# Introduction to Complex Systems

Prof. Dirk Brockmann, Winter Term 2021

# Lecture 4: Synchronization - Phase & Pulse Coupled Oscillators

## Summary

In this lecture we covered the phenomenon of Spontaneous Synchronization in which individual dynamical systems are coupled and by themselves approach a synchronous state.

You should watch videos that are part of the slide show that was presented in class:

you can also just go through the Lecture Slides one more time and watch them there.

### Systems that sync

We motivated the topic by Huygens observation of mechanical clocks that when attached to a beam automatically go into a synchronous state.

We discussed the Millenium Brigde incident and talked about fire flies that spontaneously synchronize in their periodic light pulses and mentioned neurons that by means of their action potentials and influence on one another can also sychronize.

### Oscillators

We introduced oscillators as a basic building block for sync phenomena, their phase $\theta(t)$ and frequency $\omega$. We then discussed phase coupled oscillators and derived the...

### ... Kuramoto Model

[ \dot{\theta}_n=\omega_n + \frac{1}{N}\sum _m K _{nm}\sin(\theta_m-\theta_n) ]

which is the most famous model for phase coupled oscillators:

We derived this model in detail and the assumptions that went into it. We then analyzed two oscillators and found conditions for the emergence of a sychronous state, for example only if the coupling is strong enough or the difference in natural frequencies is sufficiently small this happen.

In the sync state the oscillators advance at the mean natural frequency and with a locked phase difference.

### Pulse Coupled Oscillators

We finished the section on Synchronization and discussed pulse coupled oscillators. We discussed how a simply leaky integrator for a can be turned into a pulse oscillator if we say that the internal state variable is governed by the ODE:

[ \dot x = -x + h ]

where the parameter $h>1$ is a constant input. In addition to the dynamics we said that the system emits a pulse when the dynamic variable $x(t)$ crosses the threshold $\theta = 1$ and subsequently the system is reset to the state $x=0$ and the process is repeated.

We discussed that when we investigate a system of such oscillators we can couple them such that an emitted pulse by oscillator $n$ advances the internal state variable $x$ of an oscillator $m$ by an increment $\epsilon$.

## Class Notes

Lecture Notes

### Further Information

Old Lecture Notes on Synchronization