| 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 structurally stable 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 L\`{o}pez Nieto. |
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