Periodic Orbits, Lyapunov Vectors, and Singular Vectors in the Lorenz System

Trevisan, A.; Pancotti, Francesco

doi:10.1175/1520-0469(1998)055<0390:polvas>2.0.co;2

Cited by 97 publications

(115 citation statements)

References 34 publications

(39 reference statements)

Supporting

Mentioning

111

Contrasting

Order By: Relevance

“…The present results are generally consistent with the analysis by Trevisan and Pancotti (1998) of Lyapunov, Floquet, and singular vectors for a low-order unstable periodic cycle of the three-component Lorenz (1963) model. Those authors also find that the unstable Floquet mode tends to resemble the tangent along the cycle.…”

Section: Discussionsupporting

confidence: 91%

See 1 more Smart Citation

Lyapunov, Floquet, and singular vectors for baroclinic waves

Samelson

2001

Nonlin. Processes Geophys.

View full text Add to dashboard Cite

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.

show abstract

Section: Discussionsupporting

confidence: 91%

“…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.

View full text Add to dashboard Cite

show abstract

“…Analogously, the J through nth forward Lypunov vectors are a basis for the subspace in which infinitesimal errors grow, in the long-term, with a rate of no more than λ J . Then, as suggested by Legras and Vautard (1996) and detailed by Trevisan and Pancotti (1998) …”

Section: Lyapunov Vectorsmentioning

confidence: 91%

The role of model dynamics in ensemble Kalman filter performance for chaotic systems

McLaughlin

Entekhabi

et al. 2011

TELLUSA

View full text Add to dashboard Cite

A B S T R A C TThe ensemble Kalman filter (EnKF) is susceptible to losing track of observations, or 'diverging', when applied to large chaotic systems such as atmospheric and ocean models. Past studies have demonstrated the adverse impact of sampling error during the filter's update step. We examine how system dynamics affect EnKF performance, and whether the absence of certain dynamic features in the ensemble may lead to divergence. The EnKF is applied to a simple chaotic model, and ensembles are checked against singular vectors of the tangent linear model, corresponding to short-term growth and Lyapunov vectors, corresponding to long-term growth. Results show that the ensemble strongly aligns itself with the subspace spanned by unstable Lyapunov vectors. Furthermore, the filter avoids divergence only if the full linearized long-term unstable subspace is spanned. However, short-term dynamics also become important as nonlinearity in the system increases. Non-linear movement prevents errors in the long-term stable subspace from decaying indefinitely. If these errors then undergo linear intermittent growth, a small ensemble may fail to properly represent all important modes, causing filter divergence. A combination of long and short-term growth dynamics are thus critical to EnKF performance. These findings can help in developing practical robust filters based on model dynamics.

show abstract

“…That paper also conjectured that the difference between the two measures of local predictability and their relationship to the balanced dynamics may be due to the fact that the local Lyapunov number measures predictability for trajectories on the attractor of the system, while the dominant singular value can potentially be associated with trajectories which are initially off the attractor (e.g. Anderson, 1995;Legras and Vautard, 1996;Trevisan and Pancotti, 1998). In this paper, we will demonstrate that the relationship between the predictability measures and the role of balanced dynamics is no different for the elastic pendulum than for dissipative atmospheric models.…”

Section: Introductionmentioning

confidence: 99%

Local predictability in a simple model of atmospheric balance

Gyarmati

Szunyogh

Patil

2003

Nonlin. Processes Geophys.

View full text Add to dashboard Cite

Abstract. The 2 degree-of-freedom elastic pendulum equations can be considered as the lowest order analogue of interacting low-frequency (slow) Rossby-Haurwitz and highfrequency (fast) gravity waves in the atmosphere. The strength of the coupling between the low and the high frequency waves is controlled by a single coupling parameter, ε, defined by the ratio of the fast and slow characteristic time scales.In this paper, efficient, high accuracy, and symplectic structure preserving numerical solutions are designed for the elastic pendulum equation in order to study the role balanced dynamics play in local predictability. To quantify changes in the local predictability, two measures are considered: the local Lyapunov number and the leading singular value of the tangent linear map.It is shown, both based on theoretical considerations and numerical experiments, that there exist regions of the phase space where the local Lyapunov number indicates exceptionally high predictability, while the dominant singular value indicates exceptionally low predictability. It is also demonstrated that the local Lyapunov number has a tendency to choose instabilities associated with balanced motions, while the dominant singular value favors instabilities related to highly unbalanced motions. The implications of these findings for atmospheric dynamics are also discussed.

show abstract

Periodic Orbits, Lyapunov Vectors, and Singular Vectors in the Lorenz System

Cited by 97 publications

References 34 publications

Lyapunov, Floquet, and singular vectors for baroclinic waves

Lyapunov, Floquet, and singular vectors for baroclinic waves

The role of model dynamics in ensemble Kalman filter performance for chaotic systems

Local predictability in a simple model of atmospheric balance

Contact Info

Product

Resources

About