Assignment 1:
due Friday November 19th, 2021
1. Take a picture of a complex system…
In class we discussed one possible definition of a complex system:
“Simple rules generate complex structure, behavior or dynamics”
The aim of this assigment is for you to search for and identify a system, that you think exhibits complex structure, behavior of dynamics that is generated by a set of simple rules.

To this end you will go outside for a one hour walk, preferentially (but not necessarily) in a natural environment, a park, the forrest, etc. and take a photograph of a system that you identified as such a complex system.

Explain in one paragraph why you think your choice is a complex system according to the definition above.

Submit your photograph and paragraph to complexsystems2021@gmail.com.
2. The road to chaos
In class we covered the properties of the logistic map $x_{n+1}=f(x_{n})=\lambda x_{n}(1x_{n})$ and discussed that a sequence of period doubling events is a road that leads to deterministic chaos.
We also discussed that this behavior is generic. The script contains a qualitativily similar bifurcartion diagram generated by the related map [ x_{n+1}=f(x_{n})=\lambda x_{n}\exp(x_{n}) ]
 Go to the script page and find the panel that displays the bifurcation diagram for this map.
 Using the zooming function and a ruler that you place on the screen find the approxikmate values for the reproduction parameter $\lambda$ at which period doubling events occur and write those down: [ \lambda_{1},\lambda_{2},\lambda_{3},…. ]
 Make a table of these values and compute the ratio [ \alpha_{n}=\frac{\lambda_{n}\lambda_{n1}}{\lambda_{n+1}\lambda_{n}} ] for as many points as you can and try to measure the limiting value of this quotient.
Fixpoints
Analyse the following 1d dynamical systems graphically and analytically. Determine the fixpoints and their stability
 $\dot{x}=x(x1)$
 $\dot{x}=x^{3}2x$
 $\dot{x}=x^{2}x^{3}$
 $\dot{x}=\sin^2(2\pi x)$
 $\dot{x}=x(\lambdax^{2})$ when $\lambda>0$ or when $\lambda<0$ or when $\lambda=0$
 $\dot{x}=(xa)(xb)(xc)$ with $a<b<c$