A. Each point of the Mandelbrot set associated with
1. Finite-valued attractor (FVA) which is a point or cycle.
2. Basin of attraction called the "filled Julia set."
B. FVA's can be characterized by their rotation numbers, a few of which (along with the corresponding Julia sets) are shown in blue in the diagram at the right (click here to enlarge).
1. Imagine the points of a cycle (in x-y plane) on a closed loop.
2. Impose a coordinate axis as shown in the figure at the right (click here to enlarge).
3. Associate with each point an angle. Then the rotation number is the average change in angle per iteration in the limit of many iterations.
4. For periodic itineraries, the rotation number can be written as
r = p/q
where p is the number of revolutions through 360 degrees required to visit all points on the cycle and q is the number of such points.
5. Each "ball" on the M can be associated with a rotation number which, while not unique, characterizes the corresponding dynamics in an informative way.
C. Bifurcations.
1. Changing rotation numbers as one goes from "ball" to "ball" is reflective of qualitative changes in dynamics called bifurcations which occur as one varies the parameters.
a. Typical of nonlinear systems.
b. Best known examples are period-doubling and tangent bifurcations in the logistic map
shown at the right (click "here to enlarge.)Xi+1 = r Xi (1 - Xi).
1. In response to increasing r past 3.0, the equilibrium,
X* = (1 - 1/r)
loses stability and emits a two-point cycle.
2. With further increases in r, the 2-cycle is itself destabilized, at which point, a four-point cycle is born.
3. The process is repeated, etc., with the intervals between successive on up to an "accumulation point," (r = 3.56 ...,) at which juncture the period becomes infinite.
4. The sequence, or "cascade," of period-doublings is sometimes called the period-doubling route to chaos (click here to enlarge).
a. Each bifurcation produces an incremental increase in dynamical complexity.
b. As opposed to the abrupt increase that results from the homoclinic bifurcations discussed in Chapter 4.
3. A succession of periodic windows follows the initial period-doubling cascade.
a. Windows are opened by tangent bifurcations each of which creates
a pair of periodic orbits.
b. Within each window, the stable cycle undergoes its own period-doubling cascade which is
followed by a sequence of windows "within a window."
d. Picture at the right (click
here
to magnify) shows the period-3 window along with the unstable
3-cycle which eventually collides with and anihilates the period-3 attractor.
c. Windows close when the resulting chaotic attractor collides with the
unstable cycle created by the tangent bifurcation which opened the window.
2. The complex logistic map,
(with which the Mandelbrot set is associated) collapses to a transformation of the logistic when the parameter, Q, is set to zero.Xi+1 = Xi2 - Yi2 + PYi+1 = 2XiYi + Q
3. In addition to period-doubling and tangent bifurcations, one observes period-n-tupling bifurcations inn the complex logistic map.
a. In a period n-tupling bifurcation, a cycle of period m give rise to a cycle of period n x m.
b. Example: Varying (P,Q) so as to cross from the main "double cycloid" of of the Mandelbrot set to one of the two balls correspoding to three-point cycles with rotation number 1/3, induces a period-tripling bifurcation whereby a fixed point emits a three-point cycle.
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Algebraically Difficult Question (10 points; due 10/17): Derive a formula for the period-2 cycle which emerges from the fixed point. X* = (1 - 1/r) at r = 3. First Hint: Writing the logistic map
Xt+1 = r Xt(1 - Xt)
as
Xt+1 = F(Xt)
we note that points on a two-cycle satisfy the equation
Xt+2 = F(F(Xt)).
Second Hint: Remember that X* is also a 2-cycle.
Show that the two-cycle does not exist that for r < 3.
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