Dynamics of nearly inviscid Faraday waves in almost circular containers

Physica D: Nonlinear Phenomena

Volumn 201, Issue 1-2, 2005, Pages 83-120

  Higuera, M ^a   Knobloch, E ^b,c   Vega, J M ^a  

^a UNIVERSIDAD POLITÉCNICA DE MADRID   (Spain)

^b UNIVERSITY OF LEEDS   (United Kingdom)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Canards; Faraday oscillations; Forced symmetry breaking; Global bifurcations; Relaxation oscillations

Indexed keywords

ASPECT RATIO; BIFURCATION (MATHEMATICS); CYLINDERS (SHAPES); EIGENVALUES AND EIGENFUNCTIONS; NAVIER STOKES EQUATIONS; ORDINARY DIFFERENTIAL EQUATIONS; OSCILLATIONS;

CANARDS; FARADAY OSCILLATIONS; FORCED SYMMETRY BREAKING; GLOBAL BIFURCATIONS; RELAXATION OSCILLATIONS;

GRAVITY WAVES;

EID: 13244298418     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2004.11.012     Document Type: Article

References (47)

1. H. Arbell, and J. Fineberg Pattern formation in two-frequency forced parametric waves Phys. Rev. E 65 2002 036224
2. V.I. Arnol'd, V.S. Afrajmovich, Yu.S. Il'yashenko, L.P. Shil'nikov, Bifurcation theory and catastrophe theory, in: V.I. Arnol'd (Ed.), Encyclopaedia of Mathematical Sciences, vol. 5, Springer-Verlag, New York, 1994.
3. S.M. Baer, and T. Erneux Singular Hopf bifurcation to relaxation oscillations SIAM J. Appl. Math. 46 1986 721 739
4. K. Bar-Eli Coupling of identical oscillators J. Phys. Chem. 94 1990 2368 2374
5. K. Bar-Eli, and M. Brøns Period lengthening near the end of oscillations in chemical systems J. Phys. Chem. 94 1990 7170 7177
6. O. Batiste, E. Knobloch, I. Mercader, and M. Net Simulations of oscillatory binary fluid convection in large aspect ratio containers Phys. Rev. E 65 2001 016303
7. A. Bergeon, and E. Knobloch Natural doubly diffusive convection in three-dimensional enclosures Phys. Fluids 14 2002 3233 3250
8. M. Brøns, and K. Bar-Eli Canard explosions and excitation in a model of the Belousov-Zhabotinsky reaction J. Phys. Chem. 95 1991 8706 8713
9. G. Dangelmayr, and E. Knobloch Hopf bifurcation with broken circular symmetry Nonlinearity 4 1991 399 427
10. S. Douady, S. Fauve, and O. Thual Oscillatory phase modulation of parametrically forced surface waves Europhys. Lett. 10 1989 309 315
11. W. Eckhaus, Relaxation oscillations, including a standard chase on French ducks, Lecture Notes in Mathematics, vol. 985, Springer-Verlag, New York, 1983, pp. 449-494.
12. W.S. Edwards, and S. Fauve Patterns and quasi-patterns in the Faraday experiment J. Fluid Mech. 278 1994 123 148
13. N. Fenichel Geometric singular perturbation theory for ordinary differential equations J. Diff. Eq. 31 1979 53 98
14. M. Funakoshi, and S. Inoue Surface waves due to resonant horizontal oscillation J. Fluid Mech. 192 1988 219 247
15. P. Glendinning Global structure of homoclinic bifurcations: a combinatorial approach Phys. Lett. A 141 1989 391 396
16. P. Glendinning, J. Abshagen, and T. Mullin Imperfect homoclinic bifurcations Phys. Rev. E 64 2001 036208
17. J. Guckenheimer On a codimension-two bifurcation D.A. Rand L.S. Young Dynamical systems and turbulence Lecture Notes in Mathematics vol. 898 1981 Springer-Verlag New York 99 142
18. J. Guckenheimer, K. Hoffman, and W. Weckesser Numerical computation of canards Int. J. Bif. Chaos 10 2000 2669 2687
19. J. Guckenheimer, and P. Holmes Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields second ed. 1986 Springer-Verlag New York
20. J. Guckenheimer, and A.R. Willms Asymptotic analysis of subcritical Hopf-homoclinic bifurcation Physica D 139 2000 195 216
21. M. Higuera, J. Porter, and E. Knobloch Heteroclinic dynamics in the nonlocal parametrically driven nonlinear Schrödinger equation Physica D 162 2002 155 187
22. M. Higuera, J.M. Vega, and E. Knobloch Coupled amplitude-streaming flow equations for nearly inviscid Faraday waves in small aspect ratio containers J. Nonlinear Sci. 12 2002 505 551
23. P. Hirschberg, and E. Knobloch The Shil'nikov-Hopf bifurcation Physica D 62 1993 202 216
24. V. Kirk Breaking of symmetry in the saddle-node Hopf bifurcation Phys. Lett. A 154 1991 243 248
25. V. Kirk Merging of resonance tongues Physica D 66 1993 267 281
26. B. Krauskopf, and S. Wieczorek Accumulating regions of winding periodic orbits in optically driven lasers Physica D 173 2002 97 113
27. A. Kudrolli, and J.P. Gollub Patterns and spatio-temporal chaos in parametrically forced surface waves: a systematic survey at large aspect ratio Physica D 97 1997 133 154
28. A. Kudrolli, B. Pier, and J.P. Gollub Superlattice patterns in surface waves Physica D 123 1998 99 111
29. A.S. Landsberg, and E. Knobloch Direction-reversing traveling waves Phys. Lett. A 159 1991 17 20
30. V. Lapuerta, C. Martel, and J.M. Vega Weakly dissipative Faraday waves in 2D large aspect ratio annuli Physica D 173 2002 178 203
31. M.S. Longuet-Higgins Mass transport in water waves Phil. Trans. Roy. Soc. A 245 1953 535 581
32. C. Martel, J.A. Nicolás, and J.M. Vega Surface-wave damping in a brimful circular cylinder J. Fluid Mech. 360 1998 213 228 (see also Corrigendum 373 (1998) 379)
33. E. Martín, C. Martel, and J.M. Vega Drift instability of standing Faraday waves J. Fluid Mech. 467 2002 57 79
34. J.W. Miles Internally resonant surface waves in a circular cylinder J. Fluid Mech. 149 1984 1 14
35. J. Moehlis Canards in a surface oxidation reaction J. Nonlinear Sci. 12 2002 319 345
36. J. Moehlis, and E. Knobloch Forced symmetry breaking as a mechanism for bursting Phys. Rev. Lett. 80 1998 5329 5332
37. J. Porter, and M. Silber Broken symmetries and pattern formation in two-frequency forced Faraday waves Phys. Rev. Lett. 89 2002 084501
38. A.M. Rucklidge, and P.C. Matthews Analysis of the shearing instability in nonlinear convection and magnetoconvection Nonlinearity 9 1996 311 351
39. A.M. Rucklidge, M. Silber, and J. Fineberg Spatial period-multiplying instabilities of hexagonal Faraday waves J. Buescu S.B.S.D. Castro A.P.S. Dias I.S. Labouriau Bifurcations, Symmetry and Patterns 2003 Birkhäuser Basel 101 114
40. H. Schlichting Berechnung ebener periodischer Grenzschichtströmungen Phys. Z. 33 1932 327 335
41. D. Terman Chaotic spikes arising from a model of bursting in excitable membranes SIAM J. Appl. Math. 51 1991 1418 1450
42. D.P. Tse, A.M. Rucklidge, R.B. Hoyle, and M. Silber Spatial period-multiplying instabilities of hexagonal Faraday waves Physica D 146 2000 367 387
43. J.M. Vega, E. Knobloch, and C. Martel Nearly inviscid Faraday waves in annular containers of moderately large aspect ratio Physica D 154 2001 313 336
44. X.J. Wang, and J. Rinzel Oscillatory and bursting properties of neurons M.A. Arbib Brain Theory and Neural Networks 1995 MIT Press Cambridge, MA
45. S. Wiggins Global Bifurcations and Chaos 1988 Springer-Verlag New York
46. R.W. Wittenberg, and P. Holmes The limited effectiveness of normal forms: A critical review and extension of local bifurcation studies of the Brusselator PDE Physica D 100 1997 1 40
47. M.K.S. Yeung, and S.H. Strogatz Nonlinear dynamics of a solid-state laser with injection Phys. Rev. E 58 1998 4421 4435 (see also Corrigendum 61 (2000) 2154)