Multi-scale dynamical behavior of spatially distributed systems: a deterministic point of view

Abstract:

Physical and biophysical systems are spatially distributed systems. Their behavior can be observed or modelled spatially at various resolutions. In this work, a deterministic point of view is adopted to analyze multi-scale behavior taking a set of ordinary differential equation (ODE) as elementary part of the system.

To perform analyses, scenes of study are thus generated based on ensembles of identical elementary ODE systems. Without any loss of generality, their dynamics is chosen chaotic in order to ensure sensitivity to initial conditions, that is, one fundamental property of atmosphere under instable conditions [1]. The Rössler system [2] is used for this purpose for both its topological and algebraic simplicity [3,4].

Two cases are thus considered: the chaotic oscillators composing the scene of study are taken either independent, or in phase synchronization. Scale behaviors are analyzed considering the scene of study as aggregations (basically obtained by spatially averaging the signal) or as associations (obtained by concatenating the time series). The global modeling technique is used to perform the numerical analyses [5].

One important result of this work is that, under phase synchronization, a scene of aggregated dynamics can be approximated by the elementary system composing the scene, but modifying its parameterization [6]. This is shown based on numerical analyses. It is then demonstrated analytically and generalized to a larger class of ODE systems. Preliminary applications to cereal crops observed from satellite are also presented.

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