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

Farazmand, Mohammad; Sapsis, Themistoklis P.

doi:10.1103/physreve.94.032212

Cited by 55 publications

(56 citation statements)

References 69 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following result states that the two subspaces are equivalent. (15). Assume that the two subspaces spanned by the columns of U and V are initially equivalent, i.e.…”

Section: Alignment Of the Otd Modes With The Dominant Cauchy-green Stmentioning

confidence: 99%

See 1 more Smart Citation

Reduced-order description of transient instabilities and computation of finite-time Lyapunov exponents

Babaee

Farazmand

Haller

et al. 2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

Self Cite

View full text Add to dashboard Cite

High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have a finite-time duration. Because of the finite-time character of these transient events, their detection through infinite-time methods, e.g., long term averages, Lyapunov exponents or information about the statistical steady-state, is not possible. Here, we utilize a recently developed framework, the Optimally Time-Dependent (OTD) modes, to extract a time-dependent subspace that spans the modes associated with transient features associated with finite-time instabilities. As the main result, we prove that the OTD modes, under appropriate conditions, converge exponentially fast to the eigendirections of the Cauchy-Green tensor associated with the most intense finite-time instabilities. Based on this observation, we develop a reduced-order method for the computation of finite-time Lyapunov exponents (FTLE) and vectors. In high-dimensional systems, the computational cost of the reduced-order method is orders of magnitude lower than the full FTLE computation. We demonstrate the validity of the theoretical findings on two numerical examples.

show abstract

“…The following result states that the two subspaces are equivalent. (15). Assume that the two subspaces spanned by the columns of U and V are initially equivalent, i.e.…”

Section: Alignment Of the Otd Modes With The Dominant Cauchy-green Stmentioning

confidence: 99%

“…For sufficiently long times where the system reaches an equilibrium it was proven in [2] that the OTD modes converge to the most unstable directions of the system in the asymptotic limit. Moreover, the same modes were utilized in [15] to formulate predictors of extreme events for Navier-Stokes equations and nonlinear waves problems.…”

Section: Introductionmentioning

confidence: 99%

Reduced-order description of transient instabilities and computation of finite-time Lyapunov exponents

Babaee

Farazmand

Haller

et al. 2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…The amplitude is set to α = 0.04; this value is chosen so that the resulting average wave height is around 0.05, comparable to experiment G1. We investigate the effect of the spreading angle since this is the parameter that was absent in the unidirectional context considered previously by Cousins and Sapsis (2016) and Farazmand and Sapsis (2016). As listed in Table 1, we consider two values of the spreading angle, θ 0 = 40 • and 60 • (experiments J1 and J2, respectively).…”

Section: Wave Spectramentioning

confidence: 99%

Reduced-order prediction of rogue waves in two-dimensional deep-water waves

Farazmand

Sapsis

2017

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

We consider the problem of large wave prediction in two-dimensional water waves. Such waves form due to the synergistic effect of dispersive mixing of smaller wave groups and the action of localized nonlinear wave interactions that leads to focusing. Instead of a direct simulation approach, we rely on the decomposition of the wave field into a discrete set of localized wave groups with optimal length scales and amplitudes. Due to the short-term character of the prediction, these wave groups do not interact and therefore their dynamics can be characterized individually. Using direct numerical simulations of the governing envelope equations we precompute the expected maximum elevation for each of those wave groups. The combination of the wave field decomposition algorithm, which provides information about the statistics of the system, and the precomputed map for the expected wave group elevation, which encodes dynamical information, allows (i) for understanding of how the probability of occurrence of rogue waves changes as the spectrum parameters vary, (ii) the computation of a critical length scale characterizing wave groups with high probability of evolving to rogue waves, and (iii) the formulation of a robust and parsimonious reduced-order prediction scheme for large waves. We assess the validity of this scheme in several cases of ocean wave spectra.

show abstract

“…Besides originating extreme events-like behavior in dynamical systems, an urgent task is prediction of extreme events, althoigh it is very difficult. So far some attempts have been made in search of early warning of extreme events 34,35 , but with not so much success. Even in deterministic dynamical systems, this is a challenging issue of current research 16,33,36 .…”

mentioning

confidence: 99%

Intermittent large deviation of chaotic trajectory in Ikeda map: Signature of extreme events

Ray

Rakshit

Ghosh

et al. 2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

We notice signatures of extreme events-like behavior in a laser based Ikeda map. The trajectory of the system occasionally travels a large distance away from the bounded chaotic region, which appears as intermittent spiking events in the temporal dynamics. The large spiking events satisfy the conditions of extreme events as usually observed in dynamical systems. The probability density function of the large spiking events shows a long-tail distribution consistent with the characteristics of rare events. The inter-event intervals obey a Poisson-like distribution. We locate the parameter regions of extreme events in phase diagrams. Furthermore, we study two Ikeda maps to explore how and when extreme events terminate via mutual interaction. A pure diffusion of information exchange is unable to terminate extreme events where synchronous occurrence of extreme events is only possible even for large interaction. On the other hand, a threshold-activated coupling can terminate extreme events above a critical value of mutual interaction.PACS numbers: 89.75.Fb, 05.45.aRare and recurrent large amplitude deviations of normally bounded dynamics are seen in many systems. Such occasional large amplitude spiking events are larger than a nominal value and their statistical distribution of occurrence shows qualitative similarities, in dynamical sense, with data records of natural disasters, rogue waves, tsunami, flood and share market crashes. These observations draw attention of researchers to investigate similar sudden large intermittent events in dynamical systems for developing an understanding of the origin of extreme events and exploring the possibilities of prediction. A laser based Ikeda map was studied earlier to profess the origin of a new dynamical phenomenon, namely, interior crisis, that leads to a sudden expansion of a chaotic attractor. This sudden expansion of attractor is not always a permanent property of the system, and it could be intermittent, which shows similarities with extreme events and this signature was overlooked earlier. Here we explore this extreme value dynamical features of the Ikeda map to confirm the phenomenon and the statistical properties of events. An investigation with two coupled maps has also been made in search of an appropriate coupling scheme that is able to terminate these undesirable extreme events.

show abstract

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

Cited by 55 publications

References 69 publications

Reduced-order description of transient instabilities and computation of finite-time Lyapunov exponents

Reduced-order description of transient instabilities and computation of finite-time Lyapunov exponents

Reduced-order prediction of rogue waves in two-dimensional deep-water waves

Intermittent large deviation of chaotic trajectory in Ikeda map: Signature of extreme events

Contact Info

Product

Resources

About