Hopf bifurcation and multistability in a system of phase oscillators

We study the phase reduction of two coupled van der Pol oscillators with asymmetric repulsive coupling under an external harmonic force. We show that the system of two phase oscillators undergoes a Hopf bifurcation and possesses multistability on a $2\pi$-periodic phase plane. We describe the bifurcation mechanisms of formation of multistability in the phase-reduced system and show that the Andronov-Hopf bifurcation in the phase-reduced system is not an artifact of the reduction approach but, indeed, has its prototype in the nonreduced system. The bifurcational mechanisms presented in the paper enable one to describe synchronization effects in a wide class of interacting systems with repulsive coupling e.g., genetic oscillators.

Figure 1

Phase-space structure of system (2) for ${\Delta }_1=-0.1,C=0.25,\delta =0.1,g_1=0.15,g_2=-0.0616$: 1 and 2 are saddle equilibrium states, 3 is an unstable focus, and 4 is a stable focus.

Figure 2

Phase-space structure evolution for system (2) with $C=0.25,\delta =0.1,g_1=0.15,g_2=-0.0616$. (a) ${\Delta }_1=-0.04$; foci become nodes. (b) ${\Delta }_1=-0.0383$, fixed points undergo pairwise saddle-node bifurcations, and two closed invariant curves (stable, $C_s$, and unstable, $C_u$) appear as the result. (c) ${\Delta }_1=0.438$; closed invariant curves undergo tangency bifurcation, and the phase trajectory densely covers the $2\pi$-periodic phase plane.

Figure 3

Phase-space structure evolution for system (2) with $C=0.25,\delta =0.1,g_1=0.15,g_2=-0.0616$. (a) ${\Delta }_1=-0.155$; foci become nodes. (b) ${\Delta }_1=-0.162$; fixed points undergo pairwise saddle-node bifurcations, and two closed invariant curves (stable, $C_s$, and unstable, $C_u$) appear as the result. (c) ${\Delta }_1=-1.5$; closed invariant curves undergo tangency bifurcation, and the phase trajectory densely covers the $2\pi$-periodic phase plane.

Figure 4

One-parametric bifurcation diagram for system (2) with $C=0.25,\delta =0.1,g_1=0.15,g_2=-0.0616$. Region B corresponds to the structure presented in Fig. 1: there are four equilibrium states:–two foci and two saddles. Region C corresponds to the situation presented in Figs. 2a and 3a. Region D corresponds to the structure observed in Figs. 2c and 3c.

Figure 5

Phase-parametric diagram for system (2) with $C=0.25,\delta =0.1,g_1=0.15,g_2=-0.0616$. The solid black curve denotes a stable equilibrium; dashed curves denote saddles; the solid gray (red) curve denotes the repeller. 1 and 2 denote saddle-node bifurcations of equilibria 2 and 4 [see Figs. 2a and 3a]; 3 and 4 denote saddle-repeller bifurcations of equilibria 1 and 3 (see the same figures).

Figure 6

Phase-space structure of system (2) for ${\Delta }_1=-0.1,C=0.15,\delta =0.1,g_1=0.15,g_2=-0.0616$. 1 and 2 are saddle equilibrium points; 3 and 4 are unstable and stable foci, respectively; $C_u$ and $C_s$ are unstable and stable invariant closed curves of type I, respectively; and $C_s^{*}$ and $C_u^{*}$ are stable and unstable closed invariant curves of type II, respectively.

Figure 7

Phase-space structure evolution of system (2) for $C=0.15,\delta =0.1,g_1=0.15,g_2=-0.0616$. (a) ${\Delta }_1=-0.09$; the limit cycles $C_s^{*}$ and $C_u^{*}$ have reduced around foci 3 and 4, respectively, and disappeared through the Hopf bifurcation at ${\Delta }_1=-0.097$. (b) ${\Delta }_1=-0.073$; foci 3 and 4 become nodes and approach saddles 1 and 2, respectively. (c) ${\Delta }_1=-0.072$; pairwise saddle-node bifurcations take place. (d) ${\Delta }_1=0.18$; invariant closed curves $C_s$ and $C_u$ disappear after the tangency bifurcation at ${\Delta }_1=0.134$.

Figure 8

Phase-space structure evolution of system (2) for $C=0.15,\delta =0.1,g_1=0.15,g_2=-0.0616$. (a) ${\Delta }_1=-0.109$. (b) ${\Delta }_1=-0.159$; foci 3 and 4 become nodes and approach saddles 1 and 2, respectively. (c) ${\Delta }_1=-0.162$; pairwise saddle-node bifurcations take place. (d) ${\Delta }_1=-0.7$; invariant closed curves $C_s$ and $C_u$ disappear after the tangency bifurcation at ${\Delta }_1=-0.526$.

Figure 9

One-parametric bifurcation diagram for system (2) with $C=0.15,\delta =0.1,g_1=0.15,g_2=-0.0616$. Region A corresponds to the structure presented in Fig. 6: there are four equilibrium states, two invariant closed curves of type I, and two invariant closed curves of type II. Region B corresponds to the structure presented in Figs. 7b and 8b. Region C corresponds to Figs. 7c and 8c. Region D corresponds to Figs. 7d and 8d. Regions B, C, and D show qualitatively the same phase-space structures as the ones in Fig. 4.

Figure 10

Phase-parametric diagram for system (2) with $C=0.15,\delta =0.1,g_1=0.15,g_2=-0.0616$. The thin solid black curve denotes a stable equilibrium; the thin dashed curves denote unstable equilibria; the solid gray (red and blue) curves denote saddles. The thick solid black curve denotes $C_s^{*}$; the thick dashed black curve denotes $C_u^{*}$. 1, 3, 4, and 6 denote a saddle-node bifurcation [see Figs. 7b and 8b]; 2 and 5 denote Hopf bifurcations; 7 and 8 denote tangency bifurcations of closed invariant curves with the homoclinic loops of saddles.

Figure 11

Bifurcation diagram for the phase-reduced system, (2), on the $\left(g_1,g_2\right)$ parametric plane for $C=0.15,\delta =0.1,{\Delta }_1=-0.1$. The solid line denotes an Andronov-Hopf bifurcation and the dashed (red) line denotes the tangency bifurcation of the limit cycle and the homoclinic loop of the saddle.

Figure 12

Structure of the phase space of the phase-reduced system, (2), for $C=0.15,\delta =0.1,{\Delta }_1=-0.1,g_2=-0.1$, and (a) $g_1=0.1$, (b) $g_1=0.185$, and (c) $g_1=0.3$. 1 and 2 are saddles, 3 and 4 are foci, $C_{s,u}$ are type I invariant closed curves, and $C_{s,u}^{*}$ are type II closed invariant curves.

Figure 13

The Andronov-Hopf bifurcation in (2), with ${\Delta }_1=-0.08$, $g_1=0.15$, $g_2=-0.0616$, and $\delta =0.1$, and the corresponding torus emergence in the nonreduced system, (1). (a) Before the bifurcation ($C=0.16$) the reduced system, (2), possesses the stable focus 1 on the phase plane. A similar focus can be observed in the stroboscopic section of the nonreduced system, (1), in coordinates, (7). (c) This regime corresponds to the stable limit cycle 1 in the phase space of the nonreduced system, (1). After the bifurcation ($C=0.1659$) the stable limit cycle 2 appears in the reduced system, (2) (a), as well as in the stroboscopic section of (1) (b). Further increase in the control parameter leads to an increase in the limit cycle both in the reduced system, (2), and in the stroboscopic section of (1). The limit cycle for $C=0.17$ is denoted $2^{\prime }$ in (a) and (b). In the nonreduced system, (1), $2^{\prime }$ corresponds to a two-dimensional torus.