Dynamical indicators for the prediction of bursting phenomena in high-dimensional systems

Drawing upon the bursting mechanism in slow-fast systems, we propose indicators for the prediction of such rare extreme events which do not require a priori known slow and fast coordinates. The indicators are associated with functionals defined in terms of optimally time-dependent (OTD) modes. One such functional has the form of the largest eigenvalue of the symmetric part of the linearized dynamics reduced to these modes. In contrast to other choices of subspaces, the proposed modes are flow invariant and therefore a projection onto them is dynamically meaningful. We illustrate the application of these indicators on three examples: a prototype low-dimensional model, a body-forced turbulent fluid flow, and a unidirectional model of nonlinear water waves. We use Bayesian statistics to quantify the predictive power of the proposed indicators.

Figure 1

An illustration of slow-fast systems with bursting orbits homoclinic to the slow manifold. While we depict the slow manifold with a plane, it can in reality be a complicated high-dimensional manifold.

Figure 2

An illustration of the OTD modes. The OTD modes ${\mathbf{v}}_i$ remain orthonormal for all times (the dark black squares mark right angles). While differing from their images under the linear dynamics ${\mathrm{\Phi }}_{t_0}^t$, the OTD modes span the same subspace as their images.

Figure 3

A trajectory of the system (12) with parameters $\alpha =0.01, \omega =2\pi , \lambda =0.1$, and $\beta =0.1$. The initial condition is $(0,0.01,0.01)$. (a) The trajectory $\mathbf{u}\left(t\right)$ in the state space. (b) The time series of the $z$ component of the trajectory for $4\times {10}^3$ time units.

Figure 4

(a) The evolution of $\sqrt{v_{1,1}^2+v_{1,2}^2}=\sqrt{1-v_{1,3}^2}$ (blue) and $v_{1,3}$ (red), where ${\mathbf{v}}_1=(v_{1,1},v_{1,2},v_{1,3})$. (b) The evolution of the eigenvalue ${\lambda }_1$ as a function of time. The dashed red line marks ${\lambda }_1=0$. Three closeup views are shown in the insets.

Figure 5

Evolution of the energy dissipation $D$ along a trajectory of the Kolmogorov flow (14) with $n=4$ and $\text{Re}=40$.

Figure 6

The Kolmogorov flow in the asymptotically stable regime with $n=1$ and $\text{Re}=40$. (Top row) The vorticity field at times $t=0$, 2, and 100. (Middle row) Curl of the first OTD mode ${\mathbf{v}}_1$ at $t=0$, 2, and 100. (Bottom row) Curl of the second OTD mode ${\mathbf{v}}_2$ at $t=0$, 2, and 100. The colors correspond to the only nonzero component of the curls. All panels show the entire domain $[0,2\pi ]\times [0,2\pi ]$.

Figure 7

Eigenvalues of the symmetric matrix ${\mathbf{S}}_2$ along a trajectory of the Kolmogorov flow in the asymptotically stable regime: $n=1$ and $\text{Re}=40$.

Figure 8

(a) Energy input $I$ versus energy dissipation $D$ shown for a long turbulent trajectory. The dots correspond to $5\times {10}^4$ time instances each 0.2 time units apart. The trajectory spends approximately $91.8\%$ of its lifetime inside the red box. The black line is the diagonal $I=D$. (b) The probability density function (PDF) of the energy dissipation. The dashed black line marks the PDF of a Gaussian distribution with mean 0.103 and standard deviation 0.018.

Figure 9

Evolution of the eigenvalues ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_{12}$ of the reduced symmetric tensor ${\mathbf{S}}_{12}$ along a chaotic trajectory of the Kolmogorov flow.

Figure 10

Snapshots of the curl of $\mathbf{u}, {\mathbf{v}}_1, {\mathbf{v}}_2, {\mathbf{v}}_3, {\mathbf{v}}_4$, and ${\mathbf{v}}_5$ (from top left to bottom right, respectively) at time $t=34.6$. The colors correspond to the only nonzero component of the curls. All panels show the entire domain $[0,2\pi ]\times [0,2\pi ]$.

Figure 11

Evolution of the energy dissipation $D$ and the eigenvalue ${\lambda }_1$ along two different trajectories (each column corresponds to a separate trajectory). The horizontal dashed lines mark the mean, the mean plus one standard deviation and the mean minus one standard deviation of the corresponding quantity.

Figure 12

The largest eigenvalue ${\lambda }_1$ of ${\mathbf{S}}_r$ as a function of time $t$ for five different values of the subspace dimension $r$ and the Kolmogorov flow at $\text{Re}=40$. The eigenvalue ${\lambda }_1$ increases with $r$, eventually converging to an upper envelope.

Figure 13

Conditional PDF of the first three eigenvalues of ${\mathbf{S}}_8$ and the maximal dissipation ${max}_{\tau }D$, where the maximum is taken over $\tau \in [t+t_i,t+t_f]$, with $t_i=3$ and $t_f=5$.

Figure 14

(a) Probability of the extreme energy dissipation $P_{EE}$ as a function of the value of the indicator ${\lambda }_1$. The probability $P_{EE}$ is computed from the definition (27) with $t_i=3$ and $t_f=5$. (b),(c) Close-up view of two episodes of extreme energy dissipation and their corresponding probabilities $P_{EE}$.

Figure 15

Conditional PDF (a) and the probability of upcoming extreme energy dissipation (b) for Reynolds number $\text{Re}=100$.

Figure 16

Same as Fig. 13 but now the linear operator is reduced to the eight most dominant DMD modes.

Figure 17

(a) The spatial maximum of $\left|u\right|$ as a function of time $t$. An extreme event occurs at around $t=475$, where ${max}_x\left|u\right|\simeq 0.34$. (b),(c) The surface elevation $h(x,t)$ (blue color) and and the modulus of the OTD mode $|v_1|$ at times $t=400$ (b) and $t=475$ (c). The thick black curves in the plots of $h(x,t)$ mark the envelopes $\pm \left|u\right(x,t\left)\right|$.

Figure 18

(a) Conditional PDF for the maximum modulus of the OTD mode $v_1$ and the solution of the MNLS equation. The maxima are taken over $x\in [0,L]$ and $\tau \in [t+t_i,t+t_f]$, with $t_i=25$ and $t_f=26$. (b) The probability of an extreme event $P_{EE}$ computed from the conditional PDF.