Module 2: Synchronization and Collective Behaviour

Syllabus Part II

This module studies how order appears in large groups of interacting units. You start with one oscillator on the circle and one oscillator driven by a rhythm, move to the Kuramoto model of many coupled oscillators and the Millennium Bridge, and end with models of flocks, schools and crowds. The main tools are phase equations, order parameters and agent-based simulation (Steven H. Strogatz 2024; Steven H. Strogatz 2000; Vicsek et al. 1995).

Session Structure

Session 10: Synchronization: from one oscillator to many

Synchronization in Nature Flows on the Circle Phase Locking

Session 11: The Kuramoto model

The Kuramoto Model The Order Parameter Critical Coupling The Millennium Bridge

Session 12: Laboratory: Kuramoto and the Millennium Bridge

Initialization The ODE Pairwise vs Mean Field (optional) Animation Bifurcation Diagram Locating the Threshold Full Simulation (optional) Finite-Size Effects Kuramoto Assignment (graded)

Session 13: Modelling collective behaviour: Vicsek and Couzin

The Vicsek Model The Couzin Model The Social Force Model

Session 14: Laboratory: collective motion

Initialization Update Rule Vectorized Update (optional) Order Parameter Validation Animation Interactive Animation (optional) Predator Avoidance (optional) Practice: Collective Motion

Learning Goals

By the end of this module you should be able to:

  • Describe an oscillator by its phase and analyse flows on the circle, including the saddle-node bifurcation of the nonuniform oscillator.
  • Derive the locking range of an oscillator driven by a periodic stimulus.
  • Write the Kuramoto model, rewrite it in mean-field form, and state the critical coupling \(K_c = 2/(\pi g(0))\).
  • Simulate the Kuramoto model, measure \(r_\infty(K)\), and separate real synchronization from finite-size fluctuations.
  • Explain the Millennium Bridge instability as a feedback loop between crowd synchrony and bridge motion, and compute the critical crowd size.
  • Implement the Vicsek model and explain the roles of noise and density in the transition to collective motion.
  • Compare the rules and the questions of the Vicsek, Couzin and social force models.

References

Strogatz, Steven H. 2024. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Chapman; Hall/CRC.
Strogatz, Steven H. 2000. “From Kuramoto to Crawford: Exploring the Onset of Synchronization in Populations of Coupled Oscillators.” Physica D: Nonlinear Phenomena 143: 1–20.
Vicsek, Tamás, András Czirók, Eshel Ben-Jacob, Inon Cohen, and Ofer Shochet. 1995. “Novel Type of Phase Transition in a System of Self-Driven Particles.” Physical Review Letters 75 (6): 1226.