Numerical Experiments - HW #2

All Students: In this exercise you will numerically determine orbit for periods 3, 4, 6 and 8 of the logistic map and list the orbits to at least 6 digits in each case. This exercise is meant for you to play with the graphics and to get approximate values and not the analytical values. For Maple, use the program IterMap2018.mws. [Rightclick and download the file. Then load into Maple.] It is similar to the example below.

You only need to change K and r values, where K is the period and r is the map parameter. To run the code you can hit Enter from the first line to the display({p1,p2,p3,p4}) command. Keep playing with r until you get the appropriate periodic orbits. Use the solve commands to get accurate values for the orbits. Print out a sufficient amount of material to prove that you have done this and report what you have learned.

Graduate Students: Explore and numerically determine the fixed points, some periodic orbits, and discuss the behavior of these one of the other maps on the Maple worksheet as compared to the logistic map. [Just cut and paste the separate function in place of the logistic map definition and play with the graphs. Report what you see.] You can also verify your results and provide a bifurcation diagram using the code in the MATLAB Wiki.

 

Iterating Maps

> restart:with(plots): Digits:=10:

Function f(x) and Kth Composition F(x) = (f^K)(x)

> f:=x->r*x*(1-x): [Maple Math]

> C:=proc(k,g) if k=1 then g(x) else g(C(k-1,g)) fi end:

Set K here for period you are seeking and change r until you see the desired intersections in the graph.

> K:=3: F:=(unapply (C(K,f),x)):

> r:=3.9: plot({f(x),F(x),x},x=0..1,color=blue);

[Maple Plot]

Finding Roots Numerically: Use solve and seek real values in [0,1] not equal to the fixed points.

Find Fixed Points

> solve(f(x)-x);

[Maple Math]

Find Candidates for Period K

> solve(F(x)-x);

[Maple Math]

Non-fixed Point Candidates

> X:={solve(F(x)-x)} minus {solve(f(x)-x)};

[Maple Math]

Test orbits:

> X[1]; f(X[1]); f(f(X[1]));

[Maple Math]

[Maple Math]

[Maple Math]

> X[2]; f(X[2]); f(f(X[2]));

[Maple Math]

[Maple Math]

[Maple Math]

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