Part XIV: Kuramoto Synchronization
Coupled oscillator theory applied to urban rhythms — from the Kuramoto model through network synchronization to traffic green waves and commute peak breaking. Understanding how cities lock into collective temporal patterns and how to reshape them.
Part Overview
Derives the Kuramoto equation from averaging, establishes the critical coupling \(K_c = 2/(\pi g(0))\), and develops order parameter self-consistency for phase transitions. Extends to network Kuramoto with Fiedler value connections, then applies to Green Wave as a locked state and commute peak as a phase transition.
Key Topics
- • Kuramoto equation from averaging
- • Critical coupling \(K_c = 2/(\pi g(0))\)
- • Order parameter self-consistency
- • Network Kuramoto
- • Fiedler value connection
- • Green Wave as locked state
- • Commute peak as phase transition
3 chapters | From coupled oscillators to urban rhythms | Synchronization shapes cities
Chapters
Chapter 1: Kuramoto Model
Derives the Kuramoto equation from phase averaging of weakly coupled limit-cycle oscillators. Establishes the critical coupling threshold \(K_c = 2/(\pi g(0))\) and the order parameter self-consistency equation for the onset of synchronization.
Chapter 2: Network Synchronization
Extends Kuramoto to network topologies via the graph Laplacian. Connects synchronization thresholds to the Fiedler eigenvalue \(\lambda_2\) and derives conditions for partial and full frequency locking on realistic urban networks.
Chapter 3: Urban Synchronization
Applies synchronization theory to real urban phenomena: Green Wave traffic coordination as a phase-locked state, commute peak formation as a synchronization phase transition, and strategies for peak breaking through desynchronization.