Control of extreme events in the bubbling onset of wave turbulence

We show the existence of an intermittent transition from temporal chaos to turbulence in a spatially extended dynamical system, namely, the forced and damped one-dimensional nonlinear Schrödinger equation. For some values of the forcing parameter, the system dynamics intermittently switches between ordered states and turbulent states, which may be seen as extreme events in some contexts. In a Fourier phase space, the intermittency takes place due to the loss of transversal stability of unstable periodic orbits embedded in a low-dimensional subspace. We mapped these transversely unstable regions and perturbed the system in order to significantly reduce the occurrence of extreme events of turbulence.

Figure 1

Time evolution of the spatial profile of the wave amplitude $\left|\psi \right(x,t\left)\right|$ and its mass $M\left(t\right)$ (blue, upper curve) and Hamiltonian $H\left(t\right)$ (green, lower curve) for (a),(b) a temporally chaotic case ($\epsilon =0.21$) and (c),(d) a turbulent case ($\epsilon =0.44$). The time interval used in the spatial profiles [(b) and (d)] is $\Delta t=50$.

Figure 2

(a) The ten largest Lyapunov exponents as a function of the forcing parameter, showing that there is an increase both in the absolute value of the positive exponents and in the number of positive ones; and (b) spectral mean showing that the increase in chaoticity is accompanied by an increase in the number of excited Fourier modes

Figure 3

Time evolution for $\epsilon =0.3$, an intermediate parameter between a laminar and the fully turbulent state, with (a) integrals $M\left(t\right)$ (blue, upper curve) and $H\left(t\right)$ (green, lower curve) showing the global intermittent feature of the system; (b) ${\xi }_0\left(t\right)$, (c) ${\xi }_{14}\left(t\right)$ showing that the system intermittently alternates between on and off states; and (d) spatial profile showing the moment of a turbulent burst event ($\Delta t=150$).

Figure 4

Two-dimensional PDF of a phase-space projection on the plane ${\xi }_0\times {\xi }_1$ for $\epsilon =0.3$. (a) ${\rho }_{\mathsf{A}}$ [Eq. (4)], where only points inside the laminar state $\mathsf{A}$ (${\xi }_{14}<{10}^{-8}$) were taken into account; (b) for points just before the system escapes the laminar state ${\rho }_E$ [Eq. (5)]; and (c) the difference between ${\rho }_E$ and ${\rho }_{\mathsf{A}}$, where the most positive areas (green, light gray) represent an approximation of the transversely unstable regions in the laminar state.

Figure 5

PDF for the system dynamics without control (blue solid line) and with control (green dashed line) for (a) ${\xi }_0$ Fourier mode amplitude in the longitudinal direction, (b) ${\xi }_{14}$ Fourier mode amplitude in the transversal direction, (c) the mass $M$ of the system, and (d) the negative of the system Hamiltonian $H$. The percentages represent how much the probability of extreme events with control (green, light gray area) is reduced relative to the case without control (blue, dark gray areas).