic mapBifurcation diagram of logistic map

fixed point

two-cycle

four-cycle

chaotic regime

“period doublings”

Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 17

Period doublings and Feigenbaum NumberPeriod doublings and Feigenbaum Number

For increasing a, period doublingsoccur faster and faster:

period 2: a1 = 3period 4: a2 = 3.449…period 8: a3 = 3.54409…period 16: a4 = 3.5644……

The sequence of values a, whereperiod doublings occur obeys asimple law:

δ is the famous Feigenbaum number

...669.4lim1

1

nn

nn

n aa

aa

Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 18

Sensitivity to initial conditionSensitivity 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.