Projects:

1. Prove Sarkovskii’s theorem.

2. For the doubling transformation T : x Tf (x)= αf f − −α h ³ ´i where α =2.502907875 is a Feigenbaum constant, its linearized operator is ··· x x x Lf φ (x)= α f 0 f φ + φ f . − −α −α −α n h ³ ´i ³ ´ h ³ ´io Let g (x) be a fixed point of T, i.e.,

g (x)=Tg(x)

with the boundary condition: g (0) = 1. Usenumericalmethodstofind the eigenvalue of this operator at g (x) that has the value greater than one, i.e., x x x α g0 g φ + φ g = δφ(x) − −α −α −α n h ³ ´i ³ ´ h ³ ´io where δ>1 and φ (x) is the associated eigenfunction.

2 3. Consider the quadratic map xn+1 = xn + c, a) Find and classify all the fixedpointsasafunctionofc. b) Find the values of c at which the fixed points bifurcate, and classify those bifurca- tions. c) For which values of c is there a stable 2-cycle? When is it superstable? d) Plot a partial bifurcation diagram for the map. Indicate the fixed points, the 2- cycles, and their stability.

4. Show that the logistic map xn+1 = rxn (1 xn) can be transformed into the quadratic 2 − map yn+1 = yn + c by a change of variables, xn = f (yn) . (One says that the logistic and quadratic maps are conjugate. More generally, a con- jugacy is a change of variables that transforms one map into another. If two maps are conjugate, they are equivalent as far as their dynamics are concerned: you just have to translate from one set of variables to the other. Strictly speaking, the transformation should be a homeomorphism, so that all topological features are preserved, as we did in the lecture on topological conjugacy.)

5. Consider the decimal shift map on the unit interval given by

xn+1 =10xn (mod 1)

1 As usual, "mod 1 " means that we look only at the noninteger part of x. For example. 2.63(mod 1) = 0.63. a) Draw the graph of the map.

b) Find all the fixed points. (Hint: Write xn in decimal form.) c)Showthatthemaphasperiodicpointsofallperiods,butthatallofthemare unstable. (For the first part, it suffices to give an explicit example of a period-p point, for each integer p>1 .) d) Show that the map has infinitely many aperiodic orbits. e) By considering the rate of separation between two nearby orbits, show that the map has sensitive dependence on initial conditions. f) Calculate the Lyapunov exponent for the decimal shift map.

6. Deterministic Diffusion

(a) Construct a deterministic 1-D map that would provide a dynamical realization of a random walk on a 1-D lattice with probability α of moving to the right and probability β =1 α of moving to the left. − (b) Suppose α = β =1/2, compute the diffusion coefficient of the map, the escape rate from a lattice of length L,the Lyapunov exponent on the repeller and the Kolmogorov-Sinai entropy on the repeller (You can assume L 1 and you can use the escape-rate formula). À

7. Develop the renormalization theory for functions with a fourth-degree maximum, e.g., f(x, r)=r x4.Estimatethefirst few terms in the power series for the universal function g(x−). By numerical experimentation, estimate the new value of δ for the quartic case. See Briggs (1991) A precise calculation of the Feigenbaum constants, Mathemtics of Computation, 57,435,forprecisevaluesofα and δ for this fourth- degree case, as well as for all other integer degrees between 2 and 12.

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