| A low-dimensional chaotic model was recently obtained for the dynamics of cereal crops
cycles in semi-arid region [1]. This model was obtained from one single time series of
vegetation index measured from space. The global modeling approach [2] was used based on
powerful algorithms recently developed for this purpose [3]. The resulting model could be
validated by comparing its predictability (a data assimilation scheme was used for this
purpose) with a statistical prediction approach based on the search of analogous states in the
phase space [4].
The cereal crops model exhibits a weakly dissipative chaos (DKY = 2.68) and a
toroidal-like structure. At present, quite few cases of such chaos are known and these are
exclusively theoretical. The first case was introduced by Lorenz in 1984 to model
the global circulation dynamics [5], which attractor’s structure is remained poorly
understood.
Indeed, one very powerful way to characterize low-dimensional chaos is based on the
topological analysis of the attractors’ flow [6]. Unfortunately, such approach does not
apply to weakly dissipative chaos. In this work, a color tracer method is introduced
and used to perform a complete topological analysis of both the Lorenz-84 system
and the cereal crops model. The usual stretching and squeezing mechanisms are
easily detected in the attractors’ structure. A stretching taking place in the globally
contracting direction of the flow is also found in both attractors. Such stretching is
unexpected and was not reported previously. The analysis also confirms the toroidal
type of chaos and allows producing both the skeleton and algebraic descriptions of
the two attractors. Their comparison shows that the cereal crops attractor is a new
attractor.
References
[1]Mangiarotti S., Drapreau L., Letellier C., 2014. Two chaotic global models
for cereal crops cycles observed from satellite in Northern Morocco. revision
submitted.
[2]Letellier C., Aguirre L.A., Freitas U.S., 2009. Frequently asked questions about
global modeling. Chaos, 19, 023103.
[3]Mangiarotti S., Coudret R., Drapreau L., Jarlan L., 2012a. Polynomial Search
and Global Modeling – two algorithms for modelling chaos. Physical Review E,
86(4), 046205.
[4]Mangiarotti S., Mazzega P., Mougin E., Hiernaux P., 2012b. Predictability of
vegetation cycles over the semi-arid region of Gourma (Mali) from forecasts of
AVHRR-NDVI signals. Remote Sensing of Environment, 123, 246–257.
[5]Lorenz, 1984. Irregularity: a fundamental property of the atmosphere, Tellus,
36A, 98-110, 1984.
[6]Gilmore R. & Lefranc M., The topology of chaos, Alice stretch and
squeezeland. Wiley-VCH, 2002. |