980 research outputs found
A probabilistic decomposition-synthesis method for the quantification of rare events due to internal instabilities
We consider the problem of the probabilistic quantification of dynamical systems that have heavy-tailed characteristics. These heavy-tailed features are associated with rare transient responses due to the occurrence of internal instabilities. Systems with these properties can be found in a variety of areas including mechanics, fluids, and waves. Here we develop a computational method, a probabilistic decomposition-synthesis technique, that takes into account the nature of internal instabilities to inexpensively determine the non-Gaussian probability density function for any arbitrary quantity of interest. Our approach relies on the decomposition of the statistics into a 'non-extreme core', typically Gaussian, and a heavy-tailed component. This decomposition is in full correspondence with a partition of the phase space into a 'stable' region where we have no internal instabilities, and a region where non-linear instabilities lead to rare transitions with high probability. We quantify the statistics in the stable region using a Gaussian approximation approach, while the non-Gaussian distribution associated with the intermittently unstable regions of phase space is inexpensively computed through order-reduction methods that take into account the strongly nonlinear character of the dynamics. The probabilistic information in the two domains is analytically synthesized through a total probability argument. The proposed approach allows for the accurate quantification of non-Gaussian tails at more than 10 standard deviations, at a fraction of the cost associated with the direct Monte-Carlo simulations. We demonstrate the probabilistic decomposition-synthesis method for rare events for two dynamical systems exhibiting extreme events: a twodegree-of-freedom system of nonlinearly coupled oscillators, and in a nonlinear envelope equation characterizing the propagation of unidirectional water waves
Probabilistic response and rare events in Mathieu׳s equation under correlated parametric excitation
We derive an analytical approximation to the probability distribution function (pdf) for the response of Mathieu׳s equation under parametric excitation by a random process with a spectrum peaked at the main resonant frequency, motivated by the problem of large amplitude ship roll resonance in random seas. The inclusion of random stochastic excitation renders the otherwise straightforward response to a system undergoing intermittent resonances: randomly occurring large amplitude bursts. Intermittent resonance occurs precisely when the random parametric excitation pushes the system into the instability region, causing an extreme magnitude response. As a result, the statistics are characterized by heavy-tails. We apply a recently developed mathematical technique, the probabilistic decomposition-synthesis method, to derive an analytical approximation to the non-Gaussian pdf of the response. We illustrate the validity of this analytical approximation through comparisons with Monte-Carlo simulations that demonstrate our result accurately captures the strong non-Gaussianinty of the response. Keywords: Mathieu׳s equationColored stochastic excitationHeavy-tailsIntermittent instabilitiesRare eventsStochastic roll resonanceUnited States. Office of Naval Research (Grant ONR N00014- 14-1-0520)Massachusetts Institute of Technology. Naval Engineering Education Center (Grant 3002883706
A primer on noise-induced transitions in applied dynamical systems
Noise plays a fundamental role in a wide variety of physical and biological
dynamical systems. It can arise from an external forcing or due to random
dynamics internal to the system. It is well established that even weak noise
can result in large behavioral changes such as transitions between or escapes
from quasi-stable states. These transitions can correspond to critical events
such as failures or extinctions that make them essential phenomena to
understand and quantify, despite the fact that their occurrence is rare. This
article will provide an overview of the theory underlying the dynamics of rare
events for stochastic models along with some example applications
Closed-loop adaptive control of extreme events in a turbulent flow
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.Comment: In press in Phys. Rev. E. Includes minor revision
Predicting ocean rogue waves from point measurements: An experimental study for unidirectional waves
Rogue waves are strong localizations of the wave field that can develop in different branches of physics and engineering, such as water or electromagnetic waves. Here, we experimentally quantify the prediction potentials of a comprehensive rogue-wave reduced-order precursor tool that has been recently developed to predict extreme events due to spatially localized modulation instability. The laboratory tests have been conducted in two different water wave facilities and they involve unidirectional water waves; in both cases we show that the deterministic and spontaneous emergence of extreme events is well predicted through the reported scheme. Due to the interdisciplinary character of the approach, similar studies may be motivated in other nonlinear dispersive media, such as nonlinear optics, plasma, and solids, governed by similar equations, allowing the early stage of extreme wave detection.United States. Office of Naval Research (Grant N00014-15-1-2381)United States. Army Research Office (Grant W911NF-17-1-0306
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