Evidences of chaos from the dynamics of snow cover in Pyrenees, Lebanon and High- Atlas mountains le...

Evidences of chaos from the dynamics of snow cover

in Pyrenees, Lebanon and High-

Atlas mountains

le 17/07/2013

Global Modelling

• Phenomenological approach of the dynamics

• well adapted to low-dimensionnal system

• several underlying hypothesis

Few knowledge about the dynamics is requiredThe dynamics is directly investigated from a time series

It can provide a strong evidence of chaotic behaviour

System should be ergodic, deterministic,autonomous and stationnary

approach

Gouesbet & Letellier (1994)

nn XXxRX

XX

Xx

,..,,

...

21

32

21

nnn

n

n

xxxfx

xxxfx

xxxfx

,..,,

...

,..,,

,..,,

21

2122

2111

• Reformulation

Polynomialapproximation

nn XXxPXXxR ,..,,,..,, 2121

approach

Global Modelling

approach

• Sunspot cycles

• Canadian lynx cycles

• Human respiration

• Rainfed wheat

• Snow cover

- natural

- ecological

- human

- agricultural

- climatic

.. systems

Maquet et al. 2007

Letellier et al. 2006

Mangiarotti et al. 2011, 2013

Previous applicationsto ‘‘real world’’ systems

.. this work !

Letellier et al. 2013

Takens’theorem (1981)

zyxz

yxzxy

xyx

tztytx ,,

z(t)

x(t)

example: attracteur de Lorenz(1963)

(1941-2010)

y(t)

zyxz

yxzxy

xyx

diffeomorphism(differentiable &

differentiable inverse)

tx

tztytx ,,

z(t)

x(t)y(t) x(t)

x(t +

2 τ

)

x(t + τ )

(1941-2010)

Takens’theorem (1981)

zyxz

yxzxy

xyx

diffeomorphism(differentiable &

differentiable inverse)

tx

tztytx ,,

z(t)

x(t)y(t) x(t)

x(t +

2 τ

)

x(t + τ ) tx

tx

tx

Takens’theorem (1981)

Phase Space Portrait

tx1

On dériven fois

n

n xX

xX

xX

x

...

13

12

1

1x

1x

ND

VI(t

)approach

Pyrénées (0-40000km²)

Mt Liban (0-12000km²)

Atlas (0-30000km²)

Pyrénées

Mt Liban

Atlas

Portraits dynamiques

txtxtx ,,

2000-2012

Models selection algorithm interactionMain GloMo Interface

Algorithmic tools R-packages

PoMoS GloMo

Packages available on CRAN: http://cran.cict.fr

approach

Pyrénées

Mt Liban

Atlas

(…)

(…)

(…)

2000-2012

yearly analysis

Pyrénées

Pyreneous

2000-2001

Pyreneous

2001-2002

Pyreneous

2002-2003

Pyreneous

2003-2004

Pyreneous

2010-2011

Pyreneous

yearly analysis

Mont Liban

Mt Liban

2001-2002

Mt Liban

2003-2004

Mt Liban

2005-2006

Mt Liban

2006-2007

Mt Liban

2007-2008

Mt Liban

2009-2010

Mt Liban

yearly analysis

Atlas

Atlas

2001-2002

Atlas

2002-2003

Atlas

2004-2005

Atlas

2005-2006

Atlas

2006-2007

Atlas

2009-2010

Atlas