Homework Assignment 2

Part 1: The Feigenbaum

In Lecture 2 - the logistic map, we discussed the properties of ….surprise…. the logistic map:

$$ x_{n+1}=\lambda x_{n}(1-x_{n})\quad\text{with}\quad n=0,1,2,3… $$

We learned that, as we increase the reproduction rate parameter $\lambda$ the system undergoes a series of period doubling events, from a single stable fixpoint, to a stable two-cycle, a period 4 cycle, period 8 etc until the period becomes infinite.

The distances between parameter points at which these period doubling events occurs becomes smaller and smaller until a critical point is reached. This critical point is given by:

$$ \lambda_{\infty}=3.569946… $$

In the script, it is discussed that the ratio

$$ \delta=\lim_{k\rightarrow\infty}\frac{\lambda_{n}-\lambda_{n-1}}{\lambda_{n+1}-\lambda_{n}} $$

approaches the fixed value of 4.669 which is known as Feigenbaum’s constant. This is a universal constant that keeps reappearing in systems like the logistic map and is intimately connected to chaotic systems.

Assignment:

Here’s the bifurcation diagram of a map that is slightly different from the logistic map, but that is also mentioned in the script:

$$ x_{n+1} = \lambda x_{n} \exp (-x_{n}) $$

and in the panel below you see the bifurcation diagram of it.

Make a table of the parameter points $\lambda _ n$ at which bifurcations occur. You may have to zoom in sequentially to read of a precise value on the $\lambda$-axis. Once you have a table of the measured values for $\lambda _ n$ compute the sequence of ratios

$$ \delta _ n=\frac{\lambda_{n}-\lambda_{n-1}}{\lambda_{n+1}-\lambda_{n}} $$

and compute the limit. Write the sequence of measurements down and send it to me.

Hint: When you hover with your mouse pointer on the graph, you can read the lambda value off the title.

Part 2: Rhythms

In nature rhythms play a role. In social, biological, ecological, physical, chemical, physiological, technological, etc. systems rythms occur. Soon, we will be discussing the phenomenon of spontaneous synchronization. To this end we need to explore and think about where rhythms occur. Think about examples. So…. in this part of the assignment you are supposed to

• write down a list of 5 systems that exhibit rhythmic behavior.

The assignment is due Tuesday, Nov. 15th, 2022