Covering Lecture Notes and EIT Second Edition, Chapters 1 and 2.

1. Write up your Project Proposal with the following sections. The result should be 1-2 pages long.
   1. Goal: What you would like to learn?
   2. System: Describe how the dynamical system is nonlinear and time-dependent. What’s the state space? What’s the dynamic? Why is it interesting?
   3. Properties: What properties are you going to investigate?
   4. Methods: What methods will you use? Why are they appropriate?
   5. Hypothesis: What is your current guess as to what you will find?
   6. Steps: List the appropriate steps for your investigation; for example, read literature, write simulator, do mathematical analysis, estimate properties from simulation, write up report, and so on.
   7. Time: Estimate how long each step will take. Can you complete the project within one month?
2. Consider the map of the interval given by

   (1)

   Construct a Markov partition for this map, as follows.

   1. Plot x_n+1 versus x_n.
   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 p_V (∞) for the Markov chain.
   3. Last week you constructed a 3 element Markov partition _Markov 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 _gen = {0 ~ (0,),1 ~ (,1)}. Label the transitions in last week’s Markov chain with the symbols observed using _gen when making transitions between Markov partition elements.
   2. Show that the resulting representation is a deterministic 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.
   4. 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 _gen = {0 ~ (0,),1 ~ (,1)}?
   5. EIT Problem 2.6. (EIT 1st edition: was Problem 2.10.)
   6. EIT Problem 2.12. (EIT 1st edition: was Problem 2.16.)
   7. EIT Problem 2.15. (EIT 1st edition: was Problem 2.21.)
   

   Homework due one week after being assigned.