Kuramoto dynamics in Hamiltonian systems

The Kuramoto model constitutes a paradigmatic model for the dissipative collective dynamics of coupled oscillators, characterizing in particular the emergence of synchrony (phase locking). Here we present a classical Hamiltonian (and thus conservative) system with $2N$ state variables that in its action-angle representation exactly yields Kuramoto dynamics on $N$-dimensional invariant manifolds. We show that locking of the phase of one oscillator on a Kuramoto manifold to the average phase emerges where the transverse Hamiltonian action dynamics of that specific oscillator becomes unstable. Moreover, the inverse participation ratio of the Hamiltonian dynamics perturbed off the manifold indicates the global synchronization transition point for finite $N$ more precisely than the standard Kuramoto order parameter. The uncovered Kuramoto dynamics in Hamiltonian systems thus distinctly links dissipative to conservative dynamics.

Figure 1

The full Hamiltonian dynamics for $N=2$ coupled oscillators. Lines of constant energy of the Hamiltonian (2) for the frequencies ${\omega }_2=\omega =-{\omega }_1,L=0$ and coupling strength (a) below the critical one, $K<K_c=2\omega$, and (b) above. points,

Figure 2

Emergence of chaos in the full Hamiltonian dynamics? Panels display two-dimensional Poincaré section of the phase space for $N=3$, where the six-dimensional phase space reduces to effectively $2N-3=3$ dimensions [19]. Different colors are used to guide the eye. The oscillator frequencies are $({\omega }_1,{\omega }_2,{\omega }_3)=(-2,-1,3)$. (a) $K=1<K_{c1}$, (b) $K_{c1}<K=2.25<K_{c2}$, (c) $K=5>K_{c2}$.

Figure 3

Action bursts indicate onset of synchrony. Panels show the dynamics of three coupled oscillators in the regime of (a,c) no phase locking and (b,d) partial phase locking. Solid lines show the dynamics of (a,b) the phases and (c,d) the actions $I_j\left(t\right)$. The initial state is drawn randomly by perturbing the actions $I_j\left(0\right)$ off the manifold $I_j\equiv 1/2$ by a random amount of the order of ${10}^{-4}$; it is thus close to but not on $T^N$. Dashed lines show the dynamics of the phases ${\varphi }_j\left(t\right)$ on the submanifold $T^N$, i.e., Kuramoto dynamics. Parameters are $({\omega }_1,{\omega }_2,{\omega }_3)=(-2,-1,3)$ and (a,c) $K=1$ and (b,d) $K=2.25$, respectively.

Figure 4

Dynamics of a chain of $N=9$ oscillators with linearly increasing natural frequencies describing the mean-field dynamics of a BEC in a tilted optical lattice, in the regime of (a,c) no phase locking and (b,d) partial phase locking. The initial state is chosen as a homogeneous BEC with small density fluctuations, i.e., $I_j\left(0\right)=1/2+{\epsilon }_j$ with small random perturbations ${\epsilon }_j$ of the order ${10}^{-4}$, and random phases ${\varphi }_j\left(0\right)$. The initial state is thus close to but not on $T^N$. Parameters are ${\omega }_B=0.25$ and (a,c) $K=0.03,L=0.4$ and (b,d) $K=0.15,L=2$ (cf. [17]).

Figure 5

Transverse instability of actions predicts order parameter. (a) Average phase velocity $\overline{d{\varphi }_j/dt}$ of $N=100$ oscillators in the regime of partial phase locking at $K=2.2$. (b) Perturbations $|{\epsilon }_j\left(t\right)|$ grow off the Kuramoto manifold, shown after $t=10$, starting from ${\epsilon }_j\left(0\right)=\pm {10}^{-6}$ with random signs. Exact numerical results for the Hamiltonian dynamics $(\square )$ are displayed in comparison to the diagonal approximation (11, 13), $(\diamond )$. Actions grow substantially more for those oscillators that are phase-locked (shaded area). (c) The prediction of the order parameter from the Hamiltonian action dynamics (16) $(\square )$ well agrees with the actual order parameter $r$ (14) (—) directly measured from the Kuramoto phases. $[N=250$; data averaged over 100 realizations of the ${\omega }_j$; vertical lines indicate standard deviation of (16)] [32].

Figure 6

Inverse participation ratio of Hamiltonian dynamics reveals synchronization phase transition. (a) Order parameter $\overline{r}$ ($\circ$, left scale) and inverse participation ratio ${\overline{\mathcal{P}}}_2$ ($\square$, right scale) as a function of coupling strength $K$ for $N=250$ oscillators. Mean-field theory predicts the onset of phase order in the Kuramoto model at $K_c=2$ (dashed vertical line). (b) width of the transition region $\delta K$, where $\overline{r}$ increases by 50% from its critical value at $K_c$, i.e., $\overline{r}(K_c+\delta K)=1.5\overline{r}\left(K_c\right)$ as a function of the number of oscillators $N$. (c) Finite size scaling of $\overline{r}$ and ${\overline{\mathcal{P}}}_2$ for a subcritical coupling strength $K=1.8$. Initial actions are $I_j=1/2 (\circ )$ and $I_j=1/2+{\epsilon }_j (\square )$, where the ${\epsilon }_j$ are drawn from a Gaussian distribution with standard deviation ${10}^{-4}$. All data points are averages over 100 realizations each [32].