Nonlinear Dynamics Seminar
Transition Routes to Synchronization in the Classical Kuramoto Model
2026/03/05 (Thu.) 15:30~17:00
- Date : 2026/03/05 (Thu.) 15:30~17:00
- Location : Seminar Room 617, Institute of Mathematics (NTU Campus)
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Speaker : Jia-Yuan Dai (National Tsing Hua University)
- Organizer : Yi-Chiuan Chen (Academia Sinica)
- Abstract : Most forward orbits of the classical Kuramoto model with identical
frequencies lead to stable totally synchronized states. Besides, there
is a plethora of saddle partially synchronized states. In this talk,
we rigorously describe the transition routes from partial
synchronization to total synchronization.
By the gradient structure of the classical Kuramoto model, the global
attractor consists of (circles of) equilibria and heteroclinic orbits
between them. By using the permutation symmetry of indices, we
determine when two distinct equilibria can be connected by
heteroclinic orbits. This yields a connection graph ordered by
inclusion. Moreover, we show that, surprisingly, the invariant
manifolds of partially synchronized equilibria are linear.
As a consequence, the classical Kuramoto model is a Morse–Smale
system. In particular, the connection graph persists under any small
perturbations of the classical model. This is joint work with Bernold
Fiedler and Alejandro López Nieto.
