PHY 28A
Natural Computation and Self-Organization:
The Physics of Information Processing in Complex Systems

Jim Crutchfield
chaos@csc.ucdavis.edu; http://csc.ucdavis.edu/~chaos

Winter
WWW: http://csc.ucdavis.edu/~chaos/courses/poci/

Homework 7

Covering Lecture Notes.

  1. Consider the map of the interval given by
    ( |{1 - 2xn 0 ≤ xn ≤ 1 x = f (x ) = 2x - 1 1 < x ≤ 32 x ∈ [0,1] . n+1 n |( n 3 32 n 3 n 3 - 3xn 3 < xn ≤ 1
    		(1)

    Construct a Markov partition for this map, as follows.

    1. Plot xn+1 versus xn.
    2. Find the monotone pieces of f(x) by determining the locations of its extrema.
    3. Show that the segments defined by the ends of the interval and these locations form a Markov partition.
    4. Give the Markov chain induced by this partition.
    5. Calculate the asymptotic state distribution pV (∞) for the Markov chain.
  2. Last week you constructed a 3 element Markov partition PMarkov for the logistic map at the parameter setting where 2 bands merge into 1 band. Analyze a generating partition for this same situation, as follows.
    1. Consider the binary partition of the interval Pgen = {0 ~ (0,1 2),1 ~ (1 2,1)}. Label the transitions in last week’s Markov chain with the symbols observed using Pgen when making transitions between Markov partition elements.
    2. Show that the resulting representation is a unifilar hidden Markov process.
    3. How many internal (Markov) state sequences map onto the observed sequence of all 1s? Are there any other examples of many-to-one mappings of state sequences onto observed sequences?
    4. States are equivalent if the sets of sequences that follow from them are the same. Give a hidden Markov chain with fewer states than the one just constructed by identifying states that are equivalent in this sense.
  3. For the logistic map at r = 4, what sets of initial conditions lead to the sequences s3 = 001 and s3 = 111 when the map iterates are observed with the partition Pgen = {0 ~ (0,1 2),1 ~ (1 2,1)}?
  4. Consider the following “bad” partition for the logistic map at r = 4.
    1. Show that the partition P = {A ~ (0,1 2),B ~ (1 2,1)} is a Markov partition and give the induced Markov chain.
    2. Let x* be the largest of the two inverse iterates of 12: x* = max{f-1(12)}. Observe the logistic map with the binary partition Pbad = {1 ~ (0,x*),0 ~ (x*,1)}. Give the hidden Markov chain which this partition induces. As above, this is a labeling of Markov chain’s transitions according to Pbad.
    3. Is this hidden Markov chain unifilar?
    4. Give a complete summary of the state sequences (over {A,B}) that map onto the all-1s sequence.

Homework due one week after being assigned.