Diffusion-induced instability and chaos in random oscillator networks

We demonstrate that diffusively coupled limit-cycle oscillators on random networks can exhibit various complex dynamical patterns. Reducing the system to a network analog of the complex Ginzburg-Landau equation, we argue that uniform oscillations can be linearly unstable with respect to spontaneous phase modulations due to diffusional coupling—the effect corresponding to the Benjamin-Feir instability in continuous media. Numerical investigations under this instability in random scale-free networks reveal a wealth of complex dynamical regimes, including partial amplitude death, clustering, and chaos. A dynamic mean-field theory explaining different kinds of nonlinear dynamics is constructed.

Figure 1

(Color online) (a) Degree $k_j$ and (b) Laplacian eigenvalue ${\Lambda }^{\left(\alpha \right)}$ of the scale-free network used in numerical simulations. (c) Linear growth rates of perturbations $\text{Re}{\lambda }_{\pm }^{\left(\alpha \right)}$ plotted as functions of $-K{\Lambda }^{\left(\alpha \right)}$. Dashed green line shows the slope $-\left(1+c_1{c}_2\right)$ of the upper branch at the origin.

Figure 2

(Color online) [(a)–(c)] Snapshots of the complex amplitude $W_j$ on the complex plane. [(d)–(f)] Snapshots of the amplitude $\left|W_j\right|$ vs the node index $j$. Coupling strength is $K=0.02$ for (a) and (d), $K=0.04$ for (b) and (e), and $K=0.08$ for (c) and (f). Node indices are sorted in decreasing order of their degrees $\left\{k_j\right\}$ so that inequalities $k_1\ge k_2\ge \cdots \ge k_N$ hold. Solid blue curves in (d)–(f) are predictions of the mean-field theory.

Figure 3

(Color online) [(a), (d), and (g)] Evolution of real (red circles) and imaginary parts (blue crosses) of the global mean field $H\left(t\right)$ and fitting by $B\text{exp}\left(i\Omega t\right)$. [(b), (e), and (h)] Bifurcation diagrams of the sinusoidally driven oscillator. Insets show limit-cycle orbits or fixed points of $V\left(t\right)$ at the parameter values indicated by broken vertical lines. [(c), (f), and (i)] Probability density functions of $\left|W_j\right|$ compared with the mean-field approximation (yellow or light gray represents lower density, and red or dark gray represents higher density, with white representing zero). Solid blue curves represent the maximal and minimal values of the complex amplitude $\left|W_j\right|$ under the mean-field approximation. Parameters are $K=0.02$ and $B=0$ for (a)–(c); $K=0.04$, $B=0.442$, and $\Omega =-1.25$ for (d)–(f); and $K=0.08$, $B=0.532$, and $\Omega =-0.796$ for (g)–(i).