Homework Assignment 3

This assignment is about one-dimensional dynamical systems] of the nature

$$ \dot x = f(x). $$

Please review the script

One-dimensional dynamical systems

before you start working on this.

Problem 1 - Fixpoints and stability

Find the fixpoints and determine their stability of the following systems.

• $\dot x = x (x - 1) (x + 1)$
• $\dot x = x^3$
• $\dot x = - (x-1)^7$
• $\dot x = x^2-1$
• $\dot x = \sin(2\pi x)$

Hint: You may be inclined to use the analytical approach by finding the zeros of $f(x)$ and computing the derivative at the fixpoints. You can do that. But beware, this may give you wrong answers in some of the above cases. So use the graphical analysis, as discussed in class.

Problem 2 - Weird epidemic

Consider a weird infectious disease in which a transmission of a virus requires the encounter of a susceptible infividual with two infected individuals, so we have a reaction

$$ S+2I\xrightarrow{\alpha}3I $$

as opposed to the generic $S+I\rightarrow 2I$. In addition to the above equation we have the recovery

$$ I\xrightarrow{\beta}S $$

just like in the ordinary SIS-model.

• Derive a one-dimensional dynamical $\dot x=f(x)$ system for the above reactions, letting $x=I/N$ and noting that the total number of individuals $N=I+S$ is constant.
• Compute the fixpoints, their stability and discuss how their existence and stability depends on the parameters $\alpha,\beta>0$.

The assignment is due Tuesday, Nov. 22nd, 2022