Forced symmetry breaking as a mechanism for rogue bursts in a dissipative nonlinear dynamical lattice

We propose an alternative to the standard mechanisms for the formation of rogue waves in a nonconservative, nonlinear lattice dynamical system. We consider an ordinary differential equation (ODE) system that features regular periodic bursting arising from forced symmetry breaking. We then connect such potentially exploding units via a diffusive lattice coupling and investigate the resulting spatiotemporal dynamics for different types of initial conditions (localized or extended). We find that in both cases, particular oscillators undergo extremely fast and large amplitude excursions, resembling a rogue wave burst. Furthermore, the probability distribution of different amplitudes exhibits bimodality, with peaks at both vanishing and very large amplitude. While this phenomenology arises over a range of coupling strengths, for large values thereof the system eventually synchronizes and the above phenomenology is suppressed. We use both distributed (such as a synchronization order parameter) and individual oscillator diagnostics to monitor the dynamics and identify potential precursors to large amplitude excursions. We also examine similar behavior with amplitude-dependent diffusive coupling.

Figure 1

Three types of initial conditions showing the distribution of initial amplitudes ${\mathcal{A}}_i(t=0)$ in a system of $N=37$ coupled oscillators. The red line with circle markers shows a single peak initial condition, the blue line with crosses shows a sine wave initial condition, and the black line with square markers shows a uniformly distributed random initial condition. In all three cases the Riemann sums pertaining to the three curves are equal.

Figure 2

Left: Space-time plot showing the evolution of ${log}_{10}\left({\mathcal{A}}_i\right)$ starting from a sine initial condition in a system of $N=37$ coupled oscillators, $1\le i\le 37$, with $K=2.1544\times {10}^{-6}$. The diagonal arrows locate the five largest amplitude excursions in this space-time evolution. Right: Zoom of the space-time plot to provide a close-up view of the largest amplitude event in the left panel.

Figure 3

Top: Evolution of large amplitude event at $i=19$ (shown as black line) as seen in the right panel of Fig. 2 along with the amplitude evolution of its immediate neighbors at $i=18$ (red line) and $i=20$ (blue line). Bottom: Evolution of large amplitude event at $i=30$ (shown as black line) as seen in the left panel of Fig. 4 along with the amplitude evolution of its immediate neighbors at $i=29$ (red line) and $i=31$ (blue line).

Figure 4

Space-time plot showing the evolution of ${log}_{10}\left({\mathcal{A}}_i\right)$ starting from a sine initial condition in a system of $N=37$ coupled oscillators with $K=1\times {10}^{-4}$ (left panel) with maximum amplitude occurring at $i=30$, and $K=2.2\times {10}^{-3}$ (right panel) with maximum amplitude occurring at $i=10$. The left panel shows that the large amplitude feature is not localized at a single oscillator and that neighboring oscillators respond coherently. In the right panel, all oscillators are synchronized and so oscillate in spatial coherence.

Figure 5

Logarithm of the probability density function $p_{{\mathcal{A}}_{\mathrm{thresh}}}$ for different amplitude thresholds ${\mathcal{A}}_{\mathrm{thresh}}$ from the time evolution shown in Fig. 2. We observe an increase in the probability of extremely large amplitudes, indicating that the occurrence of extreme events is more probable than what is expected from an exponential distribution of amplitudes.

Figure 6

Red lines with circles show the results for the single peak initial condition, blue lines with crosses show the results for the sine wave initial condition, while black lines with square markers show results for the random initial condition. The top panel shows the variation in the maximum amplitude ${\mathcal{A}}_{max}$ (on logarithmic scale) as a function of the diffusion coefficient $K$. The middle panel shows the time $t_{max}$ taken to reach the maximal excitation, also on logarithmic scale, while the bottom panel shows the variation of the location $i=N_{max}$ of that maximal event among the individual oscillators.

Figure 7

Asymptotic value of the Kuramoto order parameter, $r_a={lim}_{t\to \infty }r\left(t\right)$, as a function of the coupling coefficient $K$ for different initial conditions: the case of a single peak initial condition (red lines with circles), a sine wave initial condition (blue lines with crosses), and a uniformly distributed random initial condition (black line with square markers).

Figure 8

Asymptotic value of the Kuramoto order parameter, $r_a={lim}_{t\to \infty }r\left(t\right)$, as a function of the coupling coefficient $K$ for the single peak initial condition (red lines with dots) and the sine wave initial condition (blue lines with crosses) starting with a larger initial amplitude distribution.

Figure 9

Large amplitude events at $i=19$ as seen in the right panel of Fig. 2 in the time domain shown superposed on locations where the precursor $P$ identifies rapid growth (brown circles) due to the alignment of the trajectory with the stable direction of the saddle solution $u_{\infty }$ (top panel) and of the $v_{\infty }$ solution (bottom panel) of a single uncoupled oscillator.

Figure 10

Top: Phase portrait in the ($\rho , \varphi$) plane during regular periodic spiking at a single node in the network with no coupling (green line). Bottom: Same view in the ($\rho , \varphi$) plane with superposed evolution close to the large amplitude event at $i=19$ (black line) in the coupled network with $K=2.1544\times {10}^{-6}$, for comparison with Fig. 9.

Figure 11

Same as Fig. 6 but for the amplitude-dependent coupling (8). Red lines with circle markers show results for the single peak initial condition while blue lines with plus markers are the results for the sine wave initial condition. The top panel shows the variation in the maximum amplitude of observed excitations (on logarithmic scale) as a function of the coupling coefficient $K_0$. The middle panel shows the variation in the time taken to reach the maximal excitation $t_{max}$ on logarithmic scale, also as a function of $K_0$. Finally, the bottom panel shows the variation of the location $N_{max}$ of the maximal amplitude among the individual oscillators as a function of $K_0$. Note that the single peak initial conditions were initialized at $i=12$.