Early detection of synchrony in coupled oscillator model

Abstract

In this paper, we study the applicability of an early warning index while studying the transitions to complete and generalized synchronizations in the coupled oscillator models using an unconventional system parameter and the coupling strength as the required control parameters. The coupled oscillator models are widely used and well-documented models for studying various aspects of nature. However, the early warning index used in this paper is an explicit function of the mutual information of the coupled oscillators and reaches two different values before the interacting oscillators yield complete and generalized synchronizations. The transitions to synchrony using the unconventional control parameter are associated with a transition to periodic dynamics of the individual oscillators from their initial chaotic dynamics. Besides, when we use the coupling strength as a control parameter, the interacting oscillator exhibits chaotic dynamics during the synchronizations. Our analysis mainly involves different examples of two low-dimensional oscillators. Finally, we extend our study to a network of interacting oscillators. The applicability of the early detection index is verified in all cases.

Appendices

Appendix

A. Coupled Chen oscillators

In this section, we consider the example of mutually coupled Chen [48] oscillators. The explicit form of the equations of motion is given by:

$$\begin{aligned} \frac{{\text{d}}{x}_1}{{\text{d}}t}= a(y_1 - x_1), \end{aligned}$$

(13a)

$$\begin{aligned} \frac{{\text{d}}{y}_1}{{\text{d}}t}= (c-a)x_1 -x_1 z_1 + cy_1 + \alpha (y_2-y_1), \end{aligned}$$

(13b)

$$\begin{aligned} \frac{{\text{d}}{z}_1}{{\text{d}}t}= y_1 x_1 - bz_1, \end{aligned}$$

(13c)

$$\begin{aligned} \frac{{\text{d}}{x}_2}{{\text{d}}t}= a(y_2 - x_2), \end{aligned}$$

(13d)

$$\begin{aligned} \frac{{\text{d}}{y}_2}{{\text{d}}t}= (c-a)x_2 -x_2 z_2 + cy_2 + \alpha (y_1-y_2), \end{aligned}$$

(13e)

$$\begin{aligned} \frac{{\text{d}}{z}_2}{{\text{d}}t}= y_2 x_2 - bz_2, \end{aligned}$$

(13f)

with $a=35$, $b = 3$, and $c = 28$. Here, we have coupled the y-coordinates of the interacting oscillators. In order to study the transition to complete synchrony, the coupling strength is chosen as a control parameter. Also, we have calculated the maximum conditional Lyapunov exponent ($\lambda ^{\bot }_{\mathrm{max}}$) [14] of Eq. 13 for different values of $\alpha$. By definition, the negativity of $\lambda ^{\bot }_{\mathrm{max}}$ implies the complete synchronization state. The maximum conditional Lyapunov exponent is a suitable measure to detect the complete synchronization between two diffusively coupled identical oscillators, where the explicit form of equations of motion is available.

Fig. 11

Early detection of complete synchrony in coupled Chen oscillators: a The maximum conditional Lyapunov exponent ($\lambda ^{\bot }_{\mathrm{max}}$) is plotted as a function of the coupling strength ($\alpha$). $\lambda ^{\bot }_{\mathrm{max}}$ becomes negative at $\alpha = 2.14$. The horizontal gray dashed-line corresponds to $\lambda ^{\bot }_{\mathrm{max} }= 0$. b The early detection index (R) is plotted as a function of the control parameter $\alpha$. R reaches zero at $\alpha \simeq 2.1$ and saturates before the coupled oscillators (Eq. 13) yield the generalized synchronization state. The vertical blue dashed-line is drawn at $\alpha = 2.1$

In analysis, the initial condition to integrate Eq. 13 is chosen as (− 10, 0, 37, − 10.1, 0.1, 37.1). The maximum conditional Lyapunov exponent becomes negative for $\alpha \ge 2.14$ (Fig. 11a). Hence, the coupled Chen oscillators yield the complete synchronized state at $\alpha = 2.14$. Besides, Fig. 11b depicts that R reaches zero at $\alpha = 2.10$ and saturates. We use the coordinates $x_1(t)$ and $x_2(t)$ to calculate R at each $\alpha$.

B. Coupled Rössler oscillators with noise

Now, the example of two coupled Rössler oscillators is considered in the presence of noise. The explicit form of the coupled oscillators are as follow:

$$\begin{aligned} \frac{{\text{d}}{x}_{1,2}}{{\text{d}}t}= -y_{1,2} - z_{1,2} + \alpha (x_{2,1} - x_{1,2}) + D \eta (t), \end{aligned}$$

(14a)

$$\begin{aligned} \frac{{\text{d}}{y}_{1,2}}{{\text{d}}t}= x_{1,2} + 0.2 y_{1,2}, \end{aligned}$$

(14b)

$$\begin{aligned} \frac{{\text{d}}{z}_{1,2}}{{\text{d}}t}= 0.2 + z_{1,2} (x_{1,2} - 5.7). \end{aligned}$$

(14c)

We introduce the noise term in Eq. 14a. Here, D represents the noise amplitude and $\eta (t)$ is adopted from a Gaussian distribution (of zero mean and unit variance) randomly, i.e., $\left\langle \eta (t) \eta (t') \right\rangle = \delta (t-t')$.

The initial condition for Eq. 14 is chosen as (− 9, 0, 0,   9.1, 0.1, 0). The maximum conditional Lyapunov exponent ($\lambda ^{\bot }_{\mathrm{max}}$) of Eq. 14 with $D = 0$ becomes negative at $\alpha \simeq 0.11$ (Fig. 12a). It implies that coupled Rössler oscillators (Eq. 14) lead the complete synchronized state for $\alpha \ge 0.11$ in the absence of noise (i.e., $D = 0$). Figure 12b depicts the variations of R as a function of $\alpha$ at $D = 1 \times 10^{-5}$. For a fixed value of $\alpha$, the plotted R in Fig. 12b is the average of all R values calculated over 50 realizations of the noise $\eta (t)$. R reaches a small number at $\alpha \simeq 0.09$ and almost saturates. Therefore, Fig. 12 helps to conclude that the applicability of R is robust in the presence of noise.

Fig. 12

Robustness of the early detection index in the presence of noise: a The maximum conditional Lyapunov exponent ($\lambda ^{\bot }_{\mathrm{max}}$) becomes negative for $\alpha \ge 1.1$. The horizontal gray dashed-lines corresponds to $\lambda ^{\bot }_{\mathrm{max}} = 0$. b The early detection index (R) is plotted as a function of the control parameter $\alpha$ in the presence of noise with noise strength $D = 1 \times 10^{-5}$. R reaches a nonzero small number and saturates before the coupled oscillators (Eq. 15) yield the complete synchronization state. The vertical blue dashed-line is drawn at $\alpha = 0.09$

C. Network of Rössler oscillators

Until now, we have studied different examples of two coupled oscillators. This section extends our study to a network of interacting oscillators. We consider a ring of N mutually coupled Rössler oscillators. Following is the equations of motion of the $i^{\mathrm{th}}$ oscillator:

$$\begin{aligned} \frac{{\text{d}}{x}_i}{{\text{d}}t}= -y_i - z_i + \alpha (x_{i+1} + x_{i-1} - 2x_i), \end{aligned}$$

(15a)

$$\begin{aligned} \frac{{\text{d}}{y}_i}{{\text{d}}t}= x_i + 0.2 y_i, \end{aligned}$$

(15b)

$$\begin{aligned} \frac{{\text{d}}{z}_i}{{\text{d}}t}= 0.2 + z_i (x_i - 5.7), \end{aligned}$$

(15c)

where $i = 1, 2, \ldots , N$, with $x_{N+1} = x_1$ and $x_{-1} = x_N$. Here, the coupling strength $\alpha$ is the required control parameter, and we vary $\alpha$ monotonically within the range [0, 0.5]. In analysis, we adopt $N = 10$ and the initial conditions of interacting 10 oscillators are chosen randomly from an uniform distribution with the boundaries $-0.1$ and 0.1. The variation of $\lambda ^{\bot }_{\mathrm{max}}$ as a function of $\alpha$ infers that all 10 oscillators yield complete synchrony for $\alpha \ge 0.41$ (Fig. 13). Besides, in Fig. 14, we have plotted the early detection index R using x-coordinates of the neighbor oscillators. An overall decreasing nature of R is detected in all cases and R saturates at zero in most cases. To this end, note that over this range of $\alpha$ (i.e., $\alpha \in [0, 0.5]$), each interacting Rössler oscillator exhibits chaotic oscillation.

Fig. 13

Transition to complete synchrony in a ring of coupled Rössler oscillators: The maximum conditional Lyapunov exponent ($\lambda ^{\bot }_{\mathrm{max}}$) of Eq. 15 is plotted as a function of the control parameter $\alpha$, and $\lambda ^{\bot }_{\mathrm{max}}$ becomes negative at $\alpha = 0.41$. It implies that the coupled oscillators (Eq. 15) yield the complete synchronization state for $\alpha \ge 0.41$. The horizontal gray dashed-line is drawn at $\lambda ^{\bot }_{\mathrm{max}} = 0$

Fig. 14

Early detection of complete synchrony in a ring of coupled Rössler oscillators: The early detection index (R) is plotted as a function of the control parameter $\alpha$ using the x-coordinates of the interacting oscillators. In most cases, R reaches zero and saturates before the coupled oscillators (Eq. 15) yield the complete synchronization state. The vertical red dashed-lines in all subplots are at $\alpha = 0.41$