March 2, 2017 --- Class 16 --- Instability and Chaos Lyapunov Exponent
Activities:
Due to the terrible noise during the class, we started out with
an exercise using Mathematica that should help you with the homework:
Here are the values of r that correspond to superstable trajectories
1 5.00000000000000e-01
2 8.09016994374972e-01
4 8.74640424831925e-01
8 8.88660215692190e-01
16 8.91666844964063e-01
Use Mathmatica to define f(x,r) = 4 r x (1-x)
Then define the second iterate f^2(x,r) = f(f(x,r),r)
Then define the 4th iterate as f^4(x,r) = f^2(f^2(x,r),r)
Then define the 8th iterate as f^8(x,r) = f^4(f^4(x,r),r)
Plot these functions along with x to see the intersection of the diagonal
line with the function.
f(x,8.09016994374972e-01)
f^2(x,8.09016994374972e-01)
f^2(x,8.74640424831925-01)
f^4(x,8.74640424831925-01)
f^4(x,8.88660215692190-01)
f^8(x,8.88660215692190-01)
Instability and Chaos
A key issue in chaotic systems is sensitivity to initial
conditions. This is discussed in CSM in Problem 6.3 on page
147. We looked at r=0.91 as described in part (a) of that
problem. It is easy to write a program that will follow two
trajectories starting at nearby values and see how they converge
or diverge. We found that for r=0.91, nearby starting points
diverge. The divergence is roughly exponential. We reproduced
Fig. 6.8 on page 158 using my code map_diff.c which is in my home
directory.
Lyapunov Exponent
The Lyapunov exponent describes the exponential growth or decay
of the difference between two nearby trajectories. Equations (6.15)
and (6.16) are the basis of a computer code to calculate the
Lyapunov exponent. We reproduced Fig. 6.9 on page 160.
Bifurcation points correspond to the Lyapunov exponent going to
zero. The big dips correspond to the superstable attractors.
The transtion to chaos occurs when the Lyapunov exponent first
becomes positive. Note some islands of stability above that
transition point where the Lyapunov exponent becomes negative.
Code for calculating the lyapunov exponent can be found in
~sg/chap6/prob9.
In that directory, you can find data files from various runs of the
Lyapunov code. You might like to see how well you can find the
bifurcation points or the value s_n that correspond to superstable
trajectories.