# Lecture 6: Two-dimensional dynamical systems

## Two-dimensional dynamical systems

### Summary

We discussed two dimensional dynamical systems that are defined by two coupled sets of ODEs like [ \dot{x} =f(x,y) ] [ \dot{y} =g(x,y). ]

We discussed how these systems can be visualized in statespace, the space defined by the two state variables $x$ and $y$.

We discussed different coupling motifs like

**mutual inhibition**,**cooperativity**or**activator-inhibitor**coupling.We learned about the fact that we can think of a 2D system as a

in state-space and...*vector-field*how a trajectory is a path through state-space.

We also learned that in a two dimensional dynamical system the asymptotics can only be fixpoints or

, a restriction imposed by the*limit-cycles*Theorem.*Poincare-Bendixon*

### Script

All the information for 2D systems is provided in detail in the script bits:

The first contains more information on 2D dynamical systems than we covered in this lecture, e.g. fixpoints and fixpoint stability. We will cover this in the next lecture.

The second script item contains an interactive tool for exploring a set of representative 2D systems.