# Introduction to Complex Systems

Prof. Dirk Brockmann, Winter Term 2021

# 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 vector-field in state-space and...

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

### Script

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

Two-dimensional Dynamical Systems

Examples of 2D Systems - Interactively

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.