Crisis at the Origin of Crisis at the Origin of Deterministic Rogue WavesDeterministic Rogue WavesPPME, Universite de la Nouvelle CaledoniePPME, Universite de la Nouvelle Caledonie

C. Metayer, A. Serres, J. Tredicce INLN, UMR 6618 UNS-CNRS FranceINLN, UMR 6618 UNS-CNRS France

S. Barland, M. GiudiciCEILAP - CITEDEF – ArgentinaCEILAP - CITEDEF – Argentina

A. Hnilo, M. KovalskiA. Hnilo, M. KovalskiUniv. Politecn. Cataluna, SpainUniv. Politecn. Cataluna, Spain

Masoller, C.Univ. Fed Pernambuco, Recife, PE BrazilUniv. Fed Pernambuco, Recife, PE Brazil

W. Barbosa, F. Menezes D’Aguiar,J. Rios Leite, Rosero E.

Are rogue waves just tales of fishermen into an Are rogue waves just tales of fishermen into an Irish Pub?Irish Pub?

Or are rogue waves a common Or are rogue waves a common feature of a complex dynamical feature of a complex dynamical

systems?systems? This fact is in contradiction with the Gaussian models This fact is in contradiction with the Gaussian models

used to describe fluctuations of the wave height in the used to describe fluctuations of the wave height in the sea*. sea*.

* M. S. Longuet-Higgins, Phil. Trans. Roy. Soc. A * M. S. Longuet-Higgins, Phil. Trans. Roy. Soc. A 249249 321 (1957).321 (1957).

S. Aberg and G. Lindgren, Height distribution of S. Aberg and G. Lindgren, Height distribution of stochastic Lagrange ocean waves, Prob. Eng. Mech. stochastic Lagrange ocean waves, Prob. Eng. Mech. 2323, 359 (2008), 359 (2008)

HOWEVER……HOWEVER……

Ferry rescue after freak wave in Irish Sea

The freighter Riverdance was hit by a giant wave during severe gales in the Irish Sea…..

But….What is the definition of a But….What is the definition of a rogue wave?rogue wave?

Old Recipe: Take the 1/3 biggest amplitude Old Recipe: Take the 1/3 biggest amplitude waves; calculate their average value; waves; calculate their average value; multiply by 2….whatever amplitude exceeds multiply by 2….whatever amplitude exceeds such value is a rogue wave!!!such value is a rogue wave!!!

Or another Recipe: Take the probability Or another Recipe: Take the probability distribution; calculate distribution; calculate ; multiply by 4; ; multiply by 4; whatever….; whatever….;

and if you want a BIG BIG rogue and if you want a BIG BIG rogue wave…multiply by 8 wave…multiply by 8

In the WEB:

It is probably sufficient to say that any wave so large that it is unexpected based on current conditions can be counted as a rogue.

There are very few photographs of rogue waves. For centuries, the best evidence for their existence was anecdotal -- the countless stories told by sailors who had survived one.

How was that “Opticians” got How was that “Opticians” got interested on Rogue Waves?interested on Rogue Waves?

A “NONLINEAR OPTICS PHYSICIST” WENT TO A “NONLINEAR OPTICS PHYSICIST” WENT TO THE IRISH PUB….and then some papers appear THE IRISH PUB….and then some papers appear in Nature or other “GO..O..D” Journalsin Nature or other “GO..O..D” Journals

D. R. Solli, C. Ropers et al, Optical rogue waves, D. R. Solli, C. Ropers et al, Optical rogue waves, Nature 450 1054 (2007).Nature 450 1054 (2007).

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev and J. M. Dudley, The Genty, N. Akhmediev and J. M. Dudley, The Peregrine soliton in nonlinear fibre optics Nature Peregrine soliton in nonlinear fibre optics Nature Phys. 6, 790 (2010).Phys. 6, 790 (2010).

A. Montina, U. Bortolozzo, S. Residori, F.T. Arecchi, A. Montina, U. Bortolozzo, S. Residori, F.T. Arecchi, Phys. Rev. Lett. 103, 173901 (2009)Phys. Rev. Lett. 103, 173901 (2009)

For us, the story begins with..

“Critical slowing down at a bifurcation” JRT, Lippi ; Mandel; and B. Charasse ; A. Chevalier; B. Picque: American Journal of Physics Vol 72  Jun 2004

Cited by a series of papers on Anticipating Critical Transitions

Scheffer, M.et al. SCIENCE  338  (2012)Dakos,V. et al. PLOS ONE  7 (2012)Scheffer, M.; et al. NATURE  461 (2009)

Our Experiments o, ….Our Experiments o, ….

1) Mode Locked Ti:Sa laser1) Mode Locked Ti:Sa laser

Hnilo et al. (Opt. Lett. November 2011)Hnilo et al. (Opt. Lett. November 2011)

2) Semiconductor Laser with Injected Signal2) Semiconductor Laser with Injected Signal

Bonatto et al. (PRL, July 2011); Zamora-Munt et al. Bonatto et al. (PRL, July 2011); Zamora-Munt et al. (PRA, March 2013). (PRA, March 2013).

3) Laser with saturable absorber Hnilo et al. 3) Laser with saturable absorber Hnilo et al. (Journal of Optics, June 2013).(Journal of Optics, June 2013).

Probability distribution of maxima Probability distribution of maxima

Our Already Published Our Already Published ConclusionsConclusions

1) Extreme Events are rare but they can be 1) Extreme Events are rare but they can be much more probable than in Gaussian much more probable than in Gaussian models when the dynamical behavior is models when the dynamical behavior is “Deterministically” Chaotic“Deterministically” Chaotic

2) There is “chaos” without rogue waves 2) There is “chaos” without rogue waves and chaos with rogue wavesand chaos with rogue waves

Some questions:Some questions:

How? How? What is the dynamical process the What is the dynamical process the laser use to generate “extreme events”?laser use to generate “extreme events”?

Can we predict deterministic extreme events Can we predict deterministic extreme events in optical systems?in optical systems?

Can we control them? Can we control them?

How?How?

a) Intermittency ….a) Intermittency ….P Gaspard and X Wang, PNAS 1988

Nicolis et al., Journal of Statistical physics 1995

b) By abrupt expansion of a chaotic attractor??

Bifurcation Diagrams

Experimental results Experimental results

Laser with Modulated parameterLaser with Modulated parameterRemembering very old « times »:Remembering very old « times »:H.G. Solari J, E. Eschenazi, R. Gilmore H.G. Solari J, E. Eschenazi, R. Gilmore et al., Opt. et al., Opt. Commun. 64, 49 (1987)Commun. 64, 49 (1987)

onon

““Crisis of chaotic attractors”Crisis of chaotic attractors”

Two ingredients: 1) chaosTwo ingredients: 1) chaos

2) Enough low dissipation in order to have 2) Enough low dissipation in order to have “generalized” multistability (several stable “generalized” multistability (several stable dynamical solutions for the same parameter values) dynamical solutions for the same parameter values)

Crisis of chaotic attractorsCrisis of chaotic attractors

External crisis in a laser with External crisis in a laser with mopdulated parametermopdulated parameter

Then extreme events appear Then extreme events appear after an external crisisafter an external crisis

Predicting “Rogue waves”? Predicting “Rogue waves”? In a deterministic system, the time of “prediction” equals the inverse of the maximum positive Lyapunov exponent

But in the laser with injected signal, the prediction time is much larger, and just looking one variable: the intensity

ConclusionsConclusions

1)1) External crisis produce abrupt expansion of External crisis produce abrupt expansion of chaotic attractors and are at the origin of chaotic attractors and are at the origin of some extreme eventssome extreme events

2)2) Deterministic extreme events could be Deterministic extreme events could be predicted with « some » anticipationpredicted with « some » anticipation

3)3) I still do not know if we are able to control I still do not know if we are able to control deterministic extreme events deterministic extreme events

BUTBUT

I am always looking for the rogue I am always looking for the rogue waves in New Caledoniawaves in New Caledonia