Relative periodic solutions of the complex ginzburg-landau equation

SIAM Journal on Applied Dynamical Systems

Volumn 4, Issue 4, 2005, Pages 1042-1075

  López, Vanessa ^a,b   Boyland, Philip ^c   Heath, Michael T ^a   Moser, Robert D ^a,d  

^a University of Illinois at Urbana Champaign   (United States)

^b LAWRENCE BERKELEY NATIONAL LABORATORY   (United States)

^c UNIVERSITY OF FLORIDA   (United States)

^d The University of Texas at Austin   (United States)

Author keywords

Chaotic pattern dynamics; Ginzburg Landau equation; Relative periodic solutions; Spectral Galerkin method

Indexed keywords

CHAOTIC PATTERN DYNAMICS; GINZBURG-LANDAU EQUATION; RELATIVE PERIODIC SOLUTIONS; SPECTRAL-GALERKIN METHOD;

CHAOS THEORY; GALERKIN METHODS; LYAPUNOV METHODS; MATHEMATICAL TRANSFORMATIONS; NONLINEAR SYSTEMS; ORBITS;

DIFFERENTIAL EQUATIONS;

EID: 33644912649     PISSN: 15360040     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/040618977     Document Type: Article

References (62)

1. I. S. ARANSON AND L. KRAMER, The world of the complex Ginzburg-Landau equation, Rev. Modern Phys., 74 (2002), pp. 99-143.
2. P. J. ASTON AND M. DELLNITZ, Symmetry breaking bifurcations of chaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), pp. 1643-1676.
3. P. J. ASTON AND C. R. LAING, Symmetry and chaos in the complex Ginzburg-Landau equation. I. Reflectional symmetries, Dyn. Stab. Syst., 14 (1999), pp. 233-253.
4. P. J. ASTON AND C. R. LAING, Symmetry and chaos in the complex Ginzburg-Landau equation. II. Translational symmetries, Phys. D, 135 (2000), pp. 79-97.
5. M. BARTUCCELLI, P. CONSTANTIN, C. R. DOERING, J. D. GIBBON, AND M. GISSELFÄLT, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Phys. D, 44 (1990), pp. 421-444.
6. P. BOYLAND, Remarks on the computation of relative periodic solutions, in preparation.
7. L. BRUSCH, A. TORCINI, M. VAN HECKE, M. G. ZIMMERMANN, AND M. BÄR, Modulated amplitude waves and defect formation in the one-dimensional complex Ginzburg-Landau equation, Phys. D, 160 (2001), pp. 127-148.
8. C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI, AND T. A. ZANG, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988.
9. H. CHATÉ, Disordered regimes of the one-dimensional complex Ginzburg-Landau equation, in Spatiotemporal Patterns in Nonequilibrium Complex Systems, Sante Fe Institute Studies in the Sciences of Complexity 21, Addison-Wesley, Reading, MA, 1994, pp. 33-50.
10. W. GYU CHOE AND J. GUCKENHEIMER, Computing periodic orbits with high accuracy, Comput. Methods Appl. Mech. Engrg., 170 (1999), pp. 331-341.
11. F. CHRISTIANSEN, P. CVITANOVIĆ, AND V. PUTKARADZE, Spatiotemporal chaos in terms of unstable recurrent patterns, Nonlinearity, 10 (1997), pp. 55-70.
12. P. CONSTANTIN, A construction of inertial manifolds, in The Connection Between Infinite-Dimensional and Finite-Dimensional Dynamical Systems (Boulder, CO, 1987), Contemp. Math. 99, AMS, Providence, RI, 1989, pp. 27-62.
13. P. CVITANOVIĆ, R. ARTUSO, R. MAINIERI, G. TANNER, AND G. VATTAY, Chaos: Classical and Quantum, Niels Bohr Institute, Copenhagen, Denmark, 2003; also available online at http://www.nbi.dk/ChaosBook.
14. P. CVITANOVIĆ, Chaotic field theory: A sketch, Phys. A, 288 (2000), pp. 61-80.
15. H. DANG-VU AND C. DELCARTE, Spatial heteroclinic bifurcations of time periodic solutions to the Ginzburg-Landau equation, Quart. Appl. Math., 57 (2000), pp. 459-472.
16. J. E. DENNIS, JR., AND ROBERT B. SCHNABEL, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Classics Appl. Math. 16, SIAM, Philadelphia, 1996 (reprint of 1983 edition).
17. C. P. DETTMANN AND P. DAHLQVIST, Computing the diffusion coefficient for intermittent maps: Resummation of stability ordered cycle expansions, Phys. Rev. E (3), 57 (1998), pp. 5303-5310.
18. C. P. DETTMANN AND G. P. MORRISS, Stability ordering of cycle expansions, Phys. Rev. Lett., 78 (1997), pp. 4201-4204.
19. E. J. DOEDEL, A. R. CHAMPNEYS, T. F. FAIRGRIEVE, YU. A. KUZNETSOV, B. SANDSTEDE, AND X. WANG, AUTO 97: Continuation and bifurcation software for ordinary differential equations (with HomCont), Technical report, Concordia University, Montreal, Canada, 1997; also available online at http://indy.cs.concordia.cay/auto/.
20. A. DOELMAN, Slow time-periodic solutions of the Ginzburg-Landau equation, Phys. D, 40 (1989), pp. 156-172.
21. A. DOELMAN AND E. S. TITI, Regularity of solutions and the convergence of the Galerkin method in the Ginzburg-Landau equation, Numer. Funct. Anal. Optim., 14 (1993), pp. 299-321.
22. C. R. DOERING, J. D. GIBBON, D. D. HOLM, AND B. NICOLAENKO, Low-dimensional behaviour in the complex Ginzburg-Landau equation, Nonlinearity, 1 (1998), pp. 279-309.
23. M. J. FIELD, Equivariant dynamical systems, Trans. Amer. Math. Soc., 259 (1980), pp. 185-205.
24. M. J. FIELD, Equivariant dynamics, in Multiparameter Bifurcation Theory (Arcata, Calif., 1985), Contemp. Math. 56, M. Golubitsky and J. M. Guckenheimer, eds., AMS, Providence, RI, 1986, pp. 69-96.
25. M. FRIGO AND S. G. JOHNSON, FFTW: An adaptive software architecture for the FFT, in ICASSP Conference Proceedings, Volume 3, http://www.fftw.org/, 1998, pp. 1381-1384.
26. J.-M. GHIDAGLIA AND B. HÉRON, Dimension of the attractors associated to the Ginzburg-Landau partial differential equation, Phys. D, 28 (1987), pp. 282-304.
27. M. GOLUBITSKY, I. STEWART, AND D. A. SCHAEFFER, Singularities and Groups in Bifurcation Theory. Volume II, Springer-Verlag, New York, 1988.
28. J. GUCKENHEIMER AND B. MELOON, Computing periodic orbits and their bifurcations with automatic differentiation, SIAM J. Sci. Comput., 22 (2000), pp. 951-985.
29. M. S. JOLLY, R. TEMAM, AND C. XIONG, Convergence of a chaotic attractor with increased spatial resolution of the Ginzburg-Landau equation, Chaos Solitons Fractals, 5 (1995), pp. 1833-1845.
30. T. KAPITULA AND S. MAIER-PAAPE, Spatial dynamics of time periodic solutions for the Ginzburg-Landau equation, Z. Angew. Math. Phys., 47 (1996), pp. 265-305.
31. A. KATOK AND B. HASSELBLATT, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl., Cambridge University Press, Cambridge, UK, 1995.
32. L. R. KEEFE, Dynamics of perturbed wavetrain solutions to the Ginzburg-Landau equation, Stud. Appl. Math., 73 (1985), pp. 91-153.
33. E. J. KOSTELICH, I. KAN, C. GREBOGI, E. OTT, AND J. A. YORKE, Unstable dimension variability: A source of nonhyperbolicity in chaotic systems, Phys. D, 109 (1997), pp. 81-90.
34. M. KRUPA, Bifurcations of relative equilibria, SIAM J. Math. Anal., 21 (1990), pp. 1453-1486.
35. J. S. W. LAMB AND C. WULFF, Reversible relative periodic orbits, J. Differential Equations, 178 (2002), pp. 60-100.
36. Y. LAN AND P. CVITANOVIĆ, Variational method for finding periodic orbits in a general flow, Phys. Rev. E, 69 (2004), 016217.
37. S. L. LAU, Y. K. CHEUNG, AND S. Y. WU, A variable parameter incrementation method for dynamic instability of linear and nonlinear elastic systems, J. Appl. Mech., 49 (1982), pp. 849-853.
38. K. LEVENBERG, A method for the solution of certain non-linear problems in least squares, Quart. Appl. Math., 2 (1963), pp. 164-168.
39. C. D. LEVERMORE AND M. OLIVER, The complex Ginzburg-Landau equation as a model problem, in Dynamical Systems and Probabilistic Methods in Partial Differential Equations, Lectures in Appl. Math. 31, AMS, Providence, RI, 1996, pp. 141-190.
40. F. H. LING AND X. X. WU, Fast Galerkin method and its application to determine periodic solutions of non-linear oscillators, Internat. J. Non-Linear Mech., 22 (1987), pp. 89-98.
41. P. MANNEVILLE, Liapunov exponents for the Kuramoto-Sivashinksy model, in Macroscopic Modelling of Turbulent Flows, U. Frisch, ed., Lecture Notes in Phys. 230, Springer-Verlag, Berlin, 1985, pp. 319326.
42. D. W. MARQUARDT, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Indust. Appl. Math., 11 (1963), pp. 431-441.
43. J. E. MARSDEN AND T. S. RATIU, Introduction to Mechanics and Symmetry, 2nd ed., Texts Appl. Math. 17, Springer-Verlag, New York, 1999.
44. A. MIELKE, The Ginzburg-Landau equation in its role as a modulation equation, in Handbook of Dynamical Systems, Vol. 2, B. Fiedler, ed., North-Holland, Amsterdam, 2002, pp. 759-834.
45. W. MILLER, JR., Symmetry Groups and Their Applications, Academic Press, New York, 1972.
46. H. T. MOON, P. HUERRE, AND G. L. REDEKOPP, Transitions to chaos in the Ginzburg-Landau equation, Phys. D, 7 (1983), pp. 135-150.
47. J. J. MORÉ, B. S. GARBOW, AND K. E. HILLSTROM, User guide for MINPACK-1, Technical report ANL-80-74, Argonne National Laboratory, Argonne, IL, 1980. http://www.netlib.org/minpack/.
48. J. J. MORÉ, The Levenberg-Marquardt algorithm: Implementation and theory, in Numerical Analysis, G. A. Watson, ed., Lecture Notes in Math. 630, Springer-Verlag, New York, 1977, pp. 105-116.
49. P. J. OLVER, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, UK, 1995.
50. E. OTT, Chaos in Dynamical Systems, Cambridge University Press, Cambridge, UK, 2002.
51. M. J. D. POWELL, A hybrid method for nonlinear equations, in Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed., Gordon and Breach, London, 1970, pp. 87-114.
52. K. PROMISLOW, Time analyticity and Gevrey regularity for solutions of a class of dissipative partial differential equations, Nonlinear Anal., 16 (1991), pp. 959-980.
53. K. PROMISLOW AND R. TEMAM, Localization and approximation of attractors for the Ginzburg-Landau equation, J. Dynam. Differential Equations, 3 (1991), pp. 491-514.
54. T. D. SAUER, Shadowing breakdown and large errors in dynamical simulations of physical systems, Phys. Rev. E (3), 65 (2002), 036220.
55. P. TAKÁČ, Bifurcations to invariant 2-tori for the complex Ginzburg-Landau equation, Appl. Math. Comput., 89 (1998), pp. 241-257.
56. R. TEMAM, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. 68, Springer-Verlag, New York, 1997.
57. M. URABE AND A. REIFER, Numerical computation of nonlinear forced oscillations by Galerkin's procedure, J. Math. Anal. Appl., 14 (1966), pp. 107-140.
58. M. VAN HECKE, Coherent and incoherent structures in systems described by the 1D CGLE: Experiments and identification, Phys. D, 174 (2003), pp. 134-151.
59. W. VAN SAARLOOS, The complex Ginzburg-Landau equation for beginners, in Spatiotemporal Patterns in Nonequilibrium Complex Systems, P. E. Cladis and P. Palffy-Muhoray, eds., Santa Fe Institute Studies in the Sciences of Complexity 21, Addison-Wesley, Reading, MA, 1994, pp. 19-31.
60. C. WULFF, J. LAMB, AND I. MELBOURNE, Bifurcation from relative periodic solutions, Ergodic Theory Dynam. Systems, 21 (2001), pp. 605-635.
61. C. WULFF AND M. ROBERTS, Hamiltonian systems near relative periodic orbits, SIAM J. Appl. Dyn. Syst., 1 (2002), pp. 1-43.
62. S. M. ZOLDI AND H. S. GHEENSIDE, Spatially localized unstable periodic orbits of a high-dimensional chaotic system, Phys. Rev. E, 57 (1998), pp. R.2511-R2514.