- 113 Orbits Of A Dynamical System 261 Of 2n Thus The Point 2 24 1 Is Periodic Of Period N And Indeed Every Point 27s 1 (139.53 KiB) Viewed 63 times
explanation
113 Orbits of a Dynamical System 261 of 2n. Thus the point 2/(24-1) is periodic of period n, and Indeed, every point 27s/(2" - 1) for every n > I and I << 2"--I is a periodic point, although the period will possibly be a proper divisor of n rather than n itsell. These points are densc in the wholc circle. Because the derivative of 7' is 2 as a map from k to R, it follows that every periodic point is repelling, There are eventually periodic points, nancly 21s/(2(2" – 1)) for p > 1. After p iterations, these points join the periods identified previously. It is not difficult to sec that this is a complete list of all the periodic and eventually periodic points. So every other point has infinite orbit. Unlike our first example, these orbits will not converge lo some period, since every periodic point is repelling. This example also has a dense set of transitive points, although we only outline the argument. Write a paint as 27t for 0 << 1. Then writo 1 in binary as 1 - (0.818283 - Simase 2. Then T6 = 2 (mod 2x), and the binary expansion is - (0.8&+&x=2€–3... ise 2. The set of possible limit points of this orbit has little to do with the first few (say) billion coefficients. So we may use these to specify O close to any point in the circlc. Now arrange the tail of the binary expansion to include all possible finite sequences of O's and I's. Then by applying 7 repeatedly, each of these finite sequences eventually appears as the initial part of the binary expansion of ty. This shows that the orbit is dense in the whole circle. Exercises for Section 11.3 A. Suppose that is a point of period . Show that if X* is attracting (or repelling) for T", then cach Tix*, 1 = 0,...,-1, is an attracting or repelling) periodic point. B. Draw a phase diagram of the dynamics of T: -0.5(x - x) for XER. C. Find the periodic points of the tripling map on the circle: TT-T given by 78 = 30. D. Consider Tx=ax-x; for x ER and a > 0. (a) Find all fixed points. Decide whether they are attracting or repelling (b) Find all points of period-2. HINT: First look first for solutions of Tr= . To factor T* - *, use the fact that each fixed point is a root lo factor out a cubic, and factor out a quadratic corresponding to the period-2 cycle already found. (c) Decide whether the period-2 points are attracting or repelling (d) Find the three bifurcation points corresponding to the changes in the period-1 and 2 points (i.c.. at which values of the parameter a do changes in the dynamics occur?). (2) Draw a phase diagram of the dynamics for u = 2.1. 12 if 05:51 E. Consider the tent map T of (0, 1) onto itself by Tx= (a) Graph 7" for 1,2,3,4. (b) Using the graphs, show there are exactly 24 fixed points for T*. How are they distributed? (c) Use (b) to show that the periodic points are dense in X. (d) Show that there are two distinct orbits of period 3. HINT: Solve Tlx = x for xe [j; 3) and foxx E C.