Exploring the dynamics of conjugate coupled Chua circuits: Simulations and experiments

Dynamics of two Chua circuits, mutually coupled via conjugate variables, have been explored both numerically and experimentally. When the two autonomous systems were placed in the oscillatory regime the conjugate coupling provoked suppression of oscillations (amplitude death) in both systems. In contrast, if the two autonomous systems were placed in quiescent (fixed-point) states, then the effect of conjugate coupling was to generate oscillatory behavior (rhythmogenesis) in both systems. These phenomena of amplitude death and rhythmogenesis were found to persist for identical as well as nonidentical systems. It was also realized that the dynamics of the coupled systems can be regulated efficiently by varying the magnitude of conjugate coupling.

Figure 1

Amplitude death in identical oscillators: (a) and (b) Autonomous dynamics of two Chua circuits; (c) and (d) dynamics of the system when the circuits were mutually coupled using conjugate variables. System parameters: $a_o$, $\sigma$, $c$, $r$, and $b$ were fixed at 1527.41, 0.66, 0.779, 0.0708, and 0.636, respectively. Amplitude death was achieved in the coupled oscillators for $\gamma =8$.

Figure 2

Amplitude death in nonidentical oscillators: (a) and (b) Autonomous dynamics of two Chua circuits; (c) and (d) dynamics of the system when the circuits were mutually coupled using conjugate variables. System parameters: $\sigma$, $c$, $r$, and $b$ were fixed at 0.66, 0.779, 0.0708, and 0.636, respectively. Here $a_o$ for the first oscillator was fixed at 1527.41, and that for the second oscillator was fixed at 1520. Amplitude death was achieved in the coupled oscillators for $\gamma =$12.

Figure 3

Bifurcation diagram of one of the coupled oscillators as a function of $\gamma$ in the case of amplitude death. For $\gamma >4.5$, the oscillator undergoes a Hopf bifurcation, making a transition from period-one limit cycle to a fixed-point state.

Figure 4

Phase diagram for identical coupled oscillators in the ($a_o$,$\gamma$) plane when conjugate coupling was induced in the identical oscillators set in an oscillatory domain. AD represents the regime for amplitude death, and O represents the regime for any of the periodic or chaotic dynamical states of the coupled oscillator.

Figure 5

Maximal Lyapunov exponent of the dynamics of a coupled oscillator in the case of amplitude death for coupling constant $\gamma =19$. Control was switched on at integration step $6\times {10}^5$. The maximal Lyapunov exponent is negative for integration step $1.5\times {10}^6$ onward, indicating fixed-point dynamics for the coupled oscillator.

Figure 6

Rhythmogenesis in identical oscillators: (a) and (b) Autonomous dynamics of two Chua circuits; (c) and (d) dynamics of the system when the circuits were mutually coupled using conjugate variables. All system parameters: $a_o$, $\sigma$, $c$, $r$, and $b$ were fixed at 1580, 0.66, 0.779, 0.0708, and 0.636, respectively. Rhythmogenesis was achieved in the coupled oscillators for $\gamma =-5$.

Figure 7

Rhythmogenesis in nonidentical oscillators: (a) and (b) Autonomous dynamics of two Chua circuits (c) and (d) dynamics of the system when the circuits were mutually coupled using conjugate variables. All system parameters: $\sigma$, $c$, $r$ and $b$, were fixed at 0.66, 0.779, 0.0708, and 0.636, respectively. Here $a_o$ for the first oscillator was set at 1570, and that for the second oscillator was set at 1580. Rhythmogenesis was achieved in the coupled oscillators for $\gamma =-6$.

Figure 8

Bifurcation diagram of one of the coupled oscillators as a function of $\gamma$ in the case of rhythmogenesis. Decreasing $\gamma$ from $-3.5$ to $-4$, the system undergoes Hopf bifurcation and starts oscillating with period-one dynamics.

Figure 9

Phase diagram for identical coupled oscillators in the ($a_o$,$\gamma$) plane when conjugate coupling was induced in the oscillators set in a fixed-point dynamical state. FP represents the regime for fixed point, and O represents the regime for any of the periodic or chaotic dynamical states (rhythmogenesis) of the coupled oscillator.

Figure 10

Maximal Lyapunov exponent of the dynamics of a coupled oscillator in the case of rhythmogenesis for coupling constant $\gamma =-7.3$. Control was switched on at the integration step $6\times {10}^5$. Maximal Lyapunov exponent is positive for integration step $8\times {10}^5$ onward, indicating the generation of chaotic dynamics in the coupled oscillator.

Figure 11

Experimental setup: Here $V=12V$,$V$($C$${}_1$)${}_1$, and $V$($C$${}_2$)${}_2$ (i.e., the potential difference across capacitors of two Chua circuits) are the conjugate variables used to couple two Chua circuits. IC no. 1 is the instrumentation amplifier, which calculates the difference of two conjugate variables, and IC nos. 2–11 are simple operational amplifiers.

Figure 12

Amplitude death in nonidentical oscillators: (a) and (b) Autonomous dynamics of two Chua circuits; (c) and (d) dynamics of the system when the circuits were mutually coupled using conjugate variables. $R$ for the first oscillator was set at 1571.4 $\Omega$, and that for second oscillator was set at 1601.6 $\Omega$. Amplitude death was reached in both oscillators for $\gamma =6.98\times {10}^{-2}$.

Figure 13

Experimental bifurcation diagram of one of the coupled oscillators as a function of coupling constant $\gamma$ in the case of amplitude death. For $\gamma >0.07$ the oscillator undergoes Hopf bifurcation and makes a transition from a period-one limit cycle to fixed-point dynamics.

Figure 14

Maximal Lyapunov exponent of the dynamics of a coupled oscillator in the case of amplitude death for coupling constant $\gamma =0.0748$. Control was switched on after 0.5 msec. Negative Lyapunov exponent after 0.5 msec indicates a fixed point in the coupled oscillator.

Figure 15

Rhythmogenesis in nonidentical oscillators: (a) and (b) Autonomous dynamics of two Chua circuits; (c) and (d) dynamics of the system when the circuits were mutually coupled using conjugate variables. $R$ for the first oscillator was set at 1606.8 $\Omega$, and that for the second oscillator was set at 1607 $\Omega$. Rhythmogenesis was reached in both oscillators for $\gamma =20.80\times {10}^{-3}$.