xn+1 = f(xn) = c xn - xn3
where x,c = real numbers. Starting from x0, we are interested in asymptotic behavior of the sequence of x's. (If you try it on a calculator, see if choosing a positive vs. negative x0 value, and absolute value change the outcome after some initial steps.)
A. Find the fixed points of the map x*(c) depending on constant c. For what values of c do they exist and what is their stability? Sketch the bifurcation diagram x*(c) of period-1 fixed points.
B. As it happens with many nonlinear maps, you will encounter a bifurcation. You may suspect that bifurcations continue and lead to period doubling at characteristic values of c.
Support this suspicion by writing f2(x) = f(f(x)) and formulating equation for its fixed points. This will be a very high-order polynomial equation, but after factoring out the common terms, you will be able to show that the factored terms are the same low-order polynomials that you encountered while hunting for period-1 fixed points. They multiply a polynomial of still high (but lower) order, whose zeros are period-2 cycles.
Pieces pointing to period-1 orbits in the f2 polynomial are always present in f2(x), because period-1 fixed points also satisfy the period-2 fixed point condition, though are not the most interesting parts when you search for doubled periods. This is because, if x=f(x), then x = f(f(f(...(x)))) = fn as well.
Identify the factor in the high-order polynomial f2(x) that specifically gives period-2 cycles via equation f2(x) = x. For that factor, show that at the point where period-1 orbits eventually lose stability, at the critical value c = c2, the factor also assumes a value satisfying condition of existence of period-2. How many period-2 fixed points follow from the equation if c slightly exceeds c2?
C. Consider a closely related, negative cubic map
xn+1 = f(xn) = -c xn + xn3
where x,c = real numbers.
Find the map's fixed points x*(c), depending on constant c. For what values
of c do they exist and what is their stability?
Sketch the bifurcation diagram, speculatively also the period-2 orbits.
xn+1 = ± (c xn - xn3)
Either write two almost identical scripts for two equations, or have a way to toggle the sign of r.h.s., in separate executions.
In each case, do the iterations for 2000 values of c spread uniformly in interval c ∊ [0.5,3.5]. For consecutive c's, alternate the staring points between x0 = -0.1 and +0.1, in order not to miss any solution branches. Iterate 5000 times, but plot in the (c,x) plane points showing only the second half of the sequence, to allow it to settle in the final cycle(s) before plotting. Whenever |xn| > 100, break the iteration. Only plot x values in the range x ∊ [-2,2].
Plot the sequences starting with different x0 using two distict colors. With the help of your results of analytical study in 4.1, interpret colors you see in different parts of the orbit diagram. Why are the two equations showing similar features but different color patterns? Why does neither plot continue past c=3; what happens if c>3 or c<-1?
Advice about plotting:
If you see that the points saturate the picture, i.e. are
overlapping, switch the symbols to smaller ones, even single pixels if necessary.
The horizontal axis of the orbit diagram is usually a linearly scaled parameter
a. But since most interesting small-scale features may be hidden in the
right-hand side of the diagram, to see them more clearly you may experiment
with showing c2 instead of c, on the horizontal axis (and label it
accordingly).
If you do that, interval 0 to 1 will take about the same amount of horizontal
space in the plot as the interval 2.8 to 3, that is you will effectively zoom in
on the r.h.s. of the picture. [In other applications, if you wish to expand the
view of the left-hand part of the horizontal axis,
you may wish to plot log(x) instead of x.]
BTW, nothing is more maddening than trying to understand figures with unlabelled
or unmarked axes, or undecipherably small axis labels, so always make sure there
is no doubt what you have plotted versus what, and in what range. Font size
in axis labels must be easily readable.
A. If the area of the starting triangle is equal 1, compute the area of Koch's flake. To do this, write the formulae for: (i) number Nn of sides after step n=1,2,3,... of the procedure, (ii) length of each side after step n, (iii) area of each constructed (added) little piece, (iv) total area added in step n, (v) total area added.
Notice: do not use trigonometry or formula for the area of an equilateral triangle with known side. Proofs of this kind are readily found on internet but any mention of factor ½√3 used in them will nullify the credit for problem 4.3. Rely instead on the scaling properties: e.g., if a regular solid object is scaled to 1/b its linear size, its area decreases by a factor 1/b2.
B. Similarly, compute the area of an inverted snow flake, object whose boundary is constructed the same way as the Koch snowflake. While the snowflake grows inside out, inverted snowflake grows outside in, as if ice was deposited on an initially straight, smooth, wall of a triangular container. What percentage of initial unit area is eventually found between the outside wall and the fractal boundary?
C. Consider an even fatter object: a surface of square flake. It starts as a
square of unit area, and gradually grows on each side, in the outward
direction, an equilateral extension. Each boundary piece has its middle
1/3 taken out and replaced with 3 new pieces of 1/3 length, put at right
angles to neighbors.
Using procedure outlined above, find the length of the corrugated perimeter,
and the area of the square flake. What is the shape of the boundary? That is,
what shape is the box into which we can efficiently fit the square flake,
and what is its size?
D. What is the similarity and box dimension of the boundary of the 3 objects described in points A-C? And what dimension d has the body of yet another object, which is constructed just like in section C, but in each step also has the middle 1/9th of each full square removed?
Look at the last two Etudes , use two methods, one treating $n$ as integer variable, the other as a continuous variable (like $x$). Perform analytical and/or numerical investigation (analytical is more impressive when it gives simple results, but numerical is more widely applicable and informative, in the end :-)