Introduction to Complex Systems
Lecture 3 - One-dimensional dynamical systems & Introduction to Synchronization
In this lecture we discussed one dimensional dynamical systems like
$$ \dot x = f(x) $$
We discussed the assumptions that are implicitly made for systems that can be modeled by differential equations like the above. We discussed how the asymptotics (The behavior as $t\rightarrow\infty$) can be understood easily when we draw $f(x)$ vs. $x$.
We explained how to derive one-dimensional dynamical systems like the above for a few systems e.g.:
- the two particle annihilation process
- an extended system that also involves reproduction
- an epidemic model (the SIS-model)
All of the details are provided in detail in the script:
Read it, there’s interactive stuff in it. It’s really important that you get what’s in it.
Synchronization - Part 1
We also started our journey into synchronization phenomena. We went over a could to different systems that exhibit synchrony, we reasoned that we need oscillatory behavior for many of those systems, and the challenges of distinguishing snychronous behavior that is driven by an external force, and synchrony that emerges slowly and as an emergent behavior, just like in the pendulim clocks of Christiaan Huygens.
The slides that I went over in class are available here: