Lecture 20 - Thu 2023 11 16 One dimensional maps % =============================================================================== INTRODUCTION and MOTIVATION. We begin the analysis of the Rossler attractor by studying the dynamics in the "almost" 1-D "bottom plate". We can do this by looking at the Poincar\'e map of a cros-section back to itself. Later we will look at the baklava's details. This yields a 1-D map from an interval to itself: x_{n+1} = f(x_n); map from R to R or from interval to interval. f smooth (or, at least, continuous). The Rossler Poincare map, as a function of the parameter c. See video. Describe what it does. Another way to get the map: look at the sequence of local maximums of one of the variables, say: x = x(t), {x_n}. Then plot x_{n+1} versus x_n. This gives an "almost" curve. Why? Well, remember that the attractor is very flat, almost a surface. Hence, for a short time at least, it will behave as a dynamical system in the plane, where two initial conditions determine the trajectory. In this case the conditions are x = x_n and \dot{x} = 0. This "prescription" works for any attractor that is very flat, almost a surface. In particular, the Lorenz attractor --> The Lorenz map. Show how the map looks then. NOTE: the idea of a map from x_n to x_{n+1} is due to Lorenz. % ----------------------------------------------------------------------------- % % RECALL THE COMPUTER PROBLEM ASSIGNED IN PROBLEM SET #1. We will revisit it. % ----------------------------------------------------------------------------- % SIMPLEST EXAMPLE OF CHAOS. Because discrete trajectories can cross, 1-D maps have a much greater dynamical range than 1D continuous systems. In fact, they "contain" much of the dynamics of 3D systems such as Rossler, or Lorenz [3-D with nearly 2-D attractor]. % % ----------------------------------------------------------------------------- % % ANALYSIS. Fixed points 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 a transcendental equation. But -1 < lambda < 0. Stable but oscillating. Show cobweb. Later we will see that bifurcations happen as abs(lambda) crosses 1. % % ----------------------------------------------------------------------------- % LOGISTIC MAP. 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 (and 1-1/r is unstable). 1 < r < r_1 = 3 .......... x_* = 1 - 1/r becomes stable fixed point. Transcritical bifurcation happens at 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 ... % % ----------------------------------------------------------------------------- % LOGISTIC MAP ANALYSIS. At r = 3, the fixed point x_* = 1-1/r goes unstable, with a "flip bifurcation". Here lambda 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. 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, the ones for f). The two cycle points are thus: x_\pm = [(r+1) \pm \sqrt{(r-3)(r+1)}]/(2 r). Two cycle stability: given by f'(x_+) f'(x_-) stable 3 < r < 1+\sqrt{6} = 3.449... % % ----------------------------------------------------------------------------- % [see attached picture in web page]. Orbit Diagram. 1) Show class 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 (see #1), the observed "sequence" (U-sequence) of periodic orbits is always the same. #1 Unimodal: y = r*g(x), where g is concave with a single maximum. 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. Furthermore: Let R_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 #2) For r = R-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. #2 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