Electron phase-space trajectories in a linearly polarized free electron laser

Abstract. Global climate variability affects important local hydro-meteorological variables like precipitation and temperature. The Southern Oscillation (SO) is an easily quantifiable major driving force that gives impact on regional and local climate. The relationships between SO and local climate variation are, however, characterized by strongly nonlinear processes. Due to this, teleconnections between global-scale hydro-meteorological variables and local climate are not well understood. In this paper, we suggest to study these processes in terms of nonlinear dynamics. Consequently, the nonlinear dynamic relationship between the Southern Oscillation Index (SOI), precipitation, and temperature in Fukuoka, Japan, is investigated using a nonlinear multivariable approach. This approach is based on the joint variation of these variables in the phase space. The joint phase-space variation of SOI, precipitation, and temperature is studied with the primary objective to obtain a better understanding of the dynamical evolution of local hydro-meteorological variables affected by global atmospheric-oceanic phenomena. The results from the analyses display rather clear low-order phase space trajectories when treating the time series individually. However, when plotting phase space trajectories for several time series jointly, complicated higher-order nonlinear relationships emerge between the variables. Consequently, simple data-driven prediction techniques utilizing phase-space characteristics of individual time series may prove successful. On the other hand, since either the time series are too short and/or the phase-space properties are too complex when analysing several variables jointly, it may be difficult to use multivariable statistical prediction techniques for the present investigated variables. In any case, it is essential to further pursue studies regarding links between the SOI and observed local climatic and other geophysical variables even if these links are not fully understood in physical terms.

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Invariant Manifold Detection From Phase Space Trajectories

Volume 11: Mechanical Systems and Control ◽

10.1115/imece2008-67473 ◽

2008 ◽

Author(s):

Joseph Kuehl ◽

David Chelidze

Keyword(s):

Phase Space ◽

Invariant Manifold ◽

Invariant Manifolds ◽

Basins Of Attraction ◽

Detection Methods ◽

Trajectory Data ◽

Space Trajectories ◽

Data Set ◽

Phase Space Warping ◽

Phase Space Trajectories

Invariant manifolds provide important information about the structure of flows. When basins of attraction are present, the stable invariant manifold serves as the boundary between these basins. Thus, in experimental applications such as vibrations problems, knowledge of these manifolds is essential to understanding the evolution of phase space trajectories. Most existing methods for identifying invariant manifolds of a flow rely on knowledge of the flow field. However, in experimental applications only knowledge of phase space trajectories is available. We provide modifications to several existing invariant manifold detection methods which enables them to deal with trajectory only data, as well as introduce a new method based on the concept of phase space warping. The method of Stochastic Interrogation applied to the damped, driven Duffing equation is used to generate our data set. The result is a set of trajectory data which randomly populates a phase space. Manifolds are detected from this data set using several different methods. First is a variation on manifold “growing,” and is based on distance of closest approach to a hyperbolic trajectory with “saddle like behavior.” Second, three stretching based schemes are considered. One considers the divergence of trajectory pairs, another quantifies the deformation of a nearest neighbor cloud, and the last uses flow fields calculated from the trajectory data. Finally, the new phase space warping method is introduced. This method takes advantage of the shifting (warping) experienced by a phase space as the parameters of the system are slightly varied. This results in a shift of the invariant manifolds. The region spanned by this shift, provides a means to identify the invariant manifolds. Results show that this method gives superior detection and is robust with respect to the amount of data.

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