Spatial heteroclinic bifurcation of time periodic solutions to the Ginzburg-Landau equation
Authors:
H. Dang-Vu and C. Delcarte
Journal:
Quart. Appl. Math. 58 (2000), 459-472
MSC:
Primary 35Q55; Secondary 34C37, 35B10, 37D45, 37L10
DOI:
https://doi.org/10.1090/qam/1770649
MathSciNet review:
MR1770649
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Abstract: In this paper we study the spatial structure of the time periodic solutions to the Ginzburg-Landau equation in various configurations (supercritical and subcritical). We show that spatially periodic bursting solutions or spatially heteroclinic solutions can occur depending upon the values of the coefficients. As a consequence of this study, we obtain an exact solution to the nonlinear system of Kapitula and Maier-Paape [16]. We then show that near a spatially heteroclinic solution there is an extremely intricate complex of spatially unstable solutions and KAM surfaces.
- V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. MR 0474391
G. D. Birkhoff, Nouvelles recherches sur les systèmes dynamiques, Mem. Pont. Acad. Sci. Novi Lyncei 1, 85-216 (1935)
- J. Bricmont and A. Kupiainen, Stability of moving fronts in the Ginzburg-Landau equation, Comm. Math. Phys. 159 (1994), no. 2, 287–318. MR 1256990
- F. Cariello and M. Tabor, Painlevé expansions for nonintegrable evolution equations, Phys. D 39 (1989), no. 1, 77–94. MR 1021183, DOI https://doi.org/10.1016/0167-2789%2889%2990040-7
- H. Dang-Vu and C. Delcarte, Bifurcations and chaos in the steady-state solutions of the complex Ginzburg-Landau equation, J. Tech. Phys. 37 (1996), no. 3-4, 317–325. International Conference on Nonlinear Dynamics, Chaotic and Complex Systems (Zakopane, 1995). MR 1443466
- Arjen Doelman, Slow time-periodic solutions of the Ginzburg-Landau equation, Phys. D 40 (1989), no. 2, 156–172. MR 1029461, DOI https://doi.org/10.1016/0167-2789%2889%2990060-2
- Arjen Doelman, Finite-dimensional models of the Ginzburg-Landau equation, Nonlinearity 4 (1991), no. 2, 231–250. MR 1107006
- A. Doelman, Traveling waves in the complex Ginzburg-Landau equation, J. Nonlinear Sci. 3 (1993), no. 2, 225–266. MR 1220175, DOI https://doi.org/10.1007/BF02429865
- Arjen Doelman, Breaking the hidden symmetry in the Ginzburg-Landau equation, Phys. D 97 (1996), no. 4, 398–428. MR 1412551, DOI https://doi.org/10.1016/0167-2789%2895%2900303-7
- Charles R. Doering, John D. Gibbon, Darryl D. Holm, and Basil Nicolaenko, Low-dimensional behaviour in the complex Ginzburg-Landau equation, Nonlinearity 1 (1988), no. 2, 279–309. MR 937004
- W. Eckhaus, The Ginzburg-Landau manifold is an attractor, J. Nonlinear Sci. 3 (1993), no. 3, 329–348. MR 1237095, DOI https://doi.org/10.1007/BF02429869
- J.-P. Eckmann and Th. Gallay, Front solutions for the Ginzburg-Landau equation, Comm. Math. Phys. 152 (1993), no. 2, 221–248. MR 1210167
- Michel Hénon and Carl Heiles, The applicability of the third integral of motion: Some numerical experiments, Astronom. J. 69 (1964), 73–79. MR 158746, DOI https://doi.org/10.1086/109234
- L. M. Hocking and K. Stewartson, On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance, Proc. Roy. Soc. London Ser. A 326 (1972), 289–313. MR 329419, DOI https://doi.org/10.1098/rspa.1972.0010
- Philip Holmes, Spatial structure of time-periodic solutions of the Ginzburg-Landau equation, Phys. D 23 (1986), no. 1-3, 84–90. Spatio-temporal coherence and chaos in physical systems (Los Alamos, N.M., 1986). MR 876910, DOI https://doi.org/10.1016/0167-2789%2886%2990114-4
- Todd Kapitula and Stanislaus Maier-Paape, Spatial dynamics of time periodic solutions for the Ginzburg-Landau equation, Z. Angew. Math. Phys. 47 (1996), no. 2, 265–305. MR 1385917, DOI https://doi.org/10.1007/BF00916827
- Laurence R. Keefe, Dynamics of perturbed wavetrain solutions to the Ginzburg-Landau equation, Stud. Appl. Math. 73 (1985), no. 2, 91–153. MR 804366, DOI https://doi.org/10.1002/sapm198573291
- Michael J. Landman, Solutions of the Ginzburg-Landau equation of interest in shear flow transition, Stud. Appl. Math. 76 (1987), no. 3, 187–237. MR 1039301, DOI https://doi.org/10.1002/sapm1987763187
- B. J. Matkowsky and Vl. A. Vol′pert, Stability of plane wave solutions of complex Ginzburg-Landau equations, Quart. Appl. Math. 51 (1993), no. 2, 265–281. MR 1218368, DOI https://doi.org/10.1090/qam/1218368
- H. T. Moon, P. Huerre, and L. G. Redekopp, Transitions to chaos in the Ginzburg-Landau equation, Phys. D 7 (1983), no. 1-3, 135–150. Order in chaos (Los Alamos, N.M., 1982). MR 719050, DOI https://doi.org/10.1016/0167-2789%2883%2990124-0
- Paul K. Newton and Lawrence Sirovich, Instabilities of the Ginzburg-Landau equation: periodic solutions, Quart. Appl. Math. 44 (1986), no. 1, 49–58. MR 840442, DOI https://doi.org/10.1090/S0033-569X-1986-0840442-X
- Kazuhiro Nozaki and Naoaki Bekki, Exact solutions of the generalized Ginzburg-Landau equation, J. Phys. Soc. Japan 53 (1984), no. 5, 1581–1582. MR 750609, DOI https://doi.org/10.1143/JPSJ.53.1581
- B. I. Shraiman, A. Pumir, W. van Saarloos, P. C. Hohenberg, H. Chaté, and M. Holen, Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation, Phys. D 57 (1992), no. 3-4, 241–248. MR 1182049, DOI https://doi.org/10.1016/0167-2789%2892%2990001-4
- Lawrence Sirovich and Paul K. Newton, Periodic solutions of the Ginzburg-Landau equation, Phys. D 21 (1986), no. 1, 115–125. MR 860011, DOI https://doi.org/10.1016/0167-2789%2886%2990082-5
- P. Takáč, Invariant $2$-tori in the time-dependent Ginzburg-Landau equation, Nonlinearity 5 (1992), no. 2, 289–321. MR 1158376
- Wim van Saarloos and P. C. Hohenberg, Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations, Phys. D 56 (1992), no. 4, 303–367. MR 1169610, DOI https://doi.org/10.1016/0167-2789%2892%2990175-M
- Yi Song Yang, Global spatially periodic solutions to the Ginzburg-Landau equation, Proc. Roy. Soc. Edinburgh Sect. A 110 (1988), no. 3-4, 263–273. MR 974742, DOI https://doi.org/10.1017/S0308210500022253
V. I. Arnold, Mathematical Methods in Classical Mechanics, Springer-Verlag, New York, 1978
G. D. Birkhoff, Nouvelles recherches sur les systèmes dynamiques, Mem. Pont. Acad. Sci. Novi Lyncei 1, 85-216 (1935)
J. Bricmont and A. Kupiainen, Stability of moving fronts in the Ginzburg-Landau equation, Comm. Math. Phys. 159, 287-318 (1994)
F. Cariello and M. Tabor, Painlevé expansions for nonintegrable evolution equations, Physica D 39, 77-94 (1989)
H. Dang-Vu and C. Delcarte, Bifurcations and chaos in the steady-state solutions of the Ginzburg-Landau equation in: “International Conference on Nonlinear Dynamics, Chaotic and Complex Systems", Zakopane, Poland, 7-12 November, 1995, Polish Scientific Publishers PWN, Warszawa, 1996, pp. 317-325
A. Doelman, Slow time-periodic solutions of the Ginzburg-Landau equation, Physica D 40, 156-172 (1989)
A. Doelman, Finite-dimensional models of the Ginzburg-Landau equation, Nonlinearity 4, 231-250 (1991)
A. Doelman, Traveling waves in the complex Ginzburg-Landau equation, J. Nonlinear Sci., 3, 225-266 (1993)
A. Doelman, Breaking the hidden symmetry in the Ginzburg-Landau equation, Physica D 97, 398-428 (1996)
C. R. Doering, J. D. Gibbon, D. D. Holm, and B. Nicolaenko, Low dimensional behaviour in the complex Ginzburg-Landau equations, Nonlinearity 1, 279-309 (1988)
W. Eckhaus, The Ginzburg-Landau manifold is an attractor, J. Nonlinear Sci. 3, 329-348 (1993)
J.-P. Eckmann and Th. Gallay, Front solutions for the Ginzburg-Landau equation, Comm. Math. Phys. 152, 221-248 (1993)
M. Hénon, The applicability of the third integral of motion: Some numerical experiments, Astrophys. J. 69, 73-79 (1964)
L. M. Hocking and K. Stewartson, On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance, Proc. Roy. Soc. London 326A, 289-313 (1972)
Ph. Holmes, Spatial structure of time-periodic solutions of the Ginzburg-Landau equation, Physica D 23, 84-90 (1986)
T. Kapitula and S. Maier-Paape, Spatial dynamics of time periodic solutions for the Ginzburg-Landau equation, Z. angew. Math. Phys. (ZAMP) 47, 265-305 (1996)
L. R. Keefe, Dynamics of perturbed wavetrain solutions to Ginzburg-Landau equation, Stud. Appl. Math. 73, 91-153 (1985)
M. Landman, Solutions of the Ginzburg-Landau equation of interest in shear flow transition, Stud. Appl. Math. 76, 187-237 (1987)
B. J. Matkowsky and V. Volpert, Stability of plane wave solutions of complex Ginzburg-Landau equations, Quart. Appl. Math. 51, 265-281 (1993)
H. T. Moon, P. Huerre, and L. G. Redekopp, Transitions to chaos in the Ginzburg-Landau equation, Physica D 7, 135-152 (1983)
P. K. Newton and L. Sirovich, Instabilities of the Ginzburg-Landau equation: Periodic solutions, Quart. Appl. Math. 44, 49-58 (1986)
K. Nozaki and N. Bekki, Exact solutions of the generalized Ginzburg-Landau equation, J. Phys. Soc. Jap. 53, 1581-1582 (1984)
B. I. Shraiman, A. Pumir, W. van Saarloos, P. C. Hohenberg, H. Chaté, and M. Holen, Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation, Physica D 57, 241-248 (1992)
L. Sirovich and P. K. Newton, Periodic solutions of the Ginzburg-Landau equation, Physica D 21, 115-125 (1986)
P. Takáč, Invariant 2-tori in the time-dependent Ginzburg-Landau equation, Nonlinearity 5, 289-321 (1992)
W. van Saarloos and P. C. Hohenberg, Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations, Physica D 56, 303-367 (1992)
Y. Yang, Global spatially periodic solutions to the Ginzburg-Landau equation, Proc. Roy. Soc. Edinburgh A 110, 263-273 (1988)
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