A variational approach to probing extreme events in turbulent dynamical systems

Farazmand, Mohammad; Sapsis, Themistoklis P.

doi:10.1126/sciadv.1701533

Cited by 82 publications

(100 citation statements)

References 43 publications

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“…the rescaled approximating SDE (8) can be written more compactly as (a) The control term modifies the potential V in the original SDE (2) to the effective potential (9). The term α|x − x a | 2 /2 increases the potential gap around the desirable state x a , deepening the potential well.…”

Section: Approximating Sdementioning

confidence: 99%

Mitigation of tipping point transitions by time-delay feedback control

Farazmand

2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In stochastic multistable systems driven by the gradient of a potential, transitions between equilibria is possible because of noise. We study the ability of linear delay feedback control to mitigate these transitions, ensuring that the system stays near a desirable equilibrium. For small delays, we show that the control term has two effects: i) a stabilizing effect by deepening the potential well around the desirable equilibrium, and ii) a destabilizing effect by intensifying the noise by a factor of (1 − τ α) −1/2 , where τ and α denote the delay and the control gain, respectively. As a result, successful mitigation depends on the competition between these two factors. We also derive analytical results that elucidate the choice of the appropriate control gain and delay that ensure successful mitigations. These results eliminate the need for any Monte Carlo simulations of the stochastic differential equations, and therefore significantly reduce the computational cost of determining the suitable control parameters. We demonstrate the application of our results on two examples.Many complex systems, such as the climate, stock markets, population of species, and the nervous system, have multiple stable equilibrium states. Near the tipping point of the system, small amounts of stochastic disturbances can lead to transitions between these equilibria. Typically, one of the equilibria is desirable and transitions away from it lead to catastrophic consequences. Here, we analyze the ability of a delay feedback control to mitigate transitions away from a desirable equilibrium. arXiv:1911.05292v2 [math.OC]

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Section: Approximating Sdementioning

confidence: 99%

Mitigation of tipping point transitions by time-delay feedback control

Farazmand

2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…We also stress that the method proposed here can be generalized to the full two-dimensional setting, as well as other relevant physical systems where an understanding of extreme events is important [21,22] but made challenging by the complexity of the models involved combined with the stochasticity of their evolution and the uncertainty of their parameters [21,[23][24][25][26]. In this sense our approach adds to the several methods developed specifically to explain the mechanism and likelihood of rare but important events [27][28][29][30][31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

Experimental Evidence of Hydrodynamic Instantons: The Universal Route to Rogue Waves

et al. 2019

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A statistical theory of rogue waves is proposed and tested against experimental data collected in a long water tank where random waves with different degrees of nonlinearity are mechanically generated and free to propagate along the flume. Strong evidence is given that the rogue waves observed in the tank are hydrodynamic instantons, that is, saddle point configurations of the action associated with the stochastic model of the wave system. As shown here, these hydrodynamic instantons are complex spatio-temporal wave field configurations, which can be defined using the mathematical framework of Large Deviation Theory and calculated via tailored numerical methods. These results indicate that the instantons describe equally well rogue waves that originate from a simple linear superposition mechanism (in weakly nonlinear conditions) or from a nonlinear focusing one (in strongly nonlinear conditions), paving the way for the development of a unified explanation to rogue wave formation.

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“…(6) We note that this simplified control is a scalar multiple of the external forcing f . The corresponding probability distributions of the energy dissipation D are nearly identical whether we use the full control (5) or its simplified form (6). Figure 5 shows the closeup view of an extreme event at Re = 40; it compares the uncontrolled and controlled system trajectories starting from the same initial condition.…”

Section: Navier-stokesmentioning

confidence: 96%

“…During these bursts, the energy dissipation D increases to several standard deviations above its expected value (see figure 2(a)). Using a variational method, Farazmand and Sapsis [6] showed that these bursts are preceded by a nonlinear energy transfer from the Fourier modeû (1,0) to the modeû(0, 4). Shortly before an extreme energy dissipation event is observed, most of the energy content of the Fourier modê u(1, 0) is transferred to the Fourier modeû(0, 4).…”

Section: Uncontrolled Systemmentioning

confidence: 99%

Closed-loop adaptive control of extreme events in a turbulent flow

Farazmand

Sapsis

2019

Phys. Rev. E

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Extreme events that arise spontaneously in chaotic dynamical systems often have an adverse impact on the system or the surrounding environment. As such, their mitigation is highly desirable. Here, we introduce a novel control strategy for mitigating extreme events in a turbulent shear flow. The controller combines a probabilistic prediction of the extreme events with a deterministic actuator. The predictions are used to actuate the controller only when an extreme event is imminent. When actuated, the controller only acts on the degrees of freedom that are involved in the formation of the extreme events, exerting minimal interference with the flow dynamics. As a result, the attractors of the controlled and uncontrolled systems share the same chaotic core (containing the non-extreme events) and only differ in the tail of their distributions. We propose that such adaptive low-dimensional controllers should be used to mitigate extreme events in general chaotic dynamical systems, beyond the shear flow considered here.

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A variational approach to probing extreme events in turbulent dynamical systems

Abstract: A variational framework for the analysis and data-driven prediction of extreme events is developed.

Cited by 82 publications

References 43 publications

Mitigation of tipping point transitions by time-delay feedback control

Mitigation of tipping point transitions by time-delay feedback control

Experimental Evidence of Hydrodynamic Instantons: The Universal Route to Rogue Waves

Closed-loop adaptive control of extreme events in a turbulent flow

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