Desynchronization and pattern formation in a noisy feed-forward oscillator network

We consider a one-dimensional directional array of diffusively coupled oscillators. They are perturbed by the injection of small additive noise, typically orders of magnitude smaller than the oscillation amplitude, and the system is studied in a region of the parameters that would yield deterministic synchronization. Non-normal directed couplings seed a coherent amplification of the perturbation: this latter manifests as a modulation, transversal to the limit cycle, which gains in potency node after node. If the lattice extends long enough, the initial synchrony gets eventually lost, and the system moves toward a nontrivial attractor, which can be analytically characterized as an asymptotic splay state. The noise assisted instability, ultimately vehiculated and amplified by the non-normal nature of the imposed couplings, eventually destabilizes also this second attractor. This phenomenon yields spatiotemporal patterns, which cannot be anticipated by a conventional linear stability analysis.

Figure 1

Schematic representation of our system. Each node carries an oscillator and is unidirectionally coupled to its successive neighbor. Parameter $K$ modulates the coupling.

Figure 2

Splay state representation: The radius of the limit cycles over the chain ${\rho }_j$ is depicted by the red solid line (circles), and the blue line (squares) stands for the phase difference $(mod2\pi )$ between two successive nodes. As expected, they converge to the asymptotic values ${\rho }_{\infty },{\varphi }_{\infty }$ (dashed black lines). The parameters here are $c_1=-5,c_2=4$, and $K=4$.

Figure 3

Splay state existence: the color represents the value of ${\rho }_{\infty }$ [see Eq. (5a)], and the black zone refers to the region where the splay state does not exist. The solid lines correspond to the isocurves of ${\rho }_{\infty }$. Here $K=4$.

Figure 4

Synchronized state stability: We display the value of $K_{\min }^H$ [see Eq. (7)] beyond which the synchronized state is stable. In the black region $1+c_1{c}_2\ge 0$, and the synchronized state is always stable. The solid lines correspond to the isocurves of the heat map.

Figure 5

Splay state linear stability analysis The green curve (triangles) represents the real part of the largest eigenvalue for each node $j$, and the yellow line (diamonds) the corresponding imaginary part. The parameters here are $c_1=-5,c_2=4$, and $K=4$. In this example, the splay state is characterized by $\overline{j}=1$.

Figure 6

Diagram of the existence and stability of the synchronized and of the splay states for $K=4$: in region A, whose boundaries are fixed by the condition ${\rho }_{\infty }=0$, only the synchronized state exists and is stable; in region B both states exist, but only the synchronized one is stable; in region C both states are stable; and in region D the splay state only is stable.

Figure 7

(a) Normalized power spectra of different nodes along the chain: the solid lines stands for the theoretical calculation, and the symbols correspond to numerically computed power spectra using the Euler-Maruyama algorithm. The displayed agreement confirms the validity of the analytic calculations. In the inset: normalized power spectra for longitudinal fluctuations. (b) Trajectories of ${\rho }_j$. The amplification phenomenon can be clearly appreciated. (c) Phase portrait of $(X_j\left(t\right),Y_j\left(t\right))$ where $X_j$ and $Y_j$, respectively, stand for the real and the imaginary part of the complex variable $W_j$. Oscillations extend along the radial direction and progressively alter the unperturbed limit cycle profile. The parameters here are $c_1=-5,c_2=4,\sigma ={10}^{-5}$, and $K=4$. Each color designs a specfic node: one black (circles), two blue (squares), three green (up-pointing triangles), eight red (diamonds), nine violet (down-pointing triangles).

Figure 8

Time evolution for selected amplitude ${\rho }_j\left(t\right)$. Each solid line refers to the time series of ${\rho }_j\left(t\right)$. Each color defines a specific group; in red are depicted all the nodes that remain on the synchronized attractor (from the first to the 10th node (circles)). The 11th node (blue line (squares)) is the leftmost able to escape from the homogenous attractor. The successive nodes (12th green (up-pointing triangles) and 13th violet (diamonds)) converge progressively to the asymptotic value ${\rho }_{\infty }$ of the splay state. The horizontal black dashed lines correspond to the nonhomogeneous attractor displayed in Fig. 2. The parameters here are $c_1=-5,c_2=4,\sigma ={10}^{-5}$, and $K=4$.

Figure 9

Typical spatiotemporal pattern of our system, the “space” (nodes) is the $y$ axis while time is in abscissa. One can easily recognize the transient in which all the nodes are in the synchronized state. The orange plateau stands for the splay state (node $10\to 30$) and precedes the blurred region, where the system erratically jumps from one state to another. The parameters here are $c_1=-5,c_2=4,\sigma ={10}^{-5}$, and $K=4$.