Oscillator death induced by amplitude-dependent coupling in repulsively coupled oscillators

The effects of amplitude-dependent coupling on oscillator death (OD) are investigated for two repulsively coupled Lorenz oscillators. Based on numerical simulations, it is shown that as constraint strengths on the amplitude-dependent coupling change, an oscillatory state may undergo a transition to an OD state. The parameter regimes of the OD domain are theoretically determined, which coincide well with the numerical results. An electronic circuit is set up to exhibit the transition process to the OD state with an amplitude-dependent coupling. These findings may have practical importance on chaos control and oscillation depression.

Figure 1

(a) The theoretical results of OD domains in the (${\epsilon }_2-U_C$) parameter space. The OD domain is enclosed by line 0 and lines $i,i=1-3$ (above line 0 and below line $i$); the numbers $1–3$ stand for ${\epsilon }_1=-1,-2$, and $-3$, respectively. (b) The numerical results of OD domains marked by color dots with the same parameters as those in (a).

Figure 2

(a),(b) The time series of variable $x_{1,2}\left(t\right)$ for two coexisting solutions with two different initial value sets as $(4.47238,5.00578,-1.99434,-3.60081,-3.7079,3.13942), (4.11436,4.75925,-2.53078,-1.1519,-2.7289,-3.13624)$, respectively; ${\epsilon }_1=-2, {\epsilon }_2=3$, and $U_C=5$.

Figure 3

(a)–(c) The basins of the OD states for $U_C=5$, 6, and $6.5$, respectively; ${\epsilon }_1=-2$ and ${\epsilon }_2=3$. The initial value is set as $x_1=y_1=x_1^{*}+\delta x, x_2=y_2=-x_1^{*}+\delta y,z_1=z_2={\left(x_1^{*}\right)}^2$, and all values of $\delta x$ and $\delta y$ are recorded when the OD state is finally achieved.

Figure 4

(a), (b) The basin stabilities of OD for ${\epsilon }_2=1$ and 3, respectively; ${\epsilon }_1=-1$ (hollow triangle), $-2$ (hollow square), and $-3$ (solid square). The critical boundaries of $U_C$ coincide with those in Fig. 1.

Figure 5

(a) The analytic results of the OD domains in the (${\epsilon }_2-U_C$) parameter space with a different coupling scheme considered [Eq. (13)]. The OD domains with the amplitude constraint effects are enclosed by line 0 and lines $i,i=1-3$ (above line 0 and below lines $i$) for ${\epsilon }_1=-1, -1.5$, and $-2$, respectively. The OD domains without the amplitude constraint effects are enclosed by lines 1 and 6, lines 2 and 5, and lines 3 and 4, for ${\epsilon }_1=-1, -1.5$, and $-2$, respectively. (b) The numerical results compared with the analytic results in (a).

Figure 6

(a) The electronic circuit unit of a chaotic Lorenz oscillator. (b) The circuit of the coupling channel with the amplitude constraint effect. Here $P_1$ and $P_2$ are connected with $X_1$ of the two Lorenz circuit units, and $P_3$ and $P_4$ are connected with $I{O}_1$.

Figure 7

(a) Theoretical results of critical lines for OD domain in the (${\epsilon }_2-U_C$) parameter space with parameter ${\epsilon }_1=-1$ (lines 0 and 1) and ${\epsilon }_1=-2$ (lines 0 and 2), respectively. (b) The experimental results of OD domain for ${\epsilon }_1=-1$ (light gray dots) and ${\epsilon }_1=-2$ (dark gray dots), respectively.

Figure 8

(a),(b) The time series of $x_{1,2}\left(t\right)$ of the coupled circuits for $R_{19}=200\text{K}\Omega ({\epsilon }_2=3)$ and $R_{24}=3\text{K}\Omega ({\epsilon }_1=-2)$ showing the coexistence of the OD and oscillation for different initial conditions; $U_C\approx 4.5\text{V}$ and $V_{SS}=6\text{V}$. (c),(d) The corresponding $U\left(t\right)$'s in the channel of the coupled systems. The scales of the oscilloscope are $2\text{ms}/\text{division}$ and $2\text{V}/\text{division}$.