Unimodal Maps Notes [UMN] % =============================================================================== Here we consider iterations x_{n+1} = r*f(x_n) [#1] of maps where (i) 0 \leq f \leq 1 is concave with f(0) = f(1) = 0 and f(x_m) = 1 at some 0 < x_m < 1. (ii) 0 \leq r, x \leq 1. Note: sometimes the range of r may differ, e.g.: for logistic 0 \leq r \leq 4. % % ----------------------------------------------------------------------------- % % ANALYSIS [#2] % Fixed points x_* = f(x_*) and linear stability. Multiplier: lambda = f'(x_*). Neutral and super-stable points. Example: f(x) = x^2. Note what happens at the super-stable x_*=0; x_n = x_0^{2^n}; quadratic convergence. COBWEBS. Example: f(x) = sin x. Note that x=0 is a neutral fixed point. Use cobweb to show stability. Example: f(x) = cos x. Fixed point solves transcendental equation: x_* = cos(x_*) But -1 < lambda < 0. Stable but oscillating. Show cobweb. Later we will see that bifurcations happen as abs(lambda) crosses 1. % % ----------------------------------------------------------------------------- % % LOGISTIC MAP INTRO [#3] % f(x) = r x (1-x) Discrete analog of population model. Maps I = [0 1] to I, provided 0 \leq r \leq 4. For r > 4 trajectories ---> -\infty. Period Doubling: 0 < r < 1 ................ x_* = 0 is stable. 1 < r < r_1 = 3 .......... x_* = 1 - 1/r becomes stable fixed point. Transcritical bifurcation happens at r = 1. [1-1/r was a fixed point outside domain for r < 1]. r_1 < r < r_2 ............ Stable two cycle exists. r_2 < r < r_3 ............ Stable four cycle exists. and so on. r_1 = 3; r_2 ~ 3.449; < r_3 ... < r_n ---> r_\infty ~ 3.5699 ... Convergence is geometric lim (r_{n}-r_{n-1})/(r_{n+1}-r_{n}) = delta ~ 4.669 ... % % ----------------------------------------------------------------------------- % % WHY/HOW DOES PERIOD 2 HAPPEN? [#4] % At r = 3, the fixed point x_* = 1-1/r goes unstable, with a "flip bifurcation". Here lambda = f'(x_*) crosses -1 (so linear instability), but nonlinearity is stabilizing ... so result is an oscillation (period 2). Very similar to the Hopf bifurcation, but for a discrete system. How do we prove this? Conditions for a flip bifurcation in terms of derivatives of f at x_* can be written ---- similar, but simpler, than the conditions for a Hopf. However, checking that these are satisfied *always* for a unimodal map is hard. A simpler global/geometrical argument does the trick. The argument for the logistic map: SEE PICTURES IN WEB FOR LECTURE 24. Draw plot for f^2, and note that at r = 3 a "pitchfork" happens for f^2. Easy to compute fixed points of f^2 [quartic, but know two of then: the ones for f, so getting the other two reduces to solving a quadratic] The two cycle points are thus: x_\pm = [(r+1) \pm \sqrt{(r-3)(r+1)}]/(2 r). Stability: |f'(x_+) f'(x_-)| < 1. Stable 3 < r < 1+\sqrt{6} = 3.449... Show graphically that all of this (modulo constants) is generic for unimodal. % % ----------------------------------------------------------------------------- % % ORBIT DIAGRAM [#5] % See pictures in web for lecture 24]. % 1) Show the pictures and describe how they are done. 2) Describe the attractor. In particular the period doubling route to chaos, with a stable period 2^n cycle showing up at r = r_n. Furthermore r_1 < r_2 < .... < r_n ---> r_\infty. 3) Point out period 3 window. Intermittency: The darker lines showing up in the pictures. Describe behavior there [near periodic intervals with chaotic bursts separating them]. Other examples of intermittency (turbulence). 4) Universality and the U-sequence. Note that the orbit diagram has a rather complicated set of "periodic windows" interspaced with chaotic regions. From low to high r: --- Sequence of period doublings, with period 2^n orbits. --- A period 5 window, followed by by another period doubling sequence, with period 5*2^n orbits. --- A period 3 window, followed by by another period doubling sequence, with period 3*2^n orbits. With many more windows showing up if small ranges or r are examined. Theorem (Metropolis et all). For unimodal maps, the observed "sequence" (U-sequence) of periodic orbits is always the same. 5) Universality and Feigenbaum's constants [again, for UNIMODAL maps]. The sequence of r_n's converges geometrically to r_\infty, with r_n ~ r_\infty - c*\delta^{-n}. delta = lim (r_n-r_{n-1})/(r_{n+1}-r_n) ~ 4.669 ... is always the same [Feigenbaum constant]. % % ----------------------------------------------------------------------------- % MORE ON Orbit Diagram [#6] Furthermore: Let s_n be the value of r at which the 2^n cycle is super-stable [i.e., the orbit includes x = x_m, the value where the map has its maximum --- for the logistic map, x_m = 1/2] (see [A] below) For r = s_n, let: x_* be the point in the 2^n cycle closest to x = x_m and define d_n = x_* - x_m. Then the d_n alternate signs and converge to zero geometrically, with lim -d_n/d_{n+1} = 2.5024 ... also a universal constant. [A] Recall that a 2^n cycle is a fixed point for the f_2^n iterate of the map, and that it's stability is given by (f_2^n)' = f'(x_1)*f'(x_2)* ... * f'(x_2^n). % % =============================================================================== EOF