Lecture 5 - Critical Phenomena

Critical Phenomena

In this class we discussed how systems can exhibit one behavior for one set of parameters and qualitatively different behavior for another set of parameters.

We had already skimmed this briefly in the discussion of one-dimensional dynamical systems, e.g. in the epidemic SIS model and the Kuramoto model.

Here we looked at this in closer inspection and focused on the analysis how a system behaves asymptotically ($t\rightarrow\infty$) when a parameter is slowly changed and what happens at the transition to different behavior.

Based on three different examples:

1. The two oscillator Kuramoto model
2. The epidemic SIS-model
3. The “weird” epidemi SIS-model

we discussed different types of bifurcations, for example

1. the saddle-node bifurcation
2. the transcritical bifurcation
3. the pitchfork bifurcation

and learned how to draw bifurcation diagrams. Based on these we explored the dynamic phenomena of hysteresis and tipping points.

Resources

All the information is covered in the script and the slides I showed in class:

Script: Bifurcation Analysis of One-Dimensional Dynamical Systems

Slides: Critical Phenomena