Big data analysis from a dynamical systems point of view

開催日時

2018年5月10日 10時30分 ～ 17時00分

場所

Room 3-413, Department of Mathematics

講演者

Davide Faranda (CNRS)，Sulimon Sattari (Hokkaido University)，Sosuke Ito (Hokkaido University)

*10:30-12:00 Davide Faranda (CNRS) *
*New dynamical systems tools to study atmospheric flows*

Atmospheric flows are characterized by chaotic dynamics and recurring large-scale patterns . These two characteristics point to the existence of an atmospheric attractor defined by Lorenz as: “the collection of all states that the system can assume or approach again and again, as opposed to those that it will ultimately avoid”. The average dimension D of the attractor corresponds to the number of degrees of freedom sufficient to describe the atmospheric circulation. However, obtaining reliable estimates of D has proved challenging . Moreover, D does not provide information on transient atmospheric motions, which lead to weather extremes . Using recent developments in dynamical systems theory , we show that such motions can be classified through instantaneous rather than average properties of the attractor. The instantaneous properties are uniquely determined by instantaneous dimension and stability. Their extreme values correspond to specific atmospheric patterns, and match extreme weather occurrences. We further show the existence of a significant correlation between the time series of instantaneous stability and dimension and the mean spread of sea-level pressure fields in an operational ensemble weather forecast at steps of over two weeks. We believe this method provides an efficient and practical way of evaluating and informing operational weather forecasts.

*13:30-15:00 Sulimon Sattari (Hokkaido University)*
*Computing chaotic transport and phase space structure using symbolic dynamics*

Symbolic dynamics techniques allow for viewing systems at a coarse-grained level. A firm grasp of the symbolic dynamics allows for a network representation of the system, where nodes represent regions in the state space and and edges represent allowed transitions between them in time. We demonstrate that symbolic dynamics computed from finite-length segments of invariant manifolds can be used to compute topological entropy and periodic orbits over a variety of parameter regimes in a fluid mixing system, in a classical hydrogen atom in parallel electric and magnetic fields, and in the Henon map. The periodic orbits are then used to accurately compute escape rates, and the escape rate computed from periodic orbits converges exponentially to escape rates from Monte Carlo simulations as more periodic
orbits are included. Using an easier-to-compute, grid-based symbolic dynamics, we also demonstrate that phase space structures can be extracted by computing communities in the symbolic dynamics network. The communities are computed by maximizing the modularity, which measures the strength of division of the network into modules. Certain phase space structures, such as invariant tori, tend to lie within regions of phase space that represent a given community. Furthermore, the variation of phase space structure as parameters are varied is also captured by this technique. Representing time-data as a network in this way can yield new data analysis techniques for understanding the overall structure of a system containing large amounts of data.

*15:30-17:00 Sosuke Ito (Hokkaido University) *
*Stochastic thermodynamic interpretation of information geometry*

In recent years, the unified theory of information and thermodynamics has been intensively dis- cussed in the context of stochastic thermodynamics. The unified theory reveals that information theory would be useful to understand non-stationary dynamics of systems far from equilibrium. In this letter, we have found a new link between stochastic thermodynamics and information theory well known as information geometry. By applying this link, an information geometric inequality can be interpreted as a thermodynamic uncertainty relationship between speed and thermodynamic cost. We have numerically applied an information geometric inequality to a thermodynamic model of biochemical enzyme reaction.
Reference: pdf