Bifurcation study of phase oscillator systems with attractive and repulsive interaction

We study a model of globally coupled phase oscillators that contains two groups of oscillators with positive (synchronizing) and negative (desynchronizing) incoming connections for the first and second groups, respectively. This model was previously studied by Hong and Strogatz (the Hong-Strogatz model) in the case of a large number of oscillators. We consider a generalized Hong-Strogatz model with a constant phase shift in coupling. Our approach is based on the study of invariant manifolds and bifurcation analysis of the system. In the case of zero phase shift, various invariant manifolds are analytically described and a new dynamical mode is found. In the case of a nonzero phase shift we obtained a set of bifurcation diagrams for various systems with three or four oscillators. It is shown that in these cases system dynamics can be complex enough and include multistability and chaotic oscillations.

Figure 1

(a) Connections between two groups of oscillators (solid blue circles denote conformists; open red circles denote contrarians). (b)–(i) Distributions of conformists and contrarians on the circle for different stable regimes: (b) All the oscillators are conformists ($R=r_1=1$); (c) all the oscillators are contrarians ($R=r_2=0$); (d) the $\pi$ state ($r_1=r_2=1$, $\delta =\pi$); (e) the $\pi {\mathrm{S}}_M$ state, a $\pi$-like state with $R=0$ when $N/2$ different oscillators are located at 0 and at $\pi$; (f) the blurred $\pi$ state ($N_1{r}_1=N_2{r}_2$, $\delta =\pi$); (g) the incoherent state, IS ($r_1=r_2=0$); (h) the ${\mathrm{S}}_1$ state, synchronization of a single contrarian with conformists ($R=r_1=1$); (i) the $\pi {\mathrm{S}}_1$ state, a $\pi$-like state with one renegade contrarian [$r_1=1$, $r_2=(N_2-2)/N_2$, $\delta =\pi$].

Figure 2

Schematic diagram of possible sets of stable solutions. Notation: ${\text{FP}}_{0\pi }$, fixed points of (5) with coordinates 0 and $\pi$; $\mathcal{M}$, the invariant manifold with $R=0$; $\text{P}$, the parity state with $N_1=N_2$; ${\text{S}}_1$, synchronization of one contrarian with the conformists; $\pi \text{S}$, the $\pi$ state; $\pi {\text{S}}_1$, the $\pi$-like state with one renegade contrarian; $\pi {\text{S}}_{\mathcal{M}}={\text{FP}}_{0\pi }\cap \mathcal{M}$; $\text{IS}$, the incoherent state; $\mathrm{B}\pi \mathrm{S}$, the blurred $\pi$ state; TW, the traveling wave state.

Figure 3

Projections on the plane $({\varphi }_3,{\varphi }_5)$ of some trajectories of system (5) for $N_1=N_2=3$: (a) periodic trajectory ($k=-0.21$), (b) quasiperiodic trajectory ($k=-0.216$), (c) chaotic trajectory ($k=-0.3$), (d) Poincaré section ${\varphi }_4=\pi$ ($k=-0.3$).

Figure 4

Bifurcation diagram on the $(\alpha ,k)$ bifurcation plane for two oscillators $N_1=N_2=1$: TC indicates transcritical bifurcations. The regions of stability of the antiphase state ($R=0$) are marked by light gray (light blue online) and the lines of synchronization ($R=1$) are shown in blue. The white color indicates the regions for $0<R<1$.

Figure 5

Bifurcation diagrams on the $(\alpha ,k)$ bifurcation plane for three oscillators: (top) $N_1=2,N_2=1$; (bottom) $N_1=1,N_2=2$. Notation: AH, Andronov-Hopf; PF, pitchfork; SN, saddle-node; TC, transcritical;, HC, ${\mathrm{HC}}^{*}$, heteroclinic (saddle-connection); DB, degenerate bifurcations. The lines AH and ${\mathrm{HC}}^{*}$ coincide in the diagrams but these two bifurcations occur in different places of the phase space at the same bifurcation moment.

Figure 6

Phase portraits for ${\varphi }_1,{\varphi }_2\in [0,2\pi )$ for two conformists and one contrarian. The cases a, b,..., p of the bifurcation diagram of Fig. 5 (top) are represented for different values of the parameters $\alpha$ and $k$. Point types: red, source; blue, sink; green, saddle; magenta, center; dark green, saddle node. Stable limit cycles are shown by blue; unstable limit cycles by red. Two-dimensional color regions indicate the sets of nonisolated periodic orbits.

Figure 7

Schematic representations of the invariant sets ${\mathcal{P}}_2^i$ and $\mathcal{M}$ in the torus ${\mathbb{T}}^3$: (a) $N_1=3$, (b) $N_1=2$, (c) $N_1=1$. Different invariant planes are shown in different colors. Only the edges of three invariant planes are shown in panel (c). The invariant manifold $\mathcal{M}$ (connected straight segments) is shown in dark red in all panels.

Figure 8

Bifurcation diagrams on the $(\alpha ,k)$ bifurcation plane for four oscillators: (top) $N_1=3,N_2=1$; (bottom) $N_1=N_2=2$. Notation: AH, Andronov-Hopf; SN, saddle node; PF, pitchfork; TC, transcritical; F, fold of cycles; HC, homoclinic; DB, degenerate bifurcations.

Figure 9

Poincaré sections ${\varphi }_3=\pi$ of system (5) for $N_1=3$, $N_2=1$ when (a) $k=-0.1$, $\alpha =-1.58$ and (b) $k=2$, $\alpha =\pi /2$.