Reconstruction of two-dimensional phase dynamics from experiments on coupled oscillators

Phase models are a powerful method to quantify the coupled dynamics of nonlinear oscillators from measured data. We use two phase modeling methods to quantify the dynamics of pairs of coupled electrochemical oscillators, based on the phases of the two oscillators independently and the phase difference, respectively. We discuss the benefits of the two-dimensional approach relative to the one-dimensional approach using phase difference. We quantify the dependence of the coupling functions on the coupling magnitude and coupling time delay. We show differences in synchronization predictions of the two models using a toy model. We show that the two-dimensional approach reveals behavior not detected by the one-dimensional model in a driven experimental oscillator. This approach is broadly applicable to quantify interactions between nonlinear oscillators, especially where intrinsic oscillator sensitivity and coupling evolve with time.

Figure 1

(a) Main synchronization tongue for the two-dimensional model Eq. (7) (solid line) and the tongue predicted by the one-dimensional model $q_{1,1}$ (dashed line). (b) The second tongue at ${\omega }_2\approx 2$ for the two-dimensional model (solid line) and for the one-dimensional model $q_{2,1}$ (dashed line). Here the averaging works well and the difference is pronounced only far from resonance (see inset). (c) Devil’s staircase for the toy model Eq. (7) with ${\omega }_1=1$, for $\varepsilon =0.3$; here $\Omega =\langle \dot{\phi }\rangle$ where $\langle ·\rangle$ is time average. Locking regions at $\Omega /{\omega }_2=2$, $\Omega /{\omega }_2=1$, $\Omega /{\omega }_2=2/3$, and $\Omega /{\omega }_2=1/2$ are seen. Zoom of the plot (not shown) exhibits further locking ratios, e.g., $3/2$, $4/5$, $3/4$, and $3/5$.

Figure 2

(a) Experimental apparatus with multichannel addressable voltage and feedback. (b) Time series of relaxational oscillations at $V=1.210$V. (c) Electrochemical dissolution time series showing smooth oscillations at a potential of $V=1.105$V, ZRA: zero resistance ammeter.

Figure 3

Time series in Hilbert space; ${\theta }_i$ indicates protophase (a) for the relaxational oscillator and (b) for the smooth oscillator. Note that although the phase portraits look very much alike, the distributions of ${\theta }_i$ are different, as shown in Fig. 4(b).

Figure 4

(a) Wrapped protophase of oscillator 1 (relaxational, $V_1=1.180$V) versus protophase of oscillator 2 (smooth, $V_2=1.105$V) obtained via the Hilbert transform. (b) Phase transformation function, $\sigma$. $\sigma \left({\theta }_1\right)$ is dash-dotted, and $\sigma \left({\theta }_2\right)$ is solid. (c) Wrapped genuine phases of the two oscillators, ${\phi }_1$ and ${\phi }_2$.

Figure 5

Coupling function for oscillator 2 based on phase difference, $q^{\left(2\right)}({\phi }_1-{\phi }_2)$. (a) No time delay, $\tau =0$. (b) Time delay equal to roughly three-quarters of oscillator 2 natural period, $\tau =1.8\text{s}$.

Figure 6

Coupling function in Hz for oscillator 2 based on each phase independently, $Q^{\left(2\right)}$(${\phi }_2$,${\phi }_1$). (a) No time delay, $\tau =0$. (b) Time delay equal to three-quarters the natural period of oscillator 2, $\tau =1.8\text{s}$.

Figure 7

(Top row) $Q^{\left(1\right)}$(${\phi }_1$,${\phi }_2$); (bottom row) $Q^{\left(2\right)}$(${\phi }_2$,${\phi }_1$). $k_2$ = 1.0 for all plots. [(a), (d)] $k_1$ = 1.0, [(b), (e)] $k_1$ = 0.5, and [(c), (f)] $k_1$ = 0.1.

Figure 8

(a) Relaxational oscillator: Norm versus gain $k_1$, linear fit: $\left|\right|Q^{\left(1\right)}\left|\right|$= 0.045$k_1$ + 0.003, $R^2$ = 0.998. (b) Smooth oscillator: Norm versus gain $k_2$, linear fit: $\left|\right|Q^{\left(2\right)}\left|\right|$ = 0.032 $k_2$ + 0.001, $R^2$ = 1.000, where $R^2$ is the square of the correlation coefficient.

Figure 9

Experimental results: (a) $Q^{\left(1\right)}$ with $\nu /\omega$ = 0.95, (b) $Q^{\left(1\right)}$ with $\nu /\omega$ = 1.05, (c) $q_{1,1}^{\left(1\right)}$ with $\nu /\omega$ = 0.95, and (d) $q_{1,1}^{\left(1\right)}$ with $\nu /\omega$ = 1.05.