Synchronization of nonlinearly coupled Stuart-Landau oscillators on networks

TL;DR

This paper addresses the synchronization of $N$ identical Stuart-Landau oscillators nonlinearly coupled on networks, including directed graphs. It develops a Master Stability Function-based linear stability analysis around the synchronous limit cycle $W_{LC}(t)$, treating the resonant case $a=b+1$ analytically (autonomous) and the general nonresonant case via Floquet theory for time-periodic linearization, complemented by a Jacobi–Anger based semi-analytical approximation. The authors derive explicit conditions and dispersion relations that determine when complete synchronization is achieved or lost, and show that directed networks with complex Laplacian spectra can prevent synchronization even in the resonant setting; they also provide a semi-analytical method that closely matches numerical Floquet exponents. Overall, the work extends classical synchronization theory to nonlinear interconnections on networks and lays groundwork for future exploration of higher-order interactions in complex oscillator systems.

Abstract

The dynamics of coupled Stuart-Landau oscillators play a central role in the study of synchronization phenomena. Previous works have focused on linearly coupled oscillators in different configurations, such as all-to-all or generic complex networks, allowing for both reciprocal or non-reciprocal links. The emergence of synchronization can be deduced by proving the linear stability of the limit cycle solution for the Stuart-Landau model; the linear coupling assumption allows for a complete analytical treatment of the problem, mostly because the linearized system turns out to be autonomous. In this work, we analyze Stuart-Landau oscillators coupled through nonlinear functions on both undirected and directed networks; synchronization now depends on the study of a non-autonomous linear system and thus novel tools are required to tackle the problem. We provide a complete analytical description of the system for some choices of the nonlinear coupling, e.g., in the resonant case. Otherwise, we develop a semi-analytical framework based on Jacobi-Anger expansion and Floquet theory, which allows us to derive precise conditions for the emergence of complete synchronization. The obtained results extend the classical theory of coupled oscillators and pave the way for future studies of nonlinear interactions in networks of oscillators and beyond.

Figures (7)

• Figure 1: Complete synchronization of Stuart-Landau oscillators coupled with a symmetric network, $a=b+1$. The left column corresponds to $\mu=1+i$, $a=1$ and $b=0$, the dispersion relation (blue curve and red dots, top panel) is negative and the oscillators synchronize as we can appreciate by looking at the time evolution of $\Re W_j(t)$ (bottom panel). The middle column shows the results for to $\mu=1-i$, $a=1$ and $b=0$, the curve $\lambda(x)$ is positive (blue curve, top panel) however the dispersion relation is negative (red dots, top panel) and the system still exhibits synchronization as testified by the behavior of $\Re W_j(t)$ (bottom panel). The right column represents the case $\mu=1-i$, $a=5$ and $b=4$, both the curve $\lambda(x)$ and the dispersion relation are positive (blue curve and red dots, top panel) and the system is not able to exhibit synchronization as shown by the time evolution of $\Re W_j(t)$ (bottom panel). The coupling is represented by an Erdős–Rényi symmetric network composed of $N=150$ nodes and a probability $p=0.03$ for an edge to exist among any couple of nodes. The remaining model parameters are $\sigma=0.5$ and $\beta=1+2i$.
• Figure 2: The dependence on the parameter $a$ of the nonzero root, $x_2$, of $\lambda(x)$. By using the parameters selected in the middle and right column of Fig. \ref{['fig:AutoUnidrectednetwork']}, i.e., $\sigma=0.5$, $\beta=1+2i$ and $\mu=1-i$, we show the variation of $x_2$ as a function of $a$. Because $|W_{\mathrm{LC}}|=1/\sqrt{2}<1$ the function exhibits a non-monotone behavior as mentioned in the text. We emphasized two values, $x_2=1/2$ for $a=1$ (red diamond) and $x_2\sim 0.88$ for $a=5$ (green square).
• Figure 3: Complete synchronization of Stuart-Landau oscillators coupled via a directed network, $a=b+1$. We show the results of SL oscillators coupled via a directed network composed of $N=150$ nodes obtained by using the Watts-Strogatz algorithm with $p_{WS}=0.9$ the probability to rewire any directed network starting from a undirected ring where each node has degree $2$. The model parameters have been set to $\sigma = 1.0$, $\beta=1+2i$ and $\mu=1+2i$; moreover, $a=1$ (top row) and $a=4$ (bottom row). Panels in the left column report the curve $\lambda(x)$ (blue) and the dispersion relation $\lambda(\Lambda^{(\alpha)})$ (red dots). It can be observed that now the latter deviates from the curve, this being a signature of the presence of a nonzero imaginary part of the spectrum of the Laplace matrix. Moreover, in the top panel, the dispersion relation assumes positive values, while this does not happen for the parameters corresponding to the bottom panel. The middle columns represent the region of the complex plane where the condition $S_2(x) < y^2S_1(x)$ is satisfied (green region). It can be observed that, in the top left panel, some complex eigenvalues (black dots) belong to the instability region, thus impeding the system to synchronize, as shown in the bottom left panel, where we report $\Re W_j(t)$. By increasing $a$, we can observe that the instability region shrinks (top right panel) and all the complex eigenvalues (black dots) fall outside the instability region; this allows complete synchronization (see bottom right panel).
• Figure 4: Complete synchronization of Stuart-Landau oscillators coupled via a symmetric network, $a\neq b+1$. The left column corresponds to the set of parameters $\sigma=0.5$, $\beta=1+2i$, $\mu=1+i$, $b=0$ and $a=2$, one can appreciate that the maximum Floquet exponent achieves positive values for some Laplace eigenvalue (red dot, top left panel) and the system cannot synchronize, as shown by the time evolution of $\Re W_j(t)$ (bottom left panel). The right column presents the results for $\sigma=0.5$, $\beta=1+2i$, $\mu=1+i$, $b=0$ and $a=5$. Now, the maximum Floquet exponent remains negative for all $\Lambda^{(\alpha)}$ (red dot, top right panel) and the system completely synchronizes (bottom right panel). In both top panels, the function $\zeta(x)$ is shown as eye-guide (blue curve). The underlying network is a random Erdős-Rényi graph made of $N=50$ nodes and the probability to have a link is given by $p=0.1$.
• Figure 5: Absence of complete synchronization of Stuart-Landau oscillators coupled via a directed network, $a\neq b+1$. The left panel shows the maximum Floquet exponent (red dots), together with the function $\zeta(x)$ shown as eye-guide (blue curve). We can observe that $\zeta(\Lambda^{(\alpha)})$ takes a positive value in correspondence of a given $\Lambda^{(\alpha)}$. The middle panel provides a complementary view of the former panel, by reporting the region of instability in the complex plane, i.e., where the maximum Floquet exponent is positive. There, we can identify again a couple of complex conjugated eigenvalues lying in the instability region. The system can thus not synchronize, as shown in the right panel, where we plot $\Re W_j(t)$. The model parameters are $\sigma=0.5$, $\beta=1+2i$, $\mu=1+i$, $b=0$ and $a=5$; the underlying network is a directed random Erdős-Rényi graph made of $N=50$ nodes, and the probability to have a link is given by $p=0.08$.

Theorems & Definitions (1)

• Remark 1: The case of real eigenvalues of the Laplace matrix