A hybrid method for calculation of Ruelle–Pollicott resonances

Horvat, Martin; Veble, Gregor

doi:10.1088/1751-8113/42/46/465101

Cited by 3 publications

(3 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In essence, Ulam-type methods approximate the dynamics in phase space by a Markov chain whose transition probabilities are estimated from many simultaneous iterations of the map of interest over a large ensemble of initial data (26,27). This approach can be rigorously justified for a large class of expanding or Anosov maps (27), and it can be used in the numerical estimation of RP resonances for low-dimensional models (34), although some drawbacks may arise in applications (35).…”

Section: Estimating Rp Resonances From Observablesmentioning

confidence: 99%

“…In principle, this gives the positions of the RP resonances corresponding to nonvanishing residues associated with h. Padé approximation or Prony's method are typically used (36). The main drawback of these techniques lies in the number of exponentials to be fitted to the signal, which may lead to an inaccurate estimation of RP resonance (35).…”

Section: Estimating Rp Resonances From Observablesmentioning

confidence: 99%

See 1 more Smart Citation

Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonances

Chekroun

Neelin

Kondrashov

et al. 2014

Proc. Natl. Acad. Sci. U.S.A.

137

View full text Add to dashboard Cite

Despite the importance of uncertainties encountered in climate model simulations, the fundamental mechanisms at the origin of sensitive behavior of long-term model statistics remain unclear. Variability of turbulent flows in the atmosphere and oceans exhibits recurrent large-scale patterns. These patterns, while evolving irregularly in time, manifest characteristic frequencies across a large range of time scales, from intraseasonal through interdecadal. Based on modern spectral theory of chaotic and dissipative dynamical systems, the associated low-frequency variability may be formulated in terms of Ruelle-Pollicott (RP) resonances. RP resonances encode information on the nonlinear dynamics of the system, and an approach for estimating them-as filtered through an observable of the system-is proposed. This approach relies on an appropriate Markov representation of the dynamics associated with a given observable. It is shown that, within this representation, the spectral gap-defined as the distance between the subdominant RP resonance and the unit circle-plays a major role in the roughness of parameter dependences. The model statistics are the most sensitive for the smallest spectral gaps; such small gaps turn out to correspond to regimes where the low-frequency variability is more pronounced, whereas autocorrelations decay more slowly. The present approach is applied to analyze the rough parameter dependence encountered in key statistics of an El-Niño-Southern Oscillation model of intermediate complexity. Theoretical arguments, however, strongly suggest that such links between model sensitivity and the decay of correlation properties are not limited to this particular model and could hold much more generally.climate dynamics | Markov operators | parametric dependence | sensitivity bounds | uncertainty quantification S ensitive behavior of long-term general circulation model (GCM) statistics is attracting increased attention (1-3), but its origin and fundamental mechanisms remain unclear. These sensitive-behavior issues are of practical, as well as theoretical, importance in climate dynamics and elsewhere (4). For some GCMs, involving millions of variables, circumstances have been found where certain climate observables vary smoothly through a plausible parameter range (5) or where linear response theory applies over some range (6). On the other hand, this may not hold for every observable or parameter, and concerns arise regarding the role of some type of "structural instability" in sensitive parameter dependence (1, 2, 4).The low-order Lorenz (L63) model (7) illustrates some of the relevant issues. Various statistics exhibit linear dependence over a broad range of parameters for which the dynamics is chaotic (e.g., figure 2 of ref. 8). The statistics' linear dependence coexists here with structural instability of this model's global attractor, as small variations in the parameters cause a plethora of topological changes (9). In particular, the unstable periodic orbits that appear or disappear as a parameter chan...

show abstract

Section: Estimating Rp Resonances From Observablesmentioning

confidence: 99%

Section: Estimating Rp Resonances From Observablesmentioning

confidence: 99%

Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonances

Chekroun

Neelin

Kondrashov

et al. 2014

Proc. Natl. Acad. Sci. U.S.A.

137

View full text Add to dashboard Cite

show abstract

“…This statement has been supported by numerical analysis of some DSs (see, e.g., Ref. [Horvat and Veble 2009]).…”

Section: Discussionmentioning

confidence: 59%

Transfer operators and topological field theory

Ovchinnikov

2013

Preprint

View full text Add to dashboard Cite

The transfer operator (TO) formalism of the dynamical systems (DS) theory is reformulated here in terms of the recently proposed supersymetric theory of stochastic differential equations (SDE). It turns out that the stochastically generalized TO (GTO) of the DS theory is the finitetime Fokker-Planck evolution operator. As a result comes the supersymmetric trivialization of the so-called sharp trace and sharp determinant of the GTO, with the former being the Witten index, which is also the stochastic generalization of the Lefschetz index so that it equals the Euler characteristic of the (closed) phase space for any flow vector field, noise metric, and temperature.The enabled possibility to apply the spectral theorems of the DS theory to the Fokker-Planck operators allows to extend the previous picture of the spontaneous topological supersymmetry (Q-symmetry) breaking onto the situations with negative ground state's attenuation rate. The later signifies the exponential growth of the number of periodic solutions/orbits in the large time limit, which is the unique feature of chaotic behavior proving that the spontaneous breakdown of Q-symmetry is indeed the field-theoretic definition and stochastic generalization of the concept of deterministic chaos. In addition, the previously proposed low-temperature classification of SDEs, i.e., thermodynamic equilibrium / noise-induced chaos ((anti)instanton condensation, intermittent) / ordinary chaos (non-integrability of the flow vector field), is complemented by the discussion of the high-temperature regime where the sharp boundary between the noise-induced and ordinary chaotic phases must smear out into a crossover, and at even higher temperatures the Q-symmetry is restored. The Weyl quantization is discussed in the context of the Ito-Stratonovich dilemma.

show abstract

Momentum-dependent quantum Ruelle-Pollicott resonances in translationally invariant many-body systems

Žnidarič

2024

Phys. Rev. E

View full text Add to dashboard Cite

A hybrid method for calculation of Ruelle–Pollicott resonances

Cited by 3 publications

References 19 publications

Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonances

Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonances

Transfer operators and topological field theory

Momentum-dependent quantum Ruelle-Pollicott resonances in translationally invariant many-body systems

Contact Info

Product

Resources

About