Recycling of strange sets: II. Applications

Artuso, Roberto; Aurell, Erik; Cvitanović, Predrag

doi:10.1088/0951-7715/3/2/006

Cited by 240 publications

(197 citation statements)

References 26 publications

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“…Periodic orbit theory [20,21] has proven very useful for investigations of the corresponding low dimensional system, the periodic Lorentz gas [22,23,24,25], computing properties such as the diffusion coefficient. These methods cannot generally be applied directly to high dimensional systems due to the difficulty of finding all the periodic orbits, although a notable exception is the KuramotoShivashinsky PDE in a regime where the effective number of degrees of freedom is small [26].…”

Section: Introductionmentioning

confidence: 99%

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Taniguchi

Dettmann

Morriss

2002

Journal of Statistical Physics

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The Lyapunov spectrum corresponding to a periodic orbit for a two dimensional many particle system with hard core interactions is discussed. Noting that the matrix to describe the tangent space dynamics has the block cyclic structure, the calculation of the Lyapunov spectrum is attributed to the eigenvalue problem of 16 × 16 reduced matrices regardless of the number of particles. We show that there is the thermodynamic limit of the Lyapunov spectrum in this periodic orbit. The Lyapunov spectrum has a step structure, which is explained by using symmetries of the reduced matrices.

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Section: Introductionmentioning

confidence: 99%

Untitled

Taniguchi

Dettmann

Morriss

2002

Journal of Statistical Physics

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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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“…This form is precisely part of the conventional semiclassical density of states [15,34,35]. However, the formal parameter z is also present and can be seen as a counting and ordering parameter of the various orbits in the expansion over infinitely many orbits [1, 36,37].…”

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confidence: 99%