Volcano Transition in a System of Generalized Kuramoto Oscillators with Random Frustrated Interactions

Abstract

In a system of heterogeneous (Abelian) Kuramoto oscillators with random or `frustrated' interactions, transitions from states of incoherence to partial synchronization were observed. These so-called volcano transitions are characterized by a change in the shape of a local field distribution and were discussed in connection with an oscillator glass. In this paper, we consider a different class of oscillators, namely a system of (non-Abelian) SU(2)-Lohe oscillators that can also be defined on the 3-sphere, i.e., an oscillator is generalized to be defined as a unit vector in 4D Euclidean space. We demonstrate that such higher-dimensional Kuramoto models with reciprocal and nonreciprocal random interactions represented by a low-rank matrix exhibit a volcano transition as well. We determine the critical coupling strength at which a volcano-like transition occurs, employing an Ott-Antonsen ansatz. Numerical simulations provide additional validations of our analytical findings and reveal the differences in observable collective dynamics prior to and following the transition. Furthermore, we show that a system of unit 3-vector oscillators on the 2-sphere does not possess a volcano transition.

Figures (6)

• Figure 1: (a) Phase diagram obtained from the analytic Eq. (\ref{['eq:critical_vector']}). The dashed line indicates the critical coupling strength as a function of the symmetry parameter $\eta$. For comparison, we display critical coupling strengths that are numerically obtained (black solid dots). (b-e) Normalized distributions of components of local fields (\ref{['eq:local_field_vector']}) for reciprocal interactions with $\eta=1.0$ and (b) $J=1.8$, (c) $J=2.8$ and for nonreciprocal interactions with $\eta=0.8$ and (d) $J=2.0$, (e) $J=2.6$. Note that $J_c=2.12769$ for $\eta=1.0$ and $J_c=2.37883$ for $\eta=0.8$. We perform numerical integration of Eq. (\ref{['eq:vector_governing']}) using a fourth-order Runge-Kutta method with a time step of $dt=0.05$ and a system size of $N=500$. After discarding transient steps, we collect the data between $t=1500$ and $t=2000$ for each ensemble, and then compute the average over 100 samples, each having distinct sets of natural frequencies and random interaction matrices.
• Figure 2: Cross-sections of normalized distributions of local fields (\ref{['eq:local_field_vector']}) for both reciprocal (the first row) and nonreciprocal (the second and third row) random interactions. Note that $J_c=2.12769$ for $\eta=1.0$, $J_c=2.37883$ for $\eta=0.8$, and $J_c=3.36418$ for $\eta=0.4$. Orange-colored distributions are obtained for $J<J_c$ while blue-colored distributions are for $J>J_c$. The numerical methods used are the same as those given in the caption of Fig. \ref{['Fig:volcano_4d_overview']}.
• Figure 3: Maxima of cross-sections of local field vector distributions as a function of the coupling strength $J$ for different values of $\eta$: Green square ($\eta=1.0$), orange triangle ($\eta=0.8$) and blue circle ($\eta=0.4$). Note that Eq. (\ref{['eq:critical_vector']}) predicts $J_c=2.12769$ for $\eta=1.0$, $J_c=2.37883$ for $\eta=0.8$, and $J_c=3.36418$ for $\eta=0.4$. The numerical methods used are the same as those given in the caption of Fig. \ref{['Fig:volcano_4d_overview']}.
• Figure 4: Distribution of time-averaged velocities of each component of vector oscillators: (a) $\langle \dot{x}_i^1 \rangle_t$, (b) $\langle \dot{x}_i^2 \rangle_t$, (c) $\langle \dot{x}_i^3 \rangle_t$ and (d) $\langle \dot{x}_i^4 \rangle_t$. A blue histogram: $J=2.8 > J_c$; an orange histogram: $J=1.8 <J_c$. All figures are obtained from reciprocal interactions ($\eta=1$). However, similar results for nonreciprocal interactions with $\eta<1$ (results not shown here) can be found. We perform numerical integration of Eq. (\ref{['eq:vector_governing']}) using a fourth-order Runge-Kutta method with a time step of $dt=0.05$ and a system size of $N=500$. We average the instantaneous velocities of each component for $\Delta t=1000$ up to $T_\text{max}=3000$.
• Figure 5: Densities of $S_c(\bold{x}_i,\bold{P}_j)$ (see Eq. (\ref{['eq:angle']})) are plotted against the interaction matrix element $M_{ij}$ for (a) $J=1.8<J_c$ and (b) $J=2.8>J_c$. All figures are obtained for reciprocal interactions ($\eta=1$). However, similar results for nonreciprocal interactions with $\eta<1$ (results not shown here) can be found. We perform numerical integration of Eq. (\ref{['eq:vector_governing']}) using a fourth-order Runge-Kutta method with a time step of $dt=0.05$ and a system size of $N=4000$. After discarding transient steps, we collect the data between $t=1000$ and $t=2000$. (c) Time evolution of the norm of the global Kuramoto order parameter (\ref{['eq:global_order']}) for $\eta=1.0$ in a log-log scale. Gray dashed guideline indicates an algebraic relaxation. All the considered cases exhibit an exponential relaxation: $J=1.0 <J_c$ (blue), $J=6.0 >J_c$ (green), and $J=15.0 >J_c$ (red).