Recycling of strange sets: I. Cycle expansions

Artuso, Roberto; Aurell, Erik; Cvitanović, Predrag

doi:10.1088/0951-7715/3/2/005

Cited by 397 publications

(260 citation statements)

References 42 publications

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“…The approach is partially motivated by recent work on cycle expansions for chaotic systems (e.g. Artuso et al, 1990aArtuso et al, , 1990bChristiansen et al, 1997;Cvitanović et al, 2000), and is related to a recent study based on the Lorenz system (Trevisan and Pancotti, 1998).…”

Section: Introductionmentioning

confidence: 99%

Lyapunov, Floquet, and singular vectors for baroclinic waves

Samelson

2001

Nonlin. Processes Geophys.

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Abstract. The dynamics of the growth of linear disturbances to a chaotic basic state is analyzed in an asymptotic model of weakly nonlinear, baroclinic wave-mean interaction. In this model, an ordinary differential equation for the wave amplitude is coupled to a partial differential equation for the zonal flow correction. The leading Lyapunov vector is nearly parallel to the leading Floquet vector φ 1 of the lowest-order unstable periodic orbit over most of the attractor. Departures of the Lyapunov vector from this orientation are primarily rotations of the vector in an approximate tangent plane to the large-scale attractor structure. Exponential growth and decay rates of the Lyapunov vector during individual Poincaré section returns are an order of magnitude larger than the Lyapunov exponent λ ≈ 0.016. Relatively large deviations of the Lyapunov vector from parallel to φ 1 are generally associated with relatively large transient decays. The transient growth and decay of the Lyapunov vector is well described by the transient growth and decay of the leading Floquet vectors of the set of unstable periodic orbits associated with the attractor. Each of these vectors is also nearly parallel to φ 1 . The dynamical splitting of the complete sets of Floquet vectors for the higher-order cycles follows the previous results on the lowest-order cycle, with the vectors divided into wavedynamical and decaying zonal flow modes. Singular vectors and singular values also generally follow this split. The primary difference between the leading Lyapunov and singular vectors is the contribution of decaying, inviscidly-damped wave-dynamical structures to the singular vectors.

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Section: Introductionmentioning

confidence: 99%

Lyapunov, Floquet, and singular vectors for baroclinic waves

Samelson

2001

Nonlin. Processes Geophys.

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“…In what follows, we assume that all periodic orbits are diffractive so that the poles of g(k) are the zeros of the zeta function [12][13][14] …”

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confidence: 99%

“…For geometric orbits, this is evaluated using a cycle expansion [13,14]. Here the weights t β are multiplicative so the zeta function is a finite polynomial conveniently represented as the determinant of a Markov graph [15].…”

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confidence: 99%

Semiclassical Quantization Using Diffractive Orbits

Whelan¹

1996

Phys. Rev. Lett.

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Diffraction, in the context of semiclassical mechanics, describes the manner in which quantum mechanics smooths over discontinuities in the classical mechanics. An important example is a billiard with sharp corners; its semiclassical quantisation requires the inclusion of diffractive periodic orbits in addition to classical periodic orbits. In this paper we construct the corresponding zeta function and apply it to a scattering problem which has only diffractive periodic orbits. We find that the resonances are accurately given by the zeros of the diffractive zeta function.

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“…In circumstances where there is a convenient symbolic dynamics, there may be optimal collections of short segments that through gluing provide good pseudo-orbits to shadow all more complicated orbits. For more on this, though it is not what they had in mind, see [1].…”

Section: Numerical Methods Control Of Chaos and Conclusionmentioning

confidence: 99%

Bifurcations in the Falkner-Skan equation

Sparrow

1997

Control and Chaos

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Falkner-Skan; gluing bifurcation; reversible systemThe Falkner-Skan eQuation is a reversible three dimensional system of ODEs without fixed points. A novel sequence of bifurcations, each of which creates a large set of ,Periodic and other interesting orbits 'from infinity, occurs for each. positive integer value of a parameter. Another sequence of bifurcations destroys these orbits as the parameter increases; top<;>logical constraints allow us to understand this seql!~:t;lce of bifurcations in considerable detail. While outlining these results, we can also make a number of possibly illuminating remarks connecting parts of the proof, well-known numerical techniques for locating and continuing periodic orbits, and recent ideas m the control of chaos. The Falkner-Skan equation is a reversible three dimensional system of ODEs without fixed points. A novel sequence of bifurcations, each of which creates a large set of periodic and other interesting orbits 'from infinity', occurs for each positive integer value of a parameter. Another sequence of bifurcations destroys these orbits as the parameter increases; topological constraints allow us to understand this sequence of bifurcations in considerable detail. While outlining these results, we can also make a number of possibly illuminating remarks connecting parts of the proof, well-known numerical techniques for locating and continuing periodic orbits, and recent ideas in the control of chaos.

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Recycling of strange sets: I. Cycle expansions

Abstract: The strange sets which arise in deterministic low-dimensional dynamical systems are analysed in terms of (unstable) cycles and their eigenvalues. The general formalism of cycle expansions is introduced and its convergence discussed.

Cited by 397 publications

References 42 publications

Lyapunov, Floquet, and singular vectors for baroclinic waves

Lyapunov, Floquet, and singular vectors for baroclinic waves

Semiclassical Quantization Using Diffractive Orbits

Bifurcations in the Falkner-Skan equation

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