Dynamical effects of breaking rotational symmetry in counter-rotating Stuart-Landau oscillators

Stuart-Landau oscillators can be coupled so as to either preserve or destroy the rotational symmetry that the uncoupled system possesses. We examine some of the simplest cases of such couplings for a system of two nonidentical oscillators. When the coupling breaks the rotational invariance, there is a qualitative difference between oscillators wherein the phase velocity has the same sign (termed co-rotation) or opposite signs (termed counter-rotation). In the regime of oscillation death the relative sense of the phase rotations plays a major role. In particular, when rotational invariance is broken, counter-rotation or phase velocities of opposite signs appear to destabilize existing fixed points, thereby preserving and possibly extending the range of oscillatory behavior. The dynamical “frustration” induced by counter-rotations can thus suppress oscillation quenching when coupling breaks the symmetry.

Figure 1

Lyapunov exponents of the system coupled through similar variables [Eq. (5)] are plotted as function of coupling strength $\varepsilon$ with lines (solid, long-dashed, short-dashed, and dotted lines for ${\mathrm{LE}}_{1,2,3,4}$, respectively), along with the real part of the eigenvalues $\lambda$ [filled circle, Eq. (7)] and $\mathrm{\Lambda }$ [open triangles, Eq. (13)].

Figure 2

For similar-variable coupling and co-rotating oscillators [Eq. (5)], shown in left panel is the behavior of the system in phase space ($x_i,y_i$ plane) at different coupling values. Trajectories of two oscillators are shown by solid-red and dotted-blue lines, respectively. The starting point of the two trajectories is shown by open boxes, and the final states are shown by filled circles. Corresponding time series of variables $x_{1,2}$ are plotted on the right panel using solid-red and dotted-blue lines. Panels (a)–(c), respectively, correspond to the behaviors at different coupling values $\varepsilon =0.5,2.25,3.5$, as marked in Fig. 1.

Figure 3

Lyapunov exponents (${\mathrm{LE}}_{1,2,3,4}$ are plotted with solid, long-dashed, short-dashed and dotted lines, respectively) of the coupled system for symmetry preserving coupling case [given in Eq. (15)] as a function of $\varepsilon$ with $\overline{\omega }=10$.

Figure 4

(a) Different dynamical states observed in the system as a function of $\varepsilon$ and $\mathrm{\Delta }\omega$ for co-rotating nonlinearly coupled SL oscillators, Eq. (18) (symmetry-breaking case). The dynamics is either periodic (P) or there is oscillation death (OD). (b) Lyapunov exponents (LEs) of the coupled system as a function of coupling strength $\varepsilon$ for $\mathrm{\Delta }\omega =2$.

Figure 5

For nonlinear symmetry-breaking coupling [Eq. (18)] and co-rotating oscillators, shown in the left panel is the behavior of the system in phase space ($x_i,y_i$ plane) at different coupling strengths. Trajectories of two oscillators are shown by solid-red and dotted-blue lines, respectively. The starting point of the two trajectories are shown by open boxes, and the final states are shown by filled circles. The corresponding time series of variables $x_{1,2}$ are plotted in the right panel. Panels (a)–(c), respectively, correspond to the behavior at coupling strength $\varepsilon =0.75$ (periodic and unsynchronized), 1.55 (oscillation death), and 2.0 (periodic and phase synchronized) as marked in Fig. 4.

Figure 6

Lyapunov exponents (LEs) of the system (${\mathrm{LE}}_{1,2,3,4}$ are plotted with solid, long-dashed, short-dashed, and dotted lines, respectively) as a function of the coupling strength $\varepsilon$ for counter-rotating nonlinearly coupled SL oscillators, Eq. (18) (symmetry-breaking case).

Figure 7

For nonlinear symmetry-breaking coupling [Eq. (18)] and counter-rotating oscillators, shown in left panel is the behavior of the system in phase space ($x_i,y_i$ plane) at different coupling strength. Trajectories of two oscillators are shown by solid-red and dotted-blue lines, respectively. The starting point of the two trajectories is shown by open boxes, and the final states are shown by filled circles. The corresponding time series of variables $x_{1,2}$ are plotted on the right panel. Panels (a)–(c), respectively, correspond to the periodic behaviors at different coupling strength $\varepsilon =0.75$, 1.55, and 2.0 as marked in Fig. 6.

Figure 8

Lyapunov exponents (LEs) for conjugate coupling through dissimilar variables [Eq. (22)] (${\mathrm{LE}}_{1,2,3,4}$ are plotted with solid, long-dashed, short-dashed, and dotted lines, respectively) as a function of coupling strength $\varepsilon$, along with the real part of the eigenvalues of the Jacobian $\lambda$ [filled circle, Eq. (25)], $\mathrm{\Lambda }$ [open triangles, Eq. (31)] for (a) mismatched co-rotating $\mathrm{\Delta }\omega =4,\overline{\omega }=4$ and (b) counter-rotating $\mathrm{\Delta }\omega =4,\overline{\omega }=0$ oscillators.

Figure 9

Trajectories in the phase space (left panel) and corresponding time series of variable $x_{1,2}$ (right panel) are plotted at different coupling values (a) $\varepsilon =0.5$, (b) $\varepsilon =6$, and (c) $\varepsilon =10$ for conjugate coupling and mismatched co-rotating oscillators system $\mathrm{\Delta }\omega =4,\overline{\omega }=4$. Trajectories of two oscillators are shown by solid-red and dotted-blue lines, respectively. The starting point of the two trajectories is shown by open boxes, and the final states are shown by filled circles.

Figure 10

Trajectories in the phase space (left panel), and corresponding time series of variable $x_{1,2}$ (right panel) are plotted at different coupling values (a) $\varepsilon =0.5$ and (b) $\varepsilon =6$ for conjugate coupling in counter-rotating system $\mathrm{\Delta }\omega =4,\overline{\omega }=0$. Trajectories of two oscillators are shown by solid-red and dotted-blue lines, respectively. The starting point of the two trajectories is shown by open boxes, and the final states are shown by filled circles.