January 2019, 24(1): 273-296. doi: 10.3934/dcdsb.2018095

A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection

Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1874, USA

* Corresponding author: Iuliana Oprea

Received  December 2016 Revised  October 2017 Published  March 2018

Fund Project: Work supported by NSF grant DMS-1615909

In this paper we investigate the transition from periodic solutions to spatiotemporal chaos in a system of four globally coupled Ginzburg Landau equations describing the dynamics of instabilities in the electroconvection of nematic liquid crystals, in the weakly nonlinear regime. If spatial variations are ignored, these equations reduce to the normal form for a Hopf bifurcation with $O(2) × O(2)$ symmetry. Both the amplitude system and the normal form are studied theoretically and numerically for values of the parameters including experimentally measured values of the nematic liquid crystal Merck I52. Coexistence of low dimensional and extensive spatiotemporal chaotic patterns, as well as a temporal period doubling route to spatiotemporal chaos, corresponding to a period doubling cascade towards a chaotic attractor in the normal form, and a kind of spatiotemporal intermittency that is characteristic for anisotropic systems are identified and characterized. A low-dimensional model for the intermittent dynamics is obtained by perturbing the eight-dimensional normal form by imperfection terms that break a continuous translation symmetry.

Citation: Iuliana Oprea, Gerhard Dangelmayr. A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 273-296. doi: 10.3934/dcdsb.2018095
References:
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A. Buka, N. Éber and W. Pesch, Convective patterns in liquid crystals driven by electric field, Electronic-Liquid Crystal Communications, (2005), 1-21, http://www.e-lc.org/tmp/Nandor__Eber_2005_07_12_04_29_54.pdf doi: 10.1007/1-4020-4355-4_02.

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[7]

P. Collet and J. P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhäuser, Boston, 1980.

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C. Crawford and H. Riecke, Oscillon-type structures and their interaction in a Swift-Hohenberg model, Physica D, 129 (1999), 83-92. doi: 10.1016/S0167-2789(98)00280-2.

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M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems, Cambridge University Press, 2009.

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M. Cross and P. C. Hohenberg, Pattern formation outside equilibrium, Rev. Mod. Phys., 65 (1993), 851-1112. doi: 10.1103/RevModPhys.65.851.

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G. Dangelmayr, Complex dynamics near a Hopf bifurcation with symmetry: A parameter study, Dynamical Systems, 26 (2011), 23-60. doi: 10.1080/14689367.2010.498371.

[12]

G. DangelmayrG. AcharyaJ. GleesonI. Oprea and J. Ladd, Diagnosis of spatiotemporal chaos in wave-envelopes of a nematic electroconvection pattern, Phys. Rev. E, 79 (2009), 046215. doi: 10.1103/PhysRevE.79.046215.

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G. Dangelmayr and E. Knobloch, Hopf bifurcation with broken circular symmetry, Nonlinearity, 4 (1991), 399-427. doi: 10.1088/0951-7715/4/2/010.

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G. Dangelmayr and I. Oprea, A bifurcation study of wave patterns for electroconvection in nematic liquid crystals, Mol. Cryst. Liqu. Cryst, 413 (2004), 305-320. doi: 10.1080/15421400490437051.

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G. Dangelmayr and I. Oprea, Modulational stability of travelling waves in 2D anisotropic systems, Journal of Nonlinear Science, 18 (2008), 1-56. doi: 10.1007/s00332-007-9009-3.

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M. DasB. ChakrabartiC. DasguptaS. Ramaswamy and A. K. Sood, Routes to spatiotemporal chaos in the rheology of nematogenic fluids, Phys. Rev. E, 71 (2005), 021707. doi: 10.1103/PhysRevE.71.021707.

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M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Stat. Phys., 19 (1978), 25-52. doi: 10.1007/BF01020332.

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M. Golubitsky, I. Stewart and D. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol Ⅱ, Springer Verlag, 1988.

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G. D. Granzow and H. Riecke, Ordered and disordered defect chaos, Physica A, 249 (1998), 27-35. doi: 10.1016/S0378-4371(97)00428-7.

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F. A. HopfD. L. KaplanH. M. Gibbs and R. L. Shoemaker, Bifurcations to chaos in optical bistability, Phys. Rev. A (3), 25 (1982), 2172-2182. doi: 10.1103/PhysRevA.25.2172.

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K. Kaneko, Spatio-temporal chaos in one and two-dimensional coupled map lattices, Physica D, 37 (1989), 60-82. doi: 10.1016/0167-2789(89)90117-6.

[29]

R. Kapral, Pattern formation in two-dimensional arrays of coupled, discrete-time oscillators, Phys. Rev. A, 31 (1985), 3868-3679. doi: 10.1103/PhysRevA.31.3868.

[30]

H. KookP. H. Ling and G. Schmidt, Universal behavior of coupled nonlinear systems, Phys. Rev. A, 43 (1991), 2700-2708. doi: 10.1103/PhysRevA.43.2700.

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L. KramerE. Ben-JacobH. Brand and M. C. Cross, Wavelength selection in systems far from equilibrium, Phys. Rev. Lett., 49 (1982), 1891-1894. doi: 10.1103/PhysRevLett.49.1891.

[32]

L. Kramer and W. Pesch, Convection instabilities in nematic liquid crystals, Annual Review of Fluid Mechanics, Annual Reviews, Palo Alto, CA, 27 (1995), 515-541. doi: 10.1002/9783527609284.ch13.

[33]

J. Lega, Phase diffusion and weak turbulence, in: G. Dangelmayr, I. Oprea (Eds. ), Dynamics and Bifurcation of Patterns in Dissipative Systems, World Scientific Series on Nonlinear Sciences, Series B, World Scientific Publisher, 12 (2004), 143-157.

[34]

A. Libchaber and J. Maurer, A Rayleight-Bénard experiment: Helium in a small box, in: T. Riste (Ed.), Nonlinear Phenomena at Phase Transitions and Instabilities, Plenum Press, New York, (1981), 259-286.

[35]

A. Libchaber, C. Laroche and S. Fauve, Period doubling in mercury, a quantitative measurement, J. Physique Lett. , 43 (1982), L211.

[36]

A. Libchaber, Experimental aspects of the period doubling scenario, in: L. Garrido (Ed.), Dynamical Systems and Chaos, Springer Lecture Notes in Physics, Springer, Berlin, Heidelberg, 179 (1983), 157-164. doi: 10.1007/3-540-12276-1_11.

[37]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20 (1963), 130-141.

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E. N. Lorenz, Noisy periodicity and reverse bifurcation, in: R. H. G. Helleman (Ed.): Nonlinear Dynamics, Annals of the New York Academy of Sciences, 357 (1979), 282-291.

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R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-67.

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F. MeloP. Umbanhowar and H. L. Swinney, Hexagons, kinks, and disorder in oscillated granular layers, Phys. Rev. Lett., 75 (1995), 3838-3841. doi: 10.1103/PhysRevLett.75.3838.

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S. W. MorrisE. BodenschatzD. S. Cannel and G. Ahlers, Spiral defect chaos in large aspect ratio Rayleigh Bénard convection, Phys. Rev. Lett., 71 (1993), 2026-2029. doi: 10.1103/PhysRevLett.71.2026.

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A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Wiley, New York, 1979.

[43]

I. Oprea and G. Dangelmayr, Dynamics and bifurcations in the weak electrolyte model for electroconvection of nematic liquid crystals: A Ginzburg Landau approach, European Jr. of Mechanics, B. Fluids, 27 (2008), 726-749. doi: 10.1016/j.euromechflu.2007.12.004.

[44]

I. Oprea, I. Triandaf, G. Dangelmayr and I. B. Schwartz, Quantitative and qualitative characterization of zigzag spatiotemporal chaos in a system of amplitude equations for nematic electroconvection, Chaos, 17 (2007), 023101, 12pp.

[45]

E. Ott, Chaos in Dynamical Systems, Cambridge University Press, 2002.

[46]

H. G. Paap and H. Riecke, Drifting vortices in ramped Taylor vortex flow: Quantitative results from phase equations, Physics of Fluids, 3 (1991), 1519-1532. doi: 10.1063/1.857987.

[47]

H. Riecke and G. D. Granzow, Double phase slips and bound defect pairs in parametrically driven waves, Proceedings of the 15th Symposium on Energy Engineering Sciences (chao-dyn/9707013,1997).

[48]

H. Riecke and H. G. Paap, Perfect wave-number selection and drifting patterns in ramped Taylor vortex flow, Phys. Rev. Lett., 59 (1987), 2570-2573. doi: 10.1103/PhysRevLett.59.2570.

[49]

H. Riecke and H. G. Paap, Spatio-temporal chaos through ramp-induced Eckhaus instability, Europhy. Lett., 59 (1991), 433-438. doi: 10.1209/0295-5075/14/5/008.

[50]

H. G. Schuster and W. Just, Deterministic Chaos, An Introduction, Wiley, 2005.

[51]

M. SilberH. Riecke and L. Kramer, Symmetry breaking Hopf bifurcation in anisotropic systems, Physica D, 61 (1990), 260-277. doi: 10.1016/0167-2789(92)90170-R.

[52]

M. M. SkoricM. S. Jovanovic and M. R. Rajkovic, Spatiotemporal intermittency and chaos in stimulated Raman backscattering, Europhys. Lett., 34 (1996), 19-24.

[53]

C. W. SmithM. J. Tejwanis and D. A. Farris, Bifurcation universality for first-sound subharmonic generation in superfluid Helium-4, Phys. Rev. Lett., 48 (1982), 492-494. doi: 10.1103/PhysRevLett.48.492.

[54]

C. Sparrow, The Lorenz Equations: Bifurcation, Chaos, and Strange Attractors, Springer, New York, Heidelberg, Berlin, 1982.

[55]

D. Stassinopoulos and P. Alstrom, Coupled maps: An approach to spatiotemporal chaos, Phys. Rev. A, 45 (1992), 675-691. doi: 10.1103/PhysRevA.45.675.

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J. TestaJ. Peréz and C. Jeffries, Evidence for universal chaotic behavior of a driven nonlinear oscillator, Phys. Rev. Lett., 48 (1982), 714-717. doi: 10.1103/PhysRevLett.48.714.

[57]

M. Treiber and L. Kramer, Bipolar electrodiffusion model for electroconvection in nematics, Mol. Cryst. Liqu. Cryst., 261 (1995), 311-326.

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L. S. Tsimring and I. S. Aranson, Localized and cellular patterns in a vibrated granular layer, Phys. Rev. Lett., 79 (1995), p213.

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P. UmbanhowarF. Melo and H. L. Swinney, Localized excitations in a vertically vibrated granular layer, Nature (London), 382 (1996), 793-796. doi: 10.1038/382793a0.

[60]

S. Venkataramani and E. Ott, Spatiotemporal bifurcation phenomena with temporal period doubling: Patterns in vibrating sand, Phys. Rev. Lett., 80 (1998), 3495-3498. doi: 10.1103/PhysRevLett.80.3495.

[61]

M. Wegelin, Nichtlineare Dynamik Raumzeitlicher Muster in Hierarchischen Systemen, PhD Dissertation, Department of Physics, Tubingen, 1993.

[62]

R. J. WienerG. L. SnyderM. C. PrangeD. Frediani and P. R. Diaz, Period-doubling cascade to chaotic phase dynamics in Taylor vortex flow with hourglass geometry, Phys. Rev. E, 55 (1997), 5489-5507. doi: 10.1103/PhysRevE.55.5489.

[63]

W. J. Yeh and Y. H. Kao, Universal scaling and chaotic behavior of Josephson-junction analog, Phys. Rev. Lett., 49 (1982), 1888-1891.

[64]

Y. Zou, G. Dangelmayr and I. Oprea. Intermittency and chaos near Hopf bifurcation with broken O(2) X O(2) symmetry, Int. J. Bifurcation and Chaos, 23 (2013), 1350139, 19pp.

show all references

References:
[1]

P. Alstrom and D. Stassinopoulos, Space-time renormalization at the onset of spatio-temporal chaos in coupled maps, Chaos, 2 (1992), 301-306. doi: 10.1063/1.165872.

[2]

I. S. Aranson and L. S. Tsimring, Patterns and collective behavior in granular media: Theoretical concepts, Rev. Mod. Phys., 68 (2006), 641-692.

[3]

J. ArgyrisG. Faust and M. Haase, Routes to chaos and turbulence. A computational introduction, Phil. Trans., R. Soc. Lond. A, 344 (1993), 207-234. doi: 10.1098/rsta.1993.0088.

[4]

P. AshwinE. Covas and R. Tavakol, Transverse instability for non-normal parameters, Nonlinearity, 12 (1999), 563-577. doi: 10.1088/0951-7715/12/3/009.

[5]

A. Buka, N. Éber and W. Pesch, Convective patterns in liquid crystals driven by electric field, Electronic-Liquid Crystal Communications, (2005), 1-21, http://www.e-lc.org/tmp/Nandor__Eber_2005_07_12_04_29_54.pdf doi: 10.1007/1-4020-4355-4_02.

[6]

H. Chaté and P. Manneville, Transition to turbulence via spatiotemporal intermittency, Phys. Rev. Lett., 58 (1987), 112-115.

[7]

P. Collet and J. P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhäuser, Boston, 1980.

[8]

C. Crawford and H. Riecke, Oscillon-type structures and their interaction in a Swift-Hohenberg model, Physica D, 129 (1999), 83-92. doi: 10.1016/S0167-2789(98)00280-2.

[9]

M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems, Cambridge University Press, 2009.

[10]

M. Cross and P. C. Hohenberg, Pattern formation outside equilibrium, Rev. Mod. Phys., 65 (1993), 851-1112. doi: 10.1103/RevModPhys.65.851.

[11]

G. Dangelmayr, Complex dynamics near a Hopf bifurcation with symmetry: A parameter study, Dynamical Systems, 26 (2011), 23-60. doi: 10.1080/14689367.2010.498371.

[12]

G. DangelmayrG. AcharyaJ. GleesonI. Oprea and J. Ladd, Diagnosis of spatiotemporal chaos in wave-envelopes of a nematic electroconvection pattern, Phys. Rev. E, 79 (2009), 046215. doi: 10.1103/PhysRevE.79.046215.

[13]

G. Dangelmayr, B. Fiedler, K. Kirchgässner and A. Mielke, Dynamics of Nonlinear Waves in Dissipative Systems: Reduction, Bifurcation and Stability, Addison Wesley Longman Ltd., 1996.

[14]

G. Dangelmayr and E. Knobloch, Hopf bifurcation with broken circular symmetry, Nonlinearity, 4 (1991), 399-427. doi: 10.1088/0951-7715/4/2/010.

[15]

G. Dangelmayr and I. Oprea, A bifurcation study of wave patterns for electroconvection in nematic liquid crystals, Mol. Cryst. Liqu. Cryst, 413 (2004), 305-320. doi: 10.1080/15421400490437051.

[16]

G. Dangelmayr and I. Oprea, Modulational stability of travelling waves in 2D anisotropic systems, Journal of Nonlinear Science, 18 (2008), 1-56. doi: 10.1007/s00332-007-9009-3.

[17]

G. Dangelmayr and M. Wegelin, Hopf bifurcations in anisotropic systems, in: Golubitsky, M., Luss, D., Strogatz, S. (Eds.), Pattern Formation in Continuous and Coupled Systems, IMA Vol. Math. Appl. , 115 (1999), 33-47.

[18]

K. E. DanielsO. BrauschW. Pesch and E. Bodenschatz, Competition and bistability of ordered undulations and undulation chaos in inclined layer convection, J. Fluid Mech., 597 (2008), 261-282.

[19]

M. DasB. ChakrabartiC. DasguptaS. Ramaswamy and A. K. Sood, Routes to spatiotemporal chaos in the rheology of nematogenic fluids, Phys. Rev. E, 71 (2005), 021707. doi: 10.1103/PhysRevE.71.021707.

[20]

M. DenninG. Ahlers and D. S. Cannel, Spatiotemporal chaos in electroconvection, Science, 272 (1996), 388-390. doi: 10.1126/science.272.5260.388.

[21]

W. S. Edwards and S. Fauve, Patterns and quasipatterns in the Faraday experiment, J. Fluid Mech., 278 (1994), 123-148. doi: 10.1017/S0022112094003642.

[22]

D. A. EgolfI. MelnikovW. Pesch and R. Ecke, Mechanisms of extensive spatiotemporal chaos in Rayleigh Bénard convection, Nature, 404 (200), p733. doi: 10.1038/35008013.

[23]

M. Faraday, On a peculiar class of acoustical figures; and on certain forms assumed by a group of particles upon vibrating elastic surfaces, Phil. Trans. Roy. Soc., 121 (1831), 299-318.

[24]

M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Stat. Phys., 19 (1978), 25-52. doi: 10.1007/BF01020332.

[25]

M. Golubitsky, I. Stewart and D. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol Ⅱ, Springer Verlag, 1988.

[26]

G. D. Granzow and H. Riecke, Ordered and disordered defect chaos, Physica A, 249 (1998), 27-35. doi: 10.1016/S0378-4371(97)00428-7.

[27]

F. A. HopfD. L. KaplanH. M. Gibbs and R. L. Shoemaker, Bifurcations to chaos in optical bistability, Phys. Rev. A (3), 25 (1982), 2172-2182. doi: 10.1103/PhysRevA.25.2172.

[28]

K. Kaneko, Spatio-temporal chaos in one and two-dimensional coupled map lattices, Physica D, 37 (1989), 60-82. doi: 10.1016/0167-2789(89)90117-6.

[29]

R. Kapral, Pattern formation in two-dimensional arrays of coupled, discrete-time oscillators, Phys. Rev. A, 31 (1985), 3868-3679. doi: 10.1103/PhysRevA.31.3868.

[30]

H. KookP. H. Ling and G. Schmidt, Universal behavior of coupled nonlinear systems, Phys. Rev. A, 43 (1991), 2700-2708. doi: 10.1103/PhysRevA.43.2700.

[31]

L. KramerE. Ben-JacobH. Brand and M. C. Cross, Wavelength selection in systems far from equilibrium, Phys. Rev. Lett., 49 (1982), 1891-1894. doi: 10.1103/PhysRevLett.49.1891.

[32]

L. Kramer and W. Pesch, Convection instabilities in nematic liquid crystals, Annual Review of Fluid Mechanics, Annual Reviews, Palo Alto, CA, 27 (1995), 515-541. doi: 10.1002/9783527609284.ch13.

[33]

J. Lega, Phase diffusion and weak turbulence, in: G. Dangelmayr, I. Oprea (Eds. ), Dynamics and Bifurcation of Patterns in Dissipative Systems, World Scientific Series on Nonlinear Sciences, Series B, World Scientific Publisher, 12 (2004), 143-157.

[34]

A. Libchaber and J. Maurer, A Rayleight-Bénard experiment: Helium in a small box, in: T. Riste (Ed.), Nonlinear Phenomena at Phase Transitions and Instabilities, Plenum Press, New York, (1981), 259-286.

[35]

A. Libchaber, C. Laroche and S. Fauve, Period doubling in mercury, a quantitative measurement, J. Physique Lett. , 43 (1982), L211.

[36]

A. Libchaber, Experimental aspects of the period doubling scenario, in: L. Garrido (Ed.), Dynamical Systems and Chaos, Springer Lecture Notes in Physics, Springer, Berlin, Heidelberg, 179 (1983), 157-164. doi: 10.1007/3-540-12276-1_11.

[37]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20 (1963), 130-141.

[38]

E. N. Lorenz, Noisy periodicity and reverse bifurcation, in: R. H. G. Helleman (Ed.): Nonlinear Dynamics, Annals of the New York Academy of Sciences, 357 (1979), 282-291.

[39]

R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-67.

[40]

F. MeloP. Umbanhowar and H. L. Swinney, Hexagons, kinks, and disorder in oscillated granular layers, Phys. Rev. Lett., 75 (1995), 3838-3841. doi: 10.1103/PhysRevLett.75.3838.

[41]

S. W. MorrisE. BodenschatzD. S. Cannel and G. Ahlers, Spiral defect chaos in large aspect ratio Rayleigh Bénard convection, Phys. Rev. Lett., 71 (1993), 2026-2029. doi: 10.1103/PhysRevLett.71.2026.

[42]

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Wiley, New York, 1979.

[43]

I. Oprea and G. Dangelmayr, Dynamics and bifurcations in the weak electrolyte model for electroconvection of nematic liquid crystals: A Ginzburg Landau approach, European Jr. of Mechanics, B. Fluids, 27 (2008), 726-749. doi: 10.1016/j.euromechflu.2007.12.004.

[44]

I. Oprea, I. Triandaf, G. Dangelmayr and I. B. Schwartz, Quantitative and qualitative characterization of zigzag spatiotemporal chaos in a system of amplitude equations for nematic electroconvection, Chaos, 17 (2007), 023101, 12pp.

[45]

E. Ott, Chaos in Dynamical Systems, Cambridge University Press, 2002.

[46]

H. G. Paap and H. Riecke, Drifting vortices in ramped Taylor vortex flow: Quantitative results from phase equations, Physics of Fluids, 3 (1991), 1519-1532. doi: 10.1063/1.857987.

[47]

H. Riecke and G. D. Granzow, Double phase slips and bound defect pairs in parametrically driven waves, Proceedings of the 15th Symposium on Energy Engineering Sciences (chao-dyn/9707013,1997).

[48]

H. Riecke and H. G. Paap, Perfect wave-number selection and drifting patterns in ramped Taylor vortex flow, Phys. Rev. Lett., 59 (1987), 2570-2573. doi: 10.1103/PhysRevLett.59.2570.

[49]

H. Riecke and H. G. Paap, Spatio-temporal chaos through ramp-induced Eckhaus instability, Europhy. Lett., 59 (1991), 433-438. doi: 10.1209/0295-5075/14/5/008.

[50]

H. G. Schuster and W. Just, Deterministic Chaos, An Introduction, Wiley, 2005.

[51]

M. SilberH. Riecke and L. Kramer, Symmetry breaking Hopf bifurcation in anisotropic systems, Physica D, 61 (1990), 260-277. doi: 10.1016/0167-2789(92)90170-R.

[52]

M. M. SkoricM. S. Jovanovic and M. R. Rajkovic, Spatiotemporal intermittency and chaos in stimulated Raman backscattering, Europhys. Lett., 34 (1996), 19-24.

[53]

C. W. SmithM. J. Tejwanis and D. A. Farris, Bifurcation universality for first-sound subharmonic generation in superfluid Helium-4, Phys. Rev. Lett., 48 (1982), 492-494. doi: 10.1103/PhysRevLett.48.492.

[54]

C. Sparrow, The Lorenz Equations: Bifurcation, Chaos, and Strange Attractors, Springer, New York, Heidelberg, Berlin, 1982.

[55]

D. Stassinopoulos and P. Alstrom, Coupled maps: An approach to spatiotemporal chaos, Phys. Rev. A, 45 (1992), 675-691. doi: 10.1103/PhysRevA.45.675.

[56]

J. TestaJ. Peréz and C. Jeffries, Evidence for universal chaotic behavior of a driven nonlinear oscillator, Phys. Rev. Lett., 48 (1982), 714-717. doi: 10.1103/PhysRevLett.48.714.

[57]

M. Treiber and L. Kramer, Bipolar electrodiffusion model for electroconvection in nematics, Mol. Cryst. Liqu. Cryst., 261 (1995), 311-326.

[58]

L. S. Tsimring and I. S. Aranson, Localized and cellular patterns in a vibrated granular layer, Phys. Rev. Lett., 79 (1995), p213.

[59]

P. UmbanhowarF. Melo and H. L. Swinney, Localized excitations in a vertically vibrated granular layer, Nature (London), 382 (1996), 793-796. doi: 10.1038/382793a0.

[60]

S. Venkataramani and E. Ott, Spatiotemporal bifurcation phenomena with temporal period doubling: Patterns in vibrating sand, Phys. Rev. Lett., 80 (1998), 3495-3498. doi: 10.1103/PhysRevLett.80.3495.

[61]

M. Wegelin, Nichtlineare Dynamik Raumzeitlicher Muster in Hierarchischen Systemen, PhD Dissertation, Department of Physics, Tubingen, 1993.

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Figure 1.  Time series $|A_j(T)|$, $1\leq j\leq 4$, showing the period doubling sequence, obtained from numerical simulations of (10) for (a) $\alpha = 0.024$, (b) $\alpha = 0.025$, (c) $\alpha = 0.02515$, and (d) $\alpha = 0.0252$. In (c) the upper time series in the two plots are for $|A_1|$, $|A_4|$, and the lower time series for $|A_2|$, $|A_3|$.
Figure 2.  Phase space plots of the quasiperiodic solutions and the chaotic attractor, for (a) $\alpha = 0.025$, (b) $\alpha = 0.025025$, and (c) $\alpha = 0.02515$ (see text). Left plots: $|A_4|$ versus $|A_1|$, right plots: $|A_3|$ versus $|A_2|$.
Figure 3.  Pattern snapshot $U$, equation (9), for Simulation 1.
Figure 4.  Time series and phase space plots of dominant and secondary mode amplitudes plots for Simulation 1.
Figure 5.  Averages of the mode amplitudes $|a_j(m,n,T)|$ for Simulation 2. (a): Time averages. (b) Time series of the $(m,n)$-averages for $j = 2,3$.
Figure 6.  Snapshots of (a): $|A_j|$ ($1\leq j \leq 4$) and (b): pattern $U$, equation (9), for Simulation 2.
Figure 7.  Time series of dominant and secondary Ginzburg Landau mode amplitudes for Simulation 3.
Figure 8.  Time series of the amplitudes $|z_j(T)|$, equation (14).
Figure 9.  Real parts of snapshots of simulations of the CML (15) after transients have died out for $\lambda = 2.3$, $c = 0.25$, $d = 0.05$, $\alpha = -\beta = 0.5$ and $M = N = 64$. Initial condition for (a) is a small random perturbation of a uniform state, while for (b) it is fully random.
Table 1.  The six basic periodic solutions of (10).
Name         (Shortcut) (A1, A2, A3, A4) Pattern
Travelling Waves       (TW) (A, 0, 0, 0) |A|cos$(\omega t+p_cx+q_cy)$
Travelling Rectangles-$x$ (TR$_x$) (A, 0, 0, A) 2|A|cos$(\omega t+p_cx)\cos(q_cy)$
Travelling Rectangles-$y$(TR$_y$) (A, A, 0, 0) 2|A|cos $(\omega t+q_cy)\cos(p_cx)$
Standing Waves       (SW) (A, 0, A, 0) 2|A|cos$(\omega t)\cos(p_cx+q_cy)$
Standing Rectangles     (SR) (A, A, A, A) 4|A|cos$(\omega t)\cos(p_cx)\cos(q_cy)$
Alternating Waves     (AW) (A, iA, A, iA) 2|A|$[\cos(\omega t)\cos(p_cx+q_cy)$
   $-\sin(\omega t)\cos(p_cx-q_cy)]$
Name         (Shortcut) (A1, A2, A3, A4) Pattern
Travelling Waves       (TW) (A, 0, 0, 0) |A|cos$(\omega t+p_cx+q_cy)$
Travelling Rectangles-$x$ (TR$_x$) (A, 0, 0, A) 2|A|cos$(\omega t+p_cx)\cos(q_cy)$
Travelling Rectangles-$y$(TR$_y$) (A, A, 0, 0) 2|A|cos $(\omega t+q_cy)\cos(p_cx)$
Standing Waves       (SW) (A, 0, A, 0) 2|A|cos$(\omega t)\cos(p_cx+q_cy)$
Standing Rectangles     (SR) (A, A, A, A) 4|A|cos$(\omega t)\cos(p_cx)\cos(q_cy)$
Alternating Waves     (AW) (A, iA, A, iA) 2|A|$[\cos(\omega t)\cos(p_cx+q_cy)$
   $-\sin(\omega t)\cos(p_cx-q_cy)]$
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