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.
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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.
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Explain in one paragraph why you think your choice is a complex system according to the definition above.
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Submit your photograph and paragraph to complexsystems2021@gmail.com.
2. The road to chaos
In class we covered the properties of the logistic map $x_{n+1}=f(x_{n})=\lambda x_{n}(1-x_{n})$ 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_{n-1}}{\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 1-d dynamical systems graphically and analytically. Determine the fixpoints and their stability
- $\dot{x}=x(x-1)$
- $\dot{x}=x^{3}-2x$
- $\dot{x}=x^{2}-x^{3}$
- $\dot{x}=\sin^2(2\pi x)$
- $\dot{x}=-x(\lambda-x^{2})$ when $\lambda>0$ or when $\lambda<0$ or when $\lambda=0$
- $\dot{x}=(x-a)(x-b)(x-c)$ with $a<b<c$