Probabilistic description of extreme events in intermittently unstable systems excited by correlated stochastic processes

In this work, we consider systems that are subjected to intermittent instabilities due to external stochastic excitation. These intermittent instabilities, though rare, have a large impact on the probabilistic response of the system and give rise to heavy-tailed probability distributions. By making appropriate assumptions on the form of these instabilities, which are valid for a broad range of systems, we formulate a method for the analytical approximation of the probability distribution function (pdf) of the system response (both the main probability mass and the heavy-tail structure). In particular, this method relies on conditioning the probability density of the response on the occurrence of an instability and the separate analysis of the two states of the system, the unstable and stable state. In the stable regime we employ steady state assumptions, which lead to the derivation of the conditional response pdf using standard methods for random dynamical systems. The unstable regime is inherently transient and in order to analyze this regime we characterize the statistics under the assumption of an exponential growth phase and a subsequent decay phase until the system is brought back to the stable attractor. The method we present allows us to capture the statistics associated with the dynamics that give rise to heavy-tails in the system response and the analytical approximations compare favorably with direct Monte Carlo simulations, which we illustrate for two prototype intermittent systems: an intermittently unstable mechanical oscillator excited by correlated multiplicative noise and a complex mode in a turbulent signal with fixed frequency, where multiplicative stochastic damping and additive noise model interactions between various modes.Comment: 29 pages, 15 figure

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 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

A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise

We develop a moment equation closure minimization method for the inexpensive approximation of the steady state statistical structure of nonlinear systems whose potential functions have bimodal shapes and which are subjected to correlated excitations. Our approach relies on the derivation of moment equations that describe the dynamics governing the two-time statistics. These are combined with a non-Gaussian pdf representation for the joint response-excitation statistics that has i) single time statistical structure consistent with the analytical solutions of the Fokker-Planck equation, and ii) two-time statistical structure with Gaussian characteristics. Through the adopted pdf representation, we derive a closure scheme which we formulate in terms of a consistency condition involving the second order statistics of the response, the closure constraint. A similar condition, the dynamics constraint, is also derived directly through the moment equations. These two constraints are formulated as a low-dimensional minimization problem with respect to unknown parameters of the representation, the minimization of which imposes an interplay between the dynamics and the adopted closure. The new method allows for the semi-analytical representation of the two-time, non-Gaussian structure of the solution as well as the joint statistical structure of the response-excitation over different time instants. We demonstrate its effectiveness through the application on bistable nonlinear single-degree-of-freedom energy harvesters with mechanical and electromagnetic damping, and we show that the results compare favorably with direct Monte-Carlo Simulations

A minimization principle for the description of time-dependent modes associated with transient instabilities

We introduce a minimization formulation for the determination of a finite-dimensional, time-dependent, orthonormal basis that captures directions of the phase space associated with transient instabilities. While these instabilities have finite lifetime they can play a crucial role by either altering the system dynamics through the activation of other instabilities, or by creating sudden nonlinear energy transfers that lead to extreme responses. However, their essentially transient character makes their description a particularly challenging task. We develop a minimization framework that focuses on the optimal approximation of the system dynamics in the neighborhood of the system state. This minimization formulation results in differential equations that evolve a time-dependent basis so that it optimally approximates the most unstable directions. We demonstrate the capability of the method for two families of problems: i) linear systems including the advection-diffusion operator in a strongly non-normal regime as well as the Orr-Sommerfeld/Squire operator, and ii) nonlinear problems including a low-dimensional system with transient instabilities and the vertical jet in crossflow. We demonstrate that the time-dependent subspace captures the strongly transient non-normal energy growth (in the short time regime), while for longer times the modes capture the expected asymptotic behavior