Spectrum and Statistical Properties of Chaotic Dynamics

Baladi, Viviane

doi:10.1007/978-3-0348-8268-2_11

Cited by 5 publications

(5 citation statements)

References 84 publications

(97 reference statements)

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“…It is a classical result from ergodic theory [14,15] that correlations between observables vanish for time lags going to infinity only if there is no eigenvalue of the transfer operator P µ in the complex unit circle other than the eigenvalue 1. A more difficult problem, which is still a matter of invest igation [16,17], is to characterise the rate at which correlations decay with time [18,19]. This rate depends on the position inside the unit disk of the complex plane of the eigenvalues of transfer operators acting on anisotropic Banach spaces adapted to the dynamics of contraction and expansion of chaotic systems [20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Crisis of the chaotic attractor of a climate model: a transfer operator approach

Tantet

Lucarini

Lunkeit³

et al. 2018

Nonlinearity

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The destruction of a chaotic attractor leading to rough changes in the dynamics of a dynamical system is studied. Local bifurcations are known to be characterised by a single or a pair of characteristic exponents crossing the imaginary axis. As a result, the approach of such bifurcations in the presence of noise can be inferred from the slowing down of the decay of correlations [1]. On the other hand, little is known about global bifurcations involving high-dimensional attractors with several positive Lyapunov exponents. It is known that the global stability of chaotic attractors may be characterised by the spectral properties of the Koopman [2] or the transfer operators governing the evolution of statistical ensembles. Accordingly, it has recently been shown [3] that a boundary crisis in the Lorenz flow coincides with the approach to the unit circle of the eigenvalues of these operators associated with motions about the attractor, the stable resonances. A second type of resonances, the unstable resonances, is responsible for the decay of correlations and mixing on the attractor. In the deterministic case, those cannot be expected to be affected by general boundary crises. Here, however, we give an example of chaotic system in which slowing down of the decay of correlations of some observables does occur at the approach of a boundary crisis. The system considered is a high-dimensional, chaotic climate model of physical relevance. Moreover, coarsegrained approximations of the transfer operators on a reduced space, constructed from a long time series of the system, give evidence that this behaviour is due to the approach of unstable resonances to the unit circle. That the unstable resonances are affected by the crisis can be physically understood from the fact that the process responsible for the instability, the ice-albedo feedback, is also active on the attractor. Implications regarding response theory and the design of early-warning signals are finally discussed.

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Section: Introductionmentioning

confidence: 99%

Crisis of the chaotic attractor of a climate model: a transfer operator approach

Tantet

Lucarini

Lunkeit³

et al. 2018

Nonlinearity

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“…Before that, other methods had been developed (in dimensions one and higher, and under various assumptions of expansion, hyperbolicity, and/or regularity) in the continuation of the Markov approach of Section 1. In this introductory section, we first give a very brief and incomplete presentation of some of the results obtained by these older methods between 1976 and now, referring to [Ba1,Ba3] for more general surveys; we then give a very brief presentation of the key result of Milnor and Thurston which inspired the new kneading approach.…”

Section: Kneading Theory In Dimension Onementioning

confidence: 99%

“…The spectral theory of the transfer operator is more technical (for example it is not obvious which Banach space to use!). The survey [Ba3] contains references to the results of Saussol, Buzzi, Tsujii, and others. A version of the Hofbauer tower can be constructed, and Buzzi and Keller [BK] recently used it to prove an analogue of Theorem 2 from Section 1 in the case when f is piecewise affine, piecewise expanding (in higher dimensions), and not necessarily Markov.…”

Section: Kneading Theory In Dimension Onementioning

confidence: 99%

Dynamical zeta functions

Baladi¹

2016

Preprint

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“…This type of spectrum was first put in evidence in the case of uniformly hyperbolic maps by using Markov partitions to translate the dynamics on the phase space into a simple symbolic dynamics (namely a subshift of finite type) [40]. It was later extended to more general systems, including non-uniformly hyperbolic ones [4]. In the next sections, we will introduce the Anosov diffeomorphisms on the torus, which often serve as a 'model' for deterministic chaos.…”

Section: Exponential Mixingmentioning

confidence: 99%

“…The example of irrational translations shows that ergodicity does not imply the presence of a gap. Exponential mixing was recently proven for some non-uniformly or partially hyperbolic maps (see [4] for a review of recent results), which should (?) imply a gap in the spectrum of P .…”

Section: General Conclusionmentioning

confidence: 99%

Spectral properties of noisy classical and quantum propagators

Nonnenmacher

2003

Nonlinearity

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We study classical and quantum maps on the torus phase space, in the presence of noise. We focus on the spectral properties of the noisy evolution operator, and prove that for any amount of noise, the quantum spectrum converges to the classical one in the semiclassical limit. The small-noise behaviour of the classical spectrum highly depends on the dynamics generated by the map. For a chaotic dynamics, the outer spectrum consists of isolated eigenvalues ('resonances') inside the unit circle, leading to an exponential damping of correlations. In contrast, in the case of a regular map, part of the spectrum accumulates along a one-dimensional 'string' connecting the origin with unity, yielding a diffusive behaviour. We finally study the non-commutativity between the semiclassical and small-noise limits, and illustrate this phenomenon by computing (analytically and numerically) the classical and quantum spectra for some maps.

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Spectrum and Statistical Properties of Chaotic Dynamics

Cited by 5 publications

References 84 publications

Crisis of the chaotic attractor of a climate model: a transfer operator approach

Crisis of the chaotic attractor of a climate model: a transfer operator approach

Dynamical zeta functions

Spectral properties of noisy classical and quantum propagators

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