Figure 1

Six different patterns of the dynamics of system (9). (a) Global stability of a trivial state (for ${\delta }_{1,2}<0$). (b), (c) Stability of $S_4$ when both populations are partially synchronous [conditions for this are (B2) for (b) or (B3) for (c)]. (d) Competition between clusters if the coupling is strongly suppressive (B6); here we have bistablity of states $S_{2,3}$ describing synchronous activity of one cluster and asynchronous of another one. (e) Asymmetric interaction between clusters arises under condition (B9); here always a heteroclinic trajectory from saddle point $S_3$ to stable node $S_2$ exists (red dashed line). (f) Stability of $S_2$ under condition (B8).Reuse & Permissions

Figure 2

Ten different dynamical regimes in system (10). (a), (b) Stability of the trivial state arises at conditions (B1). Case (a): $D_{12}<1$; case (b): $D_{12}>1$. (c), (d) Stability of $M_4(1,1)$ when both clusters are in the synchronized state, under condition of a weak suppressive coupling, (B4) for (c) or (B5) for (d). (e) Competition between clusters arises at strong suppressive coupling (B7). Here we have bistability of steady states $M_{2,3}$; each of these points corresponds to synchronous activity of one cluster and asynchronous activity of another one. (f) Asymmetric interaction between clusters at asymmetric coupling (B11). Here, a sequence of heteroclinic trajectories $M_3\to M_4\to M_2$ (red dashed lines) is always present. (g), (h) The situation of stability of fixed point $M_3$ while conditions (B10) are satisfied [case (g): $D_{21}>1$, case (h): $D_{21}<1$]. (i) Bistability of fully asynchronous and fully synchronous states arises if (B12) is valid. In this case, stable manifolds of the saddle point $M_5$ divide basins of attraction of stable points $M_1$, $M_4$. (j) The case of periodic behavior arises at conditions (B13).Reuse & Permissions

Figure 3

Modeling of ensemble consisting of two subpopulations of $N={10}^3$ phase oscillators. (a) Subpopulations interact via modulation of effective frequency mismatch (9). Case of competition between subpopulations for parameter values ${\Delta }_{1,2}=1$, $a_{1,2}=5$, and ${\Gamma }_{12}={\Gamma }_{21}=12$. (b) Subpopulations interact via coupling modulations (10). A periodic regime is presented at parameter values $a_1=-1$, $a_2=1$, and $A_{12}=A_{21}=-2$. To avoid a spurious clustering and to ensure validity of the Ott-Antonsen description, a small mismatch was added: ${\omega }_n$ were randomly distributed in the range $[-0.025,+0.025]$.Reuse & Permissions

Figure 4

Multistability of steady states $C_n$ corresponding to synchronous state of only one cluster for (a) system (7) ($\Delta <a$, $\Gamma >a$) and (b) system (8) ($a>0$, $A<-a$).Reuse & Permissions

Figure 5

Stable heteroclinic cycles caused by asymmetric interactions between clusters in system (7) [(a) and (c)] and in system (8) [(b) and (d)]. Parameters: (a) $a_l-{\Delta }_l>0$, ${\Gamma }_{12}>\frac{a_2(a_1-{\Delta }_1)}{a_2-{\Delta }_2}$, ${\Gamma }_{31}>\frac{a_1(a_3-{\Delta }_3)}{a_1-{\Delta }_1}$, ${\Gamma }_{23}>\frac{a_3(a_2-{\Delta }_2)}{a_3-{\Delta }_3}$, ${\Gamma }_{21}<\frac{a_1(s_2-{\Delta }_2)}{a_1-{\Delta }_1}$, ${\Gamma }_{13}<\frac{a_3(a_1-{\Delta }_1)}{a_3-{\Delta }_3}$, and ${\Gamma }_{32}<\frac{a_2(a_3-{\Delta }_3)}{a_2-{\Delta }_2}$; (b) $a_l>0$, $A_{12}<-a_1,A_{31}<-a_3,A_{23}<-a_2$, $A_{21}>-a_2,A_{13}>-a_1,$ and $A_{32}>-a_3$. Panels (a) and (b) show the phase-space portraits while time series are presented in panels (c) and (d). Parameters: (c) $a_{1,2,3}=1$, ${\Delta }_{1,2,3}=0.2$, ${\Gamma }_{21,32,13}=2$, ${\Gamma }_{12,23,31}=0.6$ (d) $a_{1,2,3}=1.0$, $A_{21,32,13}=-3.0$, and $A_{12,23,31}=-0.3$.Reuse & Permissions

Figure 6

Dynamics of the order parameters of three interacting populations of oscillators for three different sizes of populations: (a) $N=100$, (b) $N=400$, and (c) $N=10000$. Parameters: $a_{1,2,3}=1$, ${\Delta }_{1,2,3}=0.2$, ${\Gamma }_{21,32,13}=3.0$, and ${\Gamma }_{12,23,31}=0.6$ (such a set of parameters produces a heteroclinic cycle in the phase of the system in the thermodynamic limit).Reuse & Permissions

Figure 7

(a) Poincaré sections on the plane $({\upsilon }_1,{\upsilon }_2)$ demonstrating regular and chaotic dynamics at different values of $D_1$ in the system (13). (b) Time series of a chaotic regime of system (13), for parameter values $D_1=0.5$, $D_0=2.0$, ${\varepsilon }_1=-1.0$, and ${\varepsilon }_2=1.0$. (c) Lyapunov exponents calculated at different values of $D_1$, for some particular values of the Hamiltonian. From four Lyapunov exponents, two always vanish, while other two vanish for small $D_1$ (quasiperiodicity) and are nonzero for larger coupling (chaos). (d) Chaotic time series of order parameters of four subpopulations of oscillators consisting of $N={10}^3$ elements each [the coupling configuration and the parameters are like in panel (b)].Reuse & Permissions