Effects of noise near a homoclinic bifurcation in an electrochemical system

The interaction of external noise with an electrochemical oscillator (anodic dissolution of iron) has been studied experimentally on both sides of the homoclinic bifurcation. In the oscillatory regime the regularity of the limit cycle behavior was destroyed with increasing noise amplitude until no periodicity was observable any more. In contrast, the response of the excitable state (fixed point regime) became more regular at intermediate noise levels.

Figure 1

A schematic representation of the electrochemical cell with iron, copper, and saturated calomel electrodes being the anode, cathode, and reference, respectively.

Figure 2

Experimental bifurcation diagram constructed by varying (increasing and decreasing) the system parameter $\left(V\right)$. It shows that the parameter space is divided into two dynamical domains, namely, the fixed point behavior and autonomous periodic oscillations.

Figure 3

Analyzing in detail the autonomous dynamics at the upper end of the bifurcation diagram. (a) Incrementing time period $\left(T\right)$ of the autonomous period-1 behavior as the anodic voltage $\left(V\right)$ is increased. (b) The $\ln (V-V_h)$ vs $T$ curve can be fitted by a straight line indicating an underlying homoclinic bifurcation.

Figure 4

Destruction of the limit cycle behavior: $V_0=187\mathrm{mV}$. Time series of the anodic current $I$ for three different amplitudes of the superimposed noise (a) $A=0\mathrm{mV}$, (b) $A=20\mathrm{mV}$ and (c) $A=95\mathrm{mV}$. The corresponding power spectra reveal the effect of noise.

Figure 5

Emergence of coherence resonance: $V_0=400\mathrm{mV}$. The experimental normalized variance curve as the noise amplitude $\left(A\right)$ is monotonically increased. Corresponding time series of the anodic current $\left(I\right)$ for points (a) $A=245\mathrm{mV}$, (b) $A=260\mathrm{mV}$, (c) $A=365\mathrm{mV}$ of the NV curve are also shown.

Figure 6

Emergence of coherence resonance: $V_0=430\mathrm{mV}$. The experimental normalized variance curve as the noise amplitude $\left(A\right)$ is monotonically increased. Corresponding timeseries of the anodic current $\left(I\right)$ for points (a) $A=250\mathrm{mV}$, (b) $A=265\mathrm{mV}$, (c) $A=400\mathrm{mV}$ of the NV curve are also shown.

Figure 7

Emergence of coherence resonance in the numerical model. Two normalized variance curves as the noise amplitude $\left(A\right)$ is monotonically increased for two different set points. The dashed curve corresponds to the set point $v_0=29.245$ while the full curve corresponds to the set point $v_0=29.238$. Corresponding time series of the anodic current $\left(I\right)$ for points (a), (b), and (c) of the NV curve for $v_0=29.238$ are also shown. Consistent with experiments, the figure shows the inception of maximum regularity for an optimum level of noise.