Oscillation suppression in indirectly coupled limit cycle oscillators

We study the phenomena of oscillation quenching in a system of limit cycle oscillators which are coupled indirectly via a dynamic environment. The dynamics of the environment is assumed to decay exponentially with some decay parameter. We show that for appropriate coupling strength, the decay parameter of the environment plays a crucial role in the emergent dynamics such as amplitude death (AD) and oscillation death (OD). The critical curves for the regions of oscillation quenching as a function of coupling strength and decay parameter of the environment are obtained analytically using linear stability analysis and are found to be consistent with the numerics.

Figure 1

Diagram showing different phases of coupled system in the parameter plane of $\epsilon$ and $\gamma$. The different colors represent different dynamical regimes. White color corresponds to a state where oscillatory solutions coexist with HSS solutions. Other dynamical regimes are marked as OS, AD, HSS, and HSS + IHSS (for details, please see text). The black curves separating different dynamical regimes are obtained analytically using linear stability analysis at $\omega =2$.

Figure 2

(a) Bifurcation diagram (using XPPAUT) of two coupled SL oscillators is plotted as a function of coupling strength, $\epsilon$. Solid and dashed lines denote stable and unstable steady states, respectively, while filled blue (gray) circles represent stable periodic solutions. The bifurcation points such as Hopf bifurcation (HB) and pitchfork bifurcation (PB) are marked. Time series of state variable $x$ for the two oscillators at (b) $\epsilon =2.5$ showing AD, i.e., $x_{1,2}=0$; (c) $\epsilon =3.5$ showing HSS solutions where $x_1=x_2$; and (d) at $\epsilon =7$ representing both HSS (shown by red and blue [gray] lines) and IHSS solutions (dashed lines), where $x_1=-x_2$. Here, $\gamma =3>{\gamma }_c$ and $\omega =2$.

Figure 3

(a) Bifurcation diagram (using xppaut) of the two SL oscillators is plotted as a function of coupling strength, $\epsilon$. Solid and dashed lines denote stable and unstable steady states, respectively, while filled and open blue (gray) circles stand for stable and unstable periodic solutions, respectively. The bifurcation points, namely Hopf bifurcation (HB) and pitchfork bifurcation (PB), are marked. Time series of state variable $x$ of the two oscillator for $\epsilon =1.85$ for two different initial conditions showing (b) antiphase periodic solutions and (c) HSS solutions. The other parameters $\gamma =1<{\gamma }_c$ and $\omega =2$.

Figure 4

(a) Different dynamical regions plotted by different colors in $\epsilon -\mathrm{\Delta }$ plane for $\gamma =3$ and ${\omega }_2=2$. The stable steady-state solutions against $\epsilon$ are calculated numerically for (b) $\mathrm{\Delta }=0$ and (c) $\mathrm{\Delta }=0.1$. The blue (gray) line in panel (b) denotes one of the HSS solution branches splits into two solution branches due to parameter mismatch as shown by blue and green (gray) lines in panel (c). Similarly, the other HSS solution branch also splits into two branches.

Figure 5

A representative one-state cluster in (a) and (b), and a two-state cluster in (c) for a network of $N=100$ indirectly coupled identical SL oscillators. In all three diagrams the initial conditions for all oscillators are different, but the parameter values of the system are fixed to $\gamma =2.5,\epsilon =8.5,$ and $\omega =2$.