Transitions among the diverse oscillation quenching states induced by the interplay of direct and indirect coupling

We report the transitions among different oscillation quenching states induced by the interplay of diffusive (direct) coupling and environmental (indirect) coupling in coupled identical oscillators. This coupling scheme was introduced by Resmi et al. [Phys. Rev. E 84, 046212 (2011)] as a general scheme to induce amplitude death (AD) in nonlinear oscillators. Using a detailed bifurcation analysis we show that, in addition to AD, which actually occurs only in a small region of parameter space, this coupling scheme can induce other oscillation quenching states, namely oscillation death (OD) and a novel nontrvial AD (NAD) state, which is a nonzero bistable homogeneous steady state; more importantly, this coupling scheme mediates a transition from the AD state to the OD state and a new transition from the AD state to the NAD state. We identify diverse routes to the NAD state and map all the transition scenarios in the parameter space for periodic oscillators. Finally, we present the first experimental evidence of oscillation quenching states and their transitions induced by the interplay of direct and indirect coupling.

Figure 1

Bifurcation diagram in $\epsilon \text{−}d$ space for Stuart-Landau oscillator using xppaut ($\omega =2$ and $k=4$). The horizontal arrow shows the AD-NAD transition and vertical arrow indicates the AD-OD transition. AD occurs in a small region where ${\epsilon }_{\mathrm{HB}}<\epsilon <{\epsilon }_{\mathrm{PB}}$ and $d_{\mathrm{HB}}<d<d_{\mathrm{PB}}$ (for a detailed description see text).

Figure 2

(a) Transition from LC to AD (at HB1), and AD to OD (at PB1), with the variation of $d (\epsilon =4,k=4)$. Black line: unstable steady state; deep gray (violet) line: OD state; light gray (golden) line: NAD state; (green) solid circles: stable limit cycle. (b) Transition from LC to AD (at HB2) and AD to NAD (at PB2) with the variation of $\epsilon (d=1.5,k=4)$. Here the NAD state is a bistable state: Depending on initial conditions one gets $x_{1,2}=\pm x^{*}$. (c) Variation of the environment, $s$: (left panel) In the OD state $\left(\epsilon =4\right) s$ attains the stable zero steady state beyond HB1; (right panel) between HB2 and PB2, $s$ attains the stable zero steady state but beyond PB2 the environment $s$ becomes bistable: Depending on initial conditions $s$ may attain either $s+$ or $s-$ state.

Figure 3

(a) Variation of $\epsilon \left(d=0.5\right)$: The NAD state $\left(x_{1,2}=\pm x^{*}\right)$ appears through subcritical Hopf bifurcation at HBS2. Gray (red) line: stable steady state; open (blue) circles: unstable limit cycle. (b) Variation of $\epsilon \left(d=4\right)$: The OD arises at HBS1 through subcritical Hopf bifurcation; a new NAD state emerges at PBS1 through subcritical pitchfork bifurcation and coexists with OD [gray (green) filled region]. (c) Variation of $d \left(\epsilon =2\right)$: The OD state appears through subcritical Hopf bifurcation at HBS1. (d) Variation of $d \left(\epsilon =8\right)$: The OD state appears through subcritical pitchfork bifurcation at PBS2 that coexists with the NAD state [gray (green) filled region].

Figure 4

Phase diagram in $\epsilon \text{−}k$ space $\left(d=1.5\right)$. No AD (and thus AD-NAD transition) is possible for $k\le 2$.

Figure 5

(a) Two-parameter bifurcation diagram in $\epsilon \text{−}d$ space for Van der Pol oscillator $\left(k=1\right)$. The AD-NAD and AD-OD transitions are indicated with arrow. (b) Two-parameter bifurcation diagram in $\epsilon \text{−}k$ space $\left(d=0.5\right)$. (c) One-dimensional bifurcation diagram with $k$ at $\epsilon =1.5$ shows a transition among the LC-NAD-AD. Other parameter: $a=0.5$.

Figure 6

Experimental circuit diagram of directly and indirectly coupled VdP oscillators. The AS, Ad, and A1-A5 op-amps are realized with TL074 (JFET). All the unlabeled resistors have the value $R=10 \mathrm{k}\Omega$. $C=10$ nF, $R_a=200\Omega ,R_X=10 \mathrm{k}\Omega ,R_k=10 \mathrm{k}\Omega ,V_{\alpha }=0.1$ V. $\pm 12$-V power supplies are used; resistors (capacitors) have $\pm 5\% \left(\pm 1\%\right)$ tolerance. The boxes denoted by “B” are op-amp based buffers; inverters are realized with the unity gain inverting op-amps. The $\otimes$ sign indicates squarer using AD633. Inset (in the middle part) shows the oscillation from the uncoupled VdP oscillators. Upper trace (yellow) $V_{x1}$; lower trace (cyan) $V_{x2}$ ($y$ axis: 10 v/div; $x$ axis: $500 \mu \mathrm{s}$/div).

Figure 7

Experimental real time traces [(a), (c), and (e)] of $V_{x1}$ and $V_{x2}$ along with the numerical time series plots [(b), (d), and (f)] of $x_1$ and $x_2$. [(a) and (b)] Variation of environmental coupling $(R_d=30 \mathrm{k}\Omega ,d=0.33)$: limit cycle at $R_{\epsilon }=11.7 \mathrm{k}\Omega \left(\epsilon =0.85\right)$, AD at $R_{\epsilon }=9 \mathrm{k}\Omega \left(\epsilon =1.11\right)$, and NAD at $R_{\epsilon }=2.75 \mathrm{k}\Omega \left(\epsilon =3.64\right)$. Two bistable NAD states $V_{x1,2}$ and $-V_{x1,2}$ at $R_{\epsilon }=2.75 \mathrm{k}\Omega \left(\epsilon =3.64\right)$ are shown. [(c) and (d)] Variation of diffusive coupling $(R_{\epsilon }=8.33 \mathrm{k}\Omega ,\epsilon =1.2$): limit cycle at $R_d=43 \mathrm{k}\Omega \left(d=0.23\right)$, AD at $R_d=30.50 \mathrm{k}\Omega \left(d=0.33\right)$, and OD state $R_d=4.53 \mathrm{k}\Omega \left(d=2.21\right)$. [(e) and (f)] Variation of environment $\left(k\right) \mathrm{[}{R}_d=20 \mathrm{k}\Omega \left(d=0.5\right),R_{\epsilon }=6.66 \mathrm{k}\Omega \left(\epsilon =1.5\right)$]: limit cycle at $R_k=90.9 \mathrm{k}\Omega \left(k=0.11\right)$, NAD at $R_k=14.28 \mathrm{k}\Omega \left(k=0.7\right)$, and AD at $R_k=5 \mathrm{k}\Omega \left(k=2\right)$; for clarity only the $V_{x1,2}$ NAD state is shown. [(a), (c), and (e)] $y$ axis: 5 v/div; $x$ axis: $500 \mu \mathrm{s}$/div. (e) $x$ axis: $250 \mu \mathrm{s}$/div.