Abstract

Macroscopic systems subjected to injection and dissipation of energy can exhibit complex spatiotemporal behaviors as result of dissipative self-organization. Here, we report a one- and two-dimensional pattern forming setup, which exhibits a transition from stationary patterns to spatiotemporal chaotic textures, based on a nematic liquid crystal layer with spatially modulated input beam and optical feedback. Using an adequate projection of spatiotemporal diagrams, we determine the largest Lyapunov exponent. Jointly, this exponent and Fourier transform allow us to distinguish between spatiotemporal chaos and amplitude turbulence concepts, which are usually merged.

© 2016 Optical Society of America

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References

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  1. F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
    [Crossref] [PubMed]
  2. G. Huyet, M. C. Martinoni, J. Tredicce, and S. Rica, “Spatiotemporal dynamics of lasers with a large Fresnel number,” Phys. Rev. Lett. 75, 4027–4030 (1995).
    [Crossref] [PubMed]
  3. G. Huyet and J. R. Tredicce, “Spatio-temporal chaos in the transverse section of lasers,” Physica D 96, 209–214 (1996).
    [Crossref]
  4. A. V. Mamaev and M. Saffman, “Selection of unstable patterns and control of optical turbulence by Fourier plane filtering,” Phys. Rev. Lett. 80, 3499–4502 (1998).
    [Crossref]
  5. E. A. Rogers, R. Kalra, R. D. Schroll, A. Uchida, D. P. Lathrop, and R. Roy, “Generalized synchronization of spatiotemporal chaos in a liquid crystal spatial light modulator,” Phys. Rev. Lett. 93, 084101 (2004).
    [Crossref] [PubMed]
  6. S. Residori, “Patterns, fronts and structures in a liquid-crystal-light-valve with optical feedback,” Phys. Rep. 416, 201–272 (2005).
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    [Crossref]
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    [Crossref]
  13. P. Coullet, L. Gil, and J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1989).
    [Crossref] [PubMed]
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    [Crossref]
  15. W. Decker, W. Pesch, and A. Weber, “Spiral defect chaos in Rayleigh-Benard convection,” Phys. Rev. Lett. 73, 648–651 (1994).
    [Crossref] [PubMed]
  16. B. Echebarria and H. Riecke, “Defect chaos of oscillating hexagons in rotating convection,” Phys. Rev. Lett. 84, 4838–4841 (2000).
    [Crossref] [PubMed]
  17. K. E. Daniels and E. Bodenschatz, “Defect turbulence in inclined layer convection,” Phys. Rev. Lett. 88, 034501 (2002).
    [Crossref] [PubMed]
  18. M. Miranda and J. Burguete, “Experimentally observed route to spatiotemporal chaos in an extended one-dimensional array of convective oscillators,” Phys. Rev. E 79, 046201 (2009).
    [Crossref]
  19. P. Brunet and I. Limat, “Defects and spatiotemporal disorder in a pattern of falling liquid columns,” Phys. Rev. E 70, 046207 (2004).
    [Crossref]
  20. Q. Ouyang and J. M. Flesselles, “Transition from spirals to defect turbulence driven by a convective instability,” Nature 379, 143–146 (1996).
    [Crossref]
  21. A. Garfinkel, M. L. Spano, W. L. Ditto, and J. N. Weiss, “Controlling cardiac chaos,” Science 257, 1230–1235 (1992).
    [Crossref] [PubMed]
  22. S. Q. Zhou and G. Ahlers, “Spatiotemporal chaos in electroconvection of a homeotropically aligned nematic liquid crystal,” Phys. Rev. E 74, 046212 (2006).
    [Crossref]
  23. S. J. Moon, M. D. Shattuck, C. Bizon, D. I. Goldman, J. B. Swift, and H. L. Swinney, “Phase bubbles and spatiotemporal chaos in granular patterns,” Phys. Rev. E 65, 011301 (2001).
    [Crossref]
  24. N. Verschueren, U. Bortolozzo, M. G. Clerc, and S. Residori, “Spatiotemporal chaotic localized state in liquid crystal light valve experiments with optical feedback,” Phys. Rev. Lett. 110, 104101 (2013).
    [Crossref] [PubMed]
  25. N. Verschueren, U. Bortolozzo, M. G. Clerc, and S. Residori, “Chaoticon: localized pattern with permanent dynamics,” Phil. Trans. R. Soc. A. 372, 20140011 (2014).
    [Crossref] [PubMed]
  26. P. Manneville, Dissipative Structures and Weak Turbulence (Academic, 1990).
  27. L. Pastur, U. Bortolozzo, and P. L. Ramazza, “Transition to space-time chaos in an optical loop with translational transport,” Phys. Rev. E 69, 016210 (2004).
    [Crossref]
  28. M. G. Clerc, G. Gonzalez-Cortes, and M. Wilson, “Experimental Spatiotemporal Chaotic Textures in a Liquid Crystal Light Valve with Optical Feedback,” in Nonlinear Dynamics: Materials, Theory and Experiments, M. Tlidi and M. G. Clerc, eds. (Springer, 2016).
  29. M. G. Clerc, C. Falcon, M. A. Garcia-Nustes, V. Odent, and I. Ortega, “Emergence of spatiotemporal dislocation chains in drifting patterns,” Chaos 24, 023133 (2014).
    [Crossref] [PubMed]
  30. E. Louvergneaux, “Pattern-Dislocation-Type Dynamical Instability in 1D Optical Feedback Kerr Media with Gaussian Transverse Pumping,” Phys. Rev. Lett. 87, 244501 (2001).
    [Crossref] [PubMed]
  31. S. Bielawski, C. Szwaj, C. Bruni, D. Garzella, G. L. Orlandi, and M. E. Couprie, “Advection-induced spectrotemporal defects in a free-electron laser,” Phys. Rev. Lett. 95, 034801 (2005).
    [Crossref] [PubMed]
  32. G. Nicolis, Introduction to Nonlinear Science (Cambridge University, 1995).
    [Crossref]
  33. M. G. Clerc and N. Verschueren, “Quasiperiodicity route to spatiotemporal chaos in one-dimensional pattern-forming systems,” Phys. Rev. E. 88, 052916 (2013).
    [Crossref]
  34. K. E. Daniels and E. Bodenschatz, “Defect turbulence in inclined layer convection,” Phys. Rev. Lett. 88, 034501 (2002).
    [Crossref] [PubMed]
  35. Y. Kuramoto, Chemical Oscillations: Waves, and Turbulence (Springer, 1984).
    [Crossref]
  36. U. Frisch, Turbulence: the Legacy of A.N. Kolmogorov (Cambridge University, 1995).
  37. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining the Lyapunov exponents from a time series,” Physica D 16, 285–317 (1985).
    [Crossref]
  38. E. Ott, Chaos in Dynamical Systems, 2nd ed. (Cambridge University, 2002).
    [Crossref]
  39. H. Abarbanel, Analysis of Observed Chaotic Data (Springer-Verlag, 1996).
    [Crossref]

2014 (3)

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and M. G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

N. Verschueren, U. Bortolozzo, M. G. Clerc, and S. Residori, “Chaoticon: localized pattern with permanent dynamics,” Phil. Trans. R. Soc. A. 372, 20140011 (2014).
[Crossref] [PubMed]

M. G. Clerc, C. Falcon, M. A. Garcia-Nustes, V. Odent, and I. Ortega, “Emergence of spatiotemporal dislocation chains in drifting patterns,” Chaos 24, 023133 (2014).
[Crossref] [PubMed]

2013 (2)

N. Verschueren, U. Bortolozzo, M. G. Clerc, and S. Residori, “Spatiotemporal chaotic localized state in liquid crystal light valve experiments with optical feedback,” Phys. Rev. Lett. 110, 104101 (2013).
[Crossref] [PubMed]

M. G. Clerc and N. Verschueren, “Quasiperiodicity route to spatiotemporal chaos in one-dimensional pattern-forming systems,” Phys. Rev. E. 88, 052916 (2013).
[Crossref]

2009 (1)

M. Miranda and J. Burguete, “Experimentally observed route to spatiotemporal chaos in an extended one-dimensional array of convective oscillators,” Phys. Rev. E 79, 046201 (2009).
[Crossref]

2006 (1)

S. Q. Zhou and G. Ahlers, “Spatiotemporal chaos in electroconvection of a homeotropically aligned nematic liquid crystal,” Phys. Rev. E 74, 046212 (2006).
[Crossref]

2005 (2)

S. Bielawski, C. Szwaj, C. Bruni, D. Garzella, G. L. Orlandi, and M. E. Couprie, “Advection-induced spectrotemporal defects in a free-electron laser,” Phys. Rev. Lett. 95, 034801 (2005).
[Crossref] [PubMed]

S. Residori, “Patterns, fronts and structures in a liquid-crystal-light-valve with optical feedback,” Phys. Rep. 416, 201–272 (2005).
[Crossref]

2004 (3)

E. A. Rogers, R. Kalra, R. D. Schroll, A. Uchida, D. P. Lathrop, and R. Roy, “Generalized synchronization of spatiotemporal chaos in a liquid crystal spatial light modulator,” Phys. Rev. Lett. 93, 084101 (2004).
[Crossref] [PubMed]

P. Brunet and I. Limat, “Defects and spatiotemporal disorder in a pattern of falling liquid columns,” Phys. Rev. E 70, 046207 (2004).
[Crossref]

L. Pastur, U. Bortolozzo, and P. L. Ramazza, “Transition to space-time chaos in an optical loop with translational transport,” Phys. Rev. E 69, 016210 (2004).
[Crossref]

2002 (2)

K. E. Daniels and E. Bodenschatz, “Defect turbulence in inclined layer convection,” Phys. Rev. Lett. 88, 034501 (2002).
[Crossref] [PubMed]

K. E. Daniels and E. Bodenschatz, “Defect turbulence in inclined layer convection,” Phys. Rev. Lett. 88, 034501 (2002).
[Crossref] [PubMed]

2001 (2)

S. J. Moon, M. D. Shattuck, C. Bizon, D. I. Goldman, J. B. Swift, and H. L. Swinney, “Phase bubbles and spatiotemporal chaos in granular patterns,” Phys. Rev. E 65, 011301 (2001).
[Crossref]

E. Louvergneaux, “Pattern-Dislocation-Type Dynamical Instability in 1D Optical Feedback Kerr Media with Gaussian Transverse Pumping,” Phys. Rev. Lett. 87, 244501 (2001).
[Crossref] [PubMed]

2000 (1)

B. Echebarria and H. Riecke, “Defect chaos of oscillating hexagons in rotating convection,” Phys. Rev. Lett. 84, 4838–4841 (2000).
[Crossref] [PubMed]

1998 (2)

G. Goren, J. P. Eckmann, and I. Procaccia, “Scenario for the onset of space-time chaos,” Phys. Rev. E 57, 4106–4134 (1998).
[Crossref]

A. V. Mamaev and M. Saffman, “Selection of unstable patterns and control of optical turbulence by Fourier plane filtering,” Phys. Rev. Lett. 80, 3499–4502 (1998).
[Crossref]

1996 (2)

G. Huyet and J. R. Tredicce, “Spatio-temporal chaos in the transverse section of lasers,” Physica D 96, 209–214 (1996).
[Crossref]

Q. Ouyang and J. M. Flesselles, “Transition from spirals to defect turbulence driven by a convective instability,” Nature 379, 143–146 (1996).
[Crossref]

1995 (1)

G. Huyet, M. C. Martinoni, J. Tredicce, and S. Rica, “Spatiotemporal dynamics of lasers with a large Fresnel number,” Phys. Rev. Lett. 75, 4027–4030 (1995).
[Crossref] [PubMed]

1994 (1)

W. Decker, W. Pesch, and A. Weber, “Spiral defect chaos in Rayleigh-Benard convection,” Phys. Rev. Lett. 73, 648–651 (1994).
[Crossref] [PubMed]

1992 (1)

A. Garfinkel, M. L. Spano, W. L. Ditto, and J. N. Weiss, “Controlling cardiac chaos,” Science 257, 1230–1235 (1992).
[Crossref] [PubMed]

1990 (1)

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[Crossref] [PubMed]

1989 (1)

P. Coullet, L. Gil, and J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1989).
[Crossref] [PubMed]

1988 (1)

P. Coullet and J. Lega, “Defect-mediated turbulence in wave patterns,” Europhys. Lett. 7, 511–516 (1988).
[Crossref]

1985 (1)

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining the Lyapunov exponents from a time series,” Physica D 16, 285–317 (1985).
[Crossref]

Abarbanel, H.

H. Abarbanel, Analysis of Observed Chaotic Data (Springer-Verlag, 1996).
[Crossref]

Ahlers, G.

S. Q. Zhou and G. Ahlers, “Spatiotemporal chaos in electroconvection of a homeotropically aligned nematic liquid crystal,” Phys. Rev. E 74, 046212 (2006).
[Crossref]

Arecchi, F. T.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[Crossref] [PubMed]

Bielawski, S.

S. Bielawski, C. Szwaj, C. Bruni, D. Garzella, G. L. Orlandi, and M. E. Couprie, “Advection-induced spectrotemporal defects in a free-electron laser,” Phys. Rev. Lett. 95, 034801 (2005).
[Crossref] [PubMed]

Bizon, C.

S. J. Moon, M. D. Shattuck, C. Bizon, D. I. Goldman, J. B. Swift, and H. L. Swinney, “Phase bubbles and spatiotemporal chaos in granular patterns,” Phys. Rev. E 65, 011301 (2001).
[Crossref]

Bodenschatz, E.

K. E. Daniels and E. Bodenschatz, “Defect turbulence in inclined layer convection,” Phys. Rev. Lett. 88, 034501 (2002).
[Crossref] [PubMed]

K. E. Daniels and E. Bodenschatz, “Defect turbulence in inclined layer convection,” Phys. Rev. Lett. 88, 034501 (2002).
[Crossref] [PubMed]

Bortolozzo, U.

N. Verschueren, U. Bortolozzo, M. G. Clerc, and S. Residori, “Chaoticon: localized pattern with permanent dynamics,” Phil. Trans. R. Soc. A. 372, 20140011 (2014).
[Crossref] [PubMed]

N. Verschueren, U. Bortolozzo, M. G. Clerc, and S. Residori, “Spatiotemporal chaotic localized state in liquid crystal light valve experiments with optical feedback,” Phys. Rev. Lett. 110, 104101 (2013).
[Crossref] [PubMed]

L. Pastur, U. Bortolozzo, and P. L. Ramazza, “Transition to space-time chaos in an optical loop with translational transport,” Phys. Rev. E 69, 016210 (2004).
[Crossref]

Brunet, P.

P. Brunet and I. Limat, “Defects and spatiotemporal disorder in a pattern of falling liquid columns,” Phys. Rev. E 70, 046207 (2004).
[Crossref]

Bruni, C.

S. Bielawski, C. Szwaj, C. Bruni, D. Garzella, G. L. Orlandi, and M. E. Couprie, “Advection-induced spectrotemporal defects in a free-electron laser,” Phys. Rev. Lett. 95, 034801 (2005).
[Crossref] [PubMed]

Burguete, J.

M. Miranda and J. Burguete, “Experimentally observed route to spatiotemporal chaos in an extended one-dimensional array of convective oscillators,” Phys. Rev. E 79, 046201 (2009).
[Crossref]

Clerc, M. G.

M. G. Clerc, C. Falcon, M. A. Garcia-Nustes, V. Odent, and I. Ortega, “Emergence of spatiotemporal dislocation chains in drifting patterns,” Chaos 24, 023133 (2014).
[Crossref] [PubMed]

N. Verschueren, U. Bortolozzo, M. G. Clerc, and S. Residori, “Chaoticon: localized pattern with permanent dynamics,” Phil. Trans. R. Soc. A. 372, 20140011 (2014).
[Crossref] [PubMed]

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and M. G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

N. Verschueren, U. Bortolozzo, M. G. Clerc, and S. Residori, “Spatiotemporal chaotic localized state in liquid crystal light valve experiments with optical feedback,” Phys. Rev. Lett. 110, 104101 (2013).
[Crossref] [PubMed]

M. G. Clerc and N. Verschueren, “Quasiperiodicity route to spatiotemporal chaos in one-dimensional pattern-forming systems,” Phys. Rev. E. 88, 052916 (2013).
[Crossref]

M. G. Clerc, G. Gonzalez-Cortes, and M. Wilson, “Experimental Spatiotemporal Chaotic Textures in a Liquid Crystal Light Valve with Optical Feedback,” in Nonlinear Dynamics: Materials, Theory and Experiments, M. Tlidi and M. G. Clerc, eds. (Springer, 2016).

Coullet, P.

P. Coullet, L. Gil, and J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1989).
[Crossref] [PubMed]

P. Coullet and J. Lega, “Defect-mediated turbulence in wave patterns,” Europhys. Lett. 7, 511–516 (1988).
[Crossref]

Couprie, M. E.

S. Bielawski, C. Szwaj, C. Bruni, D. Garzella, G. L. Orlandi, and M. E. Couprie, “Advection-induced spectrotemporal defects in a free-electron laser,” Phys. Rev. Lett. 95, 034801 (2005).
[Crossref] [PubMed]

Daniels, K. E.

K. E. Daniels and E. Bodenschatz, “Defect turbulence in inclined layer convection,” Phys. Rev. Lett. 88, 034501 (2002).
[Crossref] [PubMed]

K. E. Daniels and E. Bodenschatz, “Defect turbulence in inclined layer convection,” Phys. Rev. Lett. 88, 034501 (2002).
[Crossref] [PubMed]

Decker, W.

W. Decker, W. Pesch, and A. Weber, “Spiral defect chaos in Rayleigh-Benard convection,” Phys. Rev. Lett. 73, 648–651 (1994).
[Crossref] [PubMed]

Ditto, W. L.

A. Garfinkel, M. L. Spano, W. L. Ditto, and J. N. Weiss, “Controlling cardiac chaos,” Science 257, 1230–1235 (1992).
[Crossref] [PubMed]

Echebarria, B.

B. Echebarria and H. Riecke, “Defect chaos of oscillating hexagons in rotating convection,” Phys. Rev. Lett. 84, 4838–4841 (2000).
[Crossref] [PubMed]

Eckmann, J. P.

G. Goren, J. P. Eckmann, and I. Procaccia, “Scenario for the onset of space-time chaos,” Phys. Rev. E 57, 4106–4134 (1998).
[Crossref]

Falcon, C.

M. G. Clerc, C. Falcon, M. A. Garcia-Nustes, V. Odent, and I. Ortega, “Emergence of spatiotemporal dislocation chains in drifting patterns,” Chaos 24, 023133 (2014).
[Crossref] [PubMed]

Flesselles, J. M.

Q. Ouyang and J. M. Flesselles, “Transition from spirals to defect turbulence driven by a convective instability,” Nature 379, 143–146 (1996).
[Crossref]

Frisch, U.

U. Frisch, Turbulence: the Legacy of A.N. Kolmogorov (Cambridge University, 1995).

Garcia-Nustes, M. A.

M. G. Clerc, C. Falcon, M. A. Garcia-Nustes, V. Odent, and I. Ortega, “Emergence of spatiotemporal dislocation chains in drifting patterns,” Chaos 24, 023133 (2014).
[Crossref] [PubMed]

Garfinkel, A.

A. Garfinkel, M. L. Spano, W. L. Ditto, and J. N. Weiss, “Controlling cardiac chaos,” Science 257, 1230–1235 (1992).
[Crossref] [PubMed]

Garzella, D.

S. Bielawski, C. Szwaj, C. Bruni, D. Garzella, G. L. Orlandi, and M. E. Couprie, “Advection-induced spectrotemporal defects in a free-electron laser,” Phys. Rev. Lett. 95, 034801 (2005).
[Crossref] [PubMed]

Giacomelli, G.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[Crossref] [PubMed]

Gil, L.

P. Coullet, L. Gil, and J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1989).
[Crossref] [PubMed]

Goldman, D. I.

S. J. Moon, M. D. Shattuck, C. Bizon, D. I. Goldman, J. B. Swift, and H. L. Swinney, “Phase bubbles and spatiotemporal chaos in granular patterns,” Phys. Rev. E 65, 011301 (2001).
[Crossref]

Gonzalez-Cortes, G.

M. G. Clerc, G. Gonzalez-Cortes, and M. Wilson, “Experimental Spatiotemporal Chaotic Textures in a Liquid Crystal Light Valve with Optical Feedback,” in Nonlinear Dynamics: Materials, Theory and Experiments, M. Tlidi and M. G. Clerc, eds. (Springer, 2016).

Goren, G.

G. Goren, J. P. Eckmann, and I. Procaccia, “Scenario for the onset of space-time chaos,” Phys. Rev. E 57, 4106–4134 (1998).
[Crossref]

Huyet, G.

G. Huyet and J. R. Tredicce, “Spatio-temporal chaos in the transverse section of lasers,” Physica D 96, 209–214 (1996).
[Crossref]

G. Huyet, M. C. Martinoni, J. Tredicce, and S. Rica, “Spatiotemporal dynamics of lasers with a large Fresnel number,” Phys. Rev. Lett. 75, 4027–4030 (1995).
[Crossref] [PubMed]

Kalra, R.

E. A. Rogers, R. Kalra, R. D. Schroll, A. Uchida, D. P. Lathrop, and R. Roy, “Generalized synchronization of spatiotemporal chaos in a liquid crystal spatial light modulator,” Phys. Rev. Lett. 93, 084101 (2004).
[Crossref] [PubMed]

Kuramoto, Y.

Y. Kuramoto, Chemical Oscillations: Waves, and Turbulence (Springer, 1984).
[Crossref]

Lathrop, D. P.

E. A. Rogers, R. Kalra, R. D. Schroll, A. Uchida, D. P. Lathrop, and R. Roy, “Generalized synchronization of spatiotemporal chaos in a liquid crystal spatial light modulator,” Phys. Rev. Lett. 93, 084101 (2004).
[Crossref] [PubMed]

Lega, J.

P. Coullet, L. Gil, and J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1989).
[Crossref] [PubMed]

P. Coullet and J. Lega, “Defect-mediated turbulence in wave patterns,” Europhys. Lett. 7, 511–516 (1988).
[Crossref]

Limat, I.

P. Brunet and I. Limat, “Defects and spatiotemporal disorder in a pattern of falling liquid columns,” Phys. Rev. E 70, 046207 (2004).
[Crossref]

Louvergneaux, E.

E. Louvergneaux, “Pattern-Dislocation-Type Dynamical Instability in 1D Optical Feedback Kerr Media with Gaussian Transverse Pumping,” Phys. Rev. Lett. 87, 244501 (2001).
[Crossref] [PubMed]

Mamaev, A. V.

A. V. Mamaev and M. Saffman, “Selection of unstable patterns and control of optical turbulence by Fourier plane filtering,” Phys. Rev. Lett. 80, 3499–4502 (1998).
[Crossref]

Manneville, P.

P. Manneville, Dissipative Structures and Weak Turbulence (Academic, 1990).

Martinoni, M. C.

G. Huyet, M. C. Martinoni, J. Tredicce, and S. Rica, “Spatiotemporal dynamics of lasers with a large Fresnel number,” Phys. Rev. Lett. 75, 4027–4030 (1995).
[Crossref] [PubMed]

Miranda, M.

M. Miranda and J. Burguete, “Experimentally observed route to spatiotemporal chaos in an extended one-dimensional array of convective oscillators,” Phys. Rev. E 79, 046201 (2009).
[Crossref]

Moon, S. J.

S. J. Moon, M. D. Shattuck, C. Bizon, D. I. Goldman, J. B. Swift, and H. L. Swinney, “Phase bubbles and spatiotemporal chaos in granular patterns,” Phys. Rev. E 65, 011301 (2001).
[Crossref]

Nicolis, G.

G. Nicolis, Introduction to Nonlinear Science (Cambridge University, 1995).
[Crossref]

G. Nicolis and I. Prigogine, Self-Organization in Non Equilibrium Systems (J. Wiley & Sons, 1977).

G. Nicolis, Introduction to Nonlinear Science (Cambridge University, 1995).
[Crossref]

Odent, V.

M. G. Clerc, C. Falcon, M. A. Garcia-Nustes, V. Odent, and I. Ortega, “Emergence of spatiotemporal dislocation chains in drifting patterns,” Chaos 24, 023133 (2014).
[Crossref] [PubMed]

Orlandi, G. L.

S. Bielawski, C. Szwaj, C. Bruni, D. Garzella, G. L. Orlandi, and M. E. Couprie, “Advection-induced spectrotemporal defects in a free-electron laser,” Phys. Rev. Lett. 95, 034801 (2005).
[Crossref] [PubMed]

Ortega, I.

M. G. Clerc, C. Falcon, M. A. Garcia-Nustes, V. Odent, and I. Ortega, “Emergence of spatiotemporal dislocation chains in drifting patterns,” Chaos 24, 023133 (2014).
[Crossref] [PubMed]

Ott, E.

E. Ott, Chaos in Dynamical Systems, 2nd ed. (Cambridge University, 2002).
[Crossref]

Ouyang, Q.

Q. Ouyang and J. M. Flesselles, “Transition from spirals to defect turbulence driven by a convective instability,” Nature 379, 143–146 (1996).
[Crossref]

Panajotov, K.

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and M. G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

Pastur, L.

L. Pastur, U. Bortolozzo, and P. L. Ramazza, “Transition to space-time chaos in an optical loop with translational transport,” Phys. Rev. E 69, 016210 (2004).
[Crossref]

Pesch, W.

W. Decker, W. Pesch, and A. Weber, “Spiral defect chaos in Rayleigh-Benard convection,” Phys. Rev. Lett. 73, 648–651 (1994).
[Crossref] [PubMed]

Prigogine, I.

G. Nicolis and I. Prigogine, Self-Organization in Non Equilibrium Systems (J. Wiley & Sons, 1977).

Procaccia, I.

G. Goren, J. P. Eckmann, and I. Procaccia, “Scenario for the onset of space-time chaos,” Phys. Rev. E 57, 4106–4134 (1998).
[Crossref]

Ramazza, P. L.

L. Pastur, U. Bortolozzo, and P. L. Ramazza, “Transition to space-time chaos in an optical loop with translational transport,” Phys. Rev. E 69, 016210 (2004).
[Crossref]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[Crossref] [PubMed]

Residori, S.

N. Verschueren, U. Bortolozzo, M. G. Clerc, and S. Residori, “Chaoticon: localized pattern with permanent dynamics,” Phil. Trans. R. Soc. A. 372, 20140011 (2014).
[Crossref] [PubMed]

N. Verschueren, U. Bortolozzo, M. G. Clerc, and S. Residori, “Spatiotemporal chaotic localized state in liquid crystal light valve experiments with optical feedback,” Phys. Rev. Lett. 110, 104101 (2013).
[Crossref] [PubMed]

S. Residori, “Patterns, fronts and structures in a liquid-crystal-light-valve with optical feedback,” Phys. Rep. 416, 201–272 (2005).
[Crossref]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[Crossref] [PubMed]

Rica, S.

G. Huyet, M. C. Martinoni, J. Tredicce, and S. Rica, “Spatiotemporal dynamics of lasers with a large Fresnel number,” Phys. Rev. Lett. 75, 4027–4030 (1995).
[Crossref] [PubMed]

Riecke, H.

B. Echebarria and H. Riecke, “Defect chaos of oscillating hexagons in rotating convection,” Phys. Rev. Lett. 84, 4838–4841 (2000).
[Crossref] [PubMed]

Rogers, E. A.

E. A. Rogers, R. Kalra, R. D. Schroll, A. Uchida, D. P. Lathrop, and R. Roy, “Generalized synchronization of spatiotemporal chaos in a liquid crystal spatial light modulator,” Phys. Rev. Lett. 93, 084101 (2004).
[Crossref] [PubMed]

Roy, R.

E. A. Rogers, R. Kalra, R. D. Schroll, A. Uchida, D. P. Lathrop, and R. Roy, “Generalized synchronization of spatiotemporal chaos in a liquid crystal spatial light modulator,” Phys. Rev. Lett. 93, 084101 (2004).
[Crossref] [PubMed]

Saffman, M.

A. V. Mamaev and M. Saffman, “Selection of unstable patterns and control of optical turbulence by Fourier plane filtering,” Phys. Rev. Lett. 80, 3499–4502 (1998).
[Crossref]

Sánchez-Morcillo, J.

K. Staliunas and J. Sánchez-Morcillo, Transverse Patterns (Springer Science & Business Media, 2003).

Schroll, R. D.

E. A. Rogers, R. Kalra, R. D. Schroll, A. Uchida, D. P. Lathrop, and R. Roy, “Generalized synchronization of spatiotemporal chaos in a liquid crystal spatial light modulator,” Phys. Rev. Lett. 93, 084101 (2004).
[Crossref] [PubMed]

Shattuck, M. D.

S. J. Moon, M. D. Shattuck, C. Bizon, D. I. Goldman, J. B. Swift, and H. L. Swinney, “Phase bubbles and spatiotemporal chaos in granular patterns,” Phys. Rev. E 65, 011301 (2001).
[Crossref]

Spano, M. L.

A. Garfinkel, M. L. Spano, W. L. Ditto, and J. N. Weiss, “Controlling cardiac chaos,” Science 257, 1230–1235 (1992).
[Crossref] [PubMed]

Staliunas, K.

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and M. G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

K. Staliunas and J. Sánchez-Morcillo, Transverse Patterns (Springer Science & Business Media, 2003).

Swift, J. B.

S. J. Moon, M. D. Shattuck, C. Bizon, D. I. Goldman, J. B. Swift, and H. L. Swinney, “Phase bubbles and spatiotemporal chaos in granular patterns,” Phys. Rev. E 65, 011301 (2001).
[Crossref]

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining the Lyapunov exponents from a time series,” Physica D 16, 285–317 (1985).
[Crossref]

Swinney, H. L.

S. J. Moon, M. D. Shattuck, C. Bizon, D. I. Goldman, J. B. Swift, and H. L. Swinney, “Phase bubbles and spatiotemporal chaos in granular patterns,” Phys. Rev. E 65, 011301 (2001).
[Crossref]

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining the Lyapunov exponents from a time series,” Physica D 16, 285–317 (1985).
[Crossref]

Szwaj, C.

S. Bielawski, C. Szwaj, C. Bruni, D. Garzella, G. L. Orlandi, and M. E. Couprie, “Advection-induced spectrotemporal defects in a free-electron laser,” Phys. Rev. Lett. 95, 034801 (2005).
[Crossref] [PubMed]

Tlidi, M.

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and M. G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

Tredicce, J.

G. Huyet, M. C. Martinoni, J. Tredicce, and S. Rica, “Spatiotemporal dynamics of lasers with a large Fresnel number,” Phys. Rev. Lett. 75, 4027–4030 (1995).
[Crossref] [PubMed]

Tredicce, J. R.

G. Huyet and J. R. Tredicce, “Spatio-temporal chaos in the transverse section of lasers,” Physica D 96, 209–214 (1996).
[Crossref]

Uchida, A.

E. A. Rogers, R. Kalra, R. D. Schroll, A. Uchida, D. P. Lathrop, and R. Roy, “Generalized synchronization of spatiotemporal chaos in a liquid crystal spatial light modulator,” Phys. Rev. Lett. 93, 084101 (2004).
[Crossref] [PubMed]

Vastano, J. A.

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining the Lyapunov exponents from a time series,” Physica D 16, 285–317 (1985).
[Crossref]

Verschueren, N.

N. Verschueren, U. Bortolozzo, M. G. Clerc, and S. Residori, “Chaoticon: localized pattern with permanent dynamics,” Phil. Trans. R. Soc. A. 372, 20140011 (2014).
[Crossref] [PubMed]

N. Verschueren, U. Bortolozzo, M. G. Clerc, and S. Residori, “Spatiotemporal chaotic localized state in liquid crystal light valve experiments with optical feedback,” Phys. Rev. Lett. 110, 104101 (2013).
[Crossref] [PubMed]

M. G. Clerc and N. Verschueren, “Quasiperiodicity route to spatiotemporal chaos in one-dimensional pattern-forming systems,” Phys. Rev. E. 88, 052916 (2013).
[Crossref]

Vladimirov, A. G.

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and M. G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

Weber, A.

W. Decker, W. Pesch, and A. Weber, “Spiral defect chaos in Rayleigh-Benard convection,” Phys. Rev. Lett. 73, 648–651 (1994).
[Crossref] [PubMed]

Weiss, J. N.

A. Garfinkel, M. L. Spano, W. L. Ditto, and J. N. Weiss, “Controlling cardiac chaos,” Science 257, 1230–1235 (1992).
[Crossref] [PubMed]

Wilson, M.

M. G. Clerc, G. Gonzalez-Cortes, and M. Wilson, “Experimental Spatiotemporal Chaotic Textures in a Liquid Crystal Light Valve with Optical Feedback,” in Nonlinear Dynamics: Materials, Theory and Experiments, M. Tlidi and M. G. Clerc, eds. (Springer, 2016).

Wolf, A.

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining the Lyapunov exponents from a time series,” Physica D 16, 285–317 (1985).
[Crossref]

Zhou, S. Q.

S. Q. Zhou and G. Ahlers, “Spatiotemporal chaos in electroconvection of a homeotropically aligned nematic liquid crystal,” Phys. Rev. E 74, 046212 (2006).
[Crossref]

Chaos (1)

M. G. Clerc, C. Falcon, M. A. Garcia-Nustes, V. Odent, and I. Ortega, “Emergence of spatiotemporal dislocation chains in drifting patterns,” Chaos 24, 023133 (2014).
[Crossref] [PubMed]

Europhys. Lett. (1)

P. Coullet and J. Lega, “Defect-mediated turbulence in wave patterns,” Europhys. Lett. 7, 511–516 (1988).
[Crossref]

Nature (1)

Q. Ouyang and J. M. Flesselles, “Transition from spirals to defect turbulence driven by a convective instability,” Nature 379, 143–146 (1996).
[Crossref]

Phil. Trans. R. Soc. A (1)

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and M. G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

Phil. Trans. R. Soc. A. (1)

N. Verschueren, U. Bortolozzo, M. G. Clerc, and S. Residori, “Chaoticon: localized pattern with permanent dynamics,” Phil. Trans. R. Soc. A. 372, 20140011 (2014).
[Crossref] [PubMed]

Phys. Rep. (1)

S. Residori, “Patterns, fronts and structures in a liquid-crystal-light-valve with optical feedback,” Phys. Rep. 416, 201–272 (2005).
[Crossref]

Phys. Rev. E (6)

M. Miranda and J. Burguete, “Experimentally observed route to spatiotemporal chaos in an extended one-dimensional array of convective oscillators,” Phys. Rev. E 79, 046201 (2009).
[Crossref]

P. Brunet and I. Limat, “Defects and spatiotemporal disorder in a pattern of falling liquid columns,” Phys. Rev. E 70, 046207 (2004).
[Crossref]

G. Goren, J. P. Eckmann, and I. Procaccia, “Scenario for the onset of space-time chaos,” Phys. Rev. E 57, 4106–4134 (1998).
[Crossref]

L. Pastur, U. Bortolozzo, and P. L. Ramazza, “Transition to space-time chaos in an optical loop with translational transport,” Phys. Rev. E 69, 016210 (2004).
[Crossref]

S. Q. Zhou and G. Ahlers, “Spatiotemporal chaos in electroconvection of a homeotropically aligned nematic liquid crystal,” Phys. Rev. E 74, 046212 (2006).
[Crossref]

S. J. Moon, M. D. Shattuck, C. Bizon, D. I. Goldman, J. B. Swift, and H. L. Swinney, “Phase bubbles and spatiotemporal chaos in granular patterns,” Phys. Rev. E 65, 011301 (2001).
[Crossref]

Phys. Rev. E. (1)

M. G. Clerc and N. Verschueren, “Quasiperiodicity route to spatiotemporal chaos in one-dimensional pattern-forming systems,” Phys. Rev. E. 88, 052916 (2013).
[Crossref]

Phys. Rev. Lett. (12)

K. E. Daniels and E. Bodenschatz, “Defect turbulence in inclined layer convection,” Phys. Rev. Lett. 88, 034501 (2002).
[Crossref] [PubMed]

E. Louvergneaux, “Pattern-Dislocation-Type Dynamical Instability in 1D Optical Feedback Kerr Media with Gaussian Transverse Pumping,” Phys. Rev. Lett. 87, 244501 (2001).
[Crossref] [PubMed]

S. Bielawski, C. Szwaj, C. Bruni, D. Garzella, G. L. Orlandi, and M. E. Couprie, “Advection-induced spectrotemporal defects in a free-electron laser,” Phys. Rev. Lett. 95, 034801 (2005).
[Crossref] [PubMed]

N. Verschueren, U. Bortolozzo, M. G. Clerc, and S. Residori, “Spatiotemporal chaotic localized state in liquid crystal light valve experiments with optical feedback,” Phys. Rev. Lett. 110, 104101 (2013).
[Crossref] [PubMed]

W. Decker, W. Pesch, and A. Weber, “Spiral defect chaos in Rayleigh-Benard convection,” Phys. Rev. Lett. 73, 648–651 (1994).
[Crossref] [PubMed]

B. Echebarria and H. Riecke, “Defect chaos of oscillating hexagons in rotating convection,” Phys. Rev. Lett. 84, 4838–4841 (2000).
[Crossref] [PubMed]

K. E. Daniels and E. Bodenschatz, “Defect turbulence in inclined layer convection,” Phys. Rev. Lett. 88, 034501 (2002).
[Crossref] [PubMed]

P. Coullet, L. Gil, and J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1989).
[Crossref] [PubMed]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[Crossref] [PubMed]

G. Huyet, M. C. Martinoni, J. Tredicce, and S. Rica, “Spatiotemporal dynamics of lasers with a large Fresnel number,” Phys. Rev. Lett. 75, 4027–4030 (1995).
[Crossref] [PubMed]

A. V. Mamaev and M. Saffman, “Selection of unstable patterns and control of optical turbulence by Fourier plane filtering,” Phys. Rev. Lett. 80, 3499–4502 (1998).
[Crossref]

E. A. Rogers, R. Kalra, R. D. Schroll, A. Uchida, D. P. Lathrop, and R. Roy, “Generalized synchronization of spatiotemporal chaos in a liquid crystal spatial light modulator,” Phys. Rev. Lett. 93, 084101 (2004).
[Crossref] [PubMed]

Physica D (2)

G. Huyet and J. R. Tredicce, “Spatio-temporal chaos in the transverse section of lasers,” Physica D 96, 209–214 (1996).
[Crossref]

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining the Lyapunov exponents from a time series,” Physica D 16, 285–317 (1985).
[Crossref]

Science (1)

A. Garfinkel, M. L. Spano, W. L. Ditto, and J. N. Weiss, “Controlling cardiac chaos,” Science 257, 1230–1235 (1992).
[Crossref] [PubMed]

Other (11)

M. G. Clerc, G. Gonzalez-Cortes, and M. Wilson, “Experimental Spatiotemporal Chaotic Textures in a Liquid Crystal Light Valve with Optical Feedback,” in Nonlinear Dynamics: Materials, Theory and Experiments, M. Tlidi and M. G. Clerc, eds. (Springer, 2016).

P. Manneville, Dissipative Structures and Weak Turbulence (Academic, 1990).

G. Nicolis, Introduction to Nonlinear Science (Cambridge University, 1995).
[Crossref]

Y. Kuramoto, Chemical Oscillations: Waves, and Turbulence (Springer, 1984).
[Crossref]

U. Frisch, Turbulence: the Legacy of A.N. Kolmogorov (Cambridge University, 1995).

K. Staliunas and J. Sánchez-Morcillo, Transverse Patterns (Springer Science & Business Media, 2003).

O. Descalzi, M. G. Clerc, S. Residori, and G. Assanto, eds., Localized States in Physics: Solitons and Patterns (Springer Science & Business Media, 2011).
[Crossref]

G. Nicolis and I. Prigogine, Self-Organization in Non Equilibrium Systems (J. Wiley & Sons, 1977).

G. Nicolis, Introduction to Nonlinear Science (Cambridge University, 1995).
[Crossref]

E. Ott, Chaos in Dynamical Systems, 2nd ed. (Cambridge University, 2002).
[Crossref]

H. Abarbanel, Analysis of Observed Chaotic Data (Springer-Verlag, 1996).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Schematic representation of the experimental setup. LCLV stands for the liquid crystal light valve, L represents the achromatic lenses with a focal distance f = 25 cm, M are the mirrors, FB is an optical fiber bundle, BS stands for the beam splitters, PC represents a polarizing cube, SLM is a spatial light modulator and FP represents the Fourier plane. F and B stand for the forward (incoming) and backward (reflected) beam respectively. d is the equivalent optical length. In the bottom left part of the image, two examples of obtained patterns and textures.
Fig. 2
Fig. 2 Spatial textures in LCLV with optical feedback at different voltage V0. Top panels correspond to spatiotemporal diagrams of observed dynamics working with one-dimensional patterns. Middle panels panels stand for projected spatiotemporal diagrams of two-dimensional textures. a) Periodic regime, b) chaotic behavior, c) quasi-periodic dynamics, and d) a 3-D spatiotemporal diagram of a complex texture found in the intermittent regime at V0 = 4.3V rms. All values were taken during 100 s and are normalized.
Fig. 3
Fig. 3 Fourier spectra for three different dynamical regimes. In gray (blue) the Fourier spectrum of a stationary pattern, in light-gray (red) a quasi-periodic regime, it can be observed the emergence of incommensurable frequencies (f′, f″ and f′″). In black, a broadened spectrum that corresponds to a chaotic texture, presenting an exponential decay of the modes marked by the dashed line.
Fig. 4
Fig. 4 LLE estimation for the LCLV with optical feedback with V0 = 4.70Vrms. a) Projected spatiotemporal diagram of the LCLV. The bottom panels account for spatiotemporal diagram with close initial conditions (lines 1 and 2), b) Intensity profiles of lines 1 and 2, c) A 3-D graph that shows evolution of differences Δ(t, V0) for different applied voltages V0, it can be noted that not always the trajectories are exponentially separated, and d) Temporal evolution of global difference Δ(t), dots stand for experimental data and the continuous curve is the exponential fitting Δ(t) ≈ aebt + cedt with a = 0.0045, b = 8.813, c = 16.57 and d = 0.2773.
Fig. 5
Fig. 5 Bifurcation diagram constructed with the estimated LLE as a function of applied voltage V0 of the LCLV with optical feedback. The diagram is clearly separated in four dynamical regimes, stationary patters before V0 = 3.5Vrms, chaos shadowed in gray (green), intermittency between chaos and quasi-periodicity in dark gray (gray) area and a window of quasi-periodicity shadowed at the end in light gray. The stars correspond to the calculated LLE while the circles show the normalized mean intensity present in the LCLV.

Equations (2)

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I 0 ( x , y ) = { I 0 + b 0 | x | < a 0 , and | y | a 0 b 0 else
λ 0 = lim t lim Δ 0 0 1 t ln [ u ( x , t ) u ( x , t ) u ( x , t o ) u ( x , t o ) ] ,

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