Hopf bifurcation with cubic symmetry and instability of ABC flow

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 459, Issue 2035, 2003, Pages 1801-1827

  Ashwin, Peter ^a   Podvigina, Olga ^a,b,c,d  

^a UNIVERSITY OF EXETER   (United Kingdom)

^b CNRS   (France)

^c INSTITUTE OF EARTHQUAKE PREDICTION THEORY AND MATHEMATICAL GEOPHYSICS   (Russian Federation)

^d MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

ABC flow; Fluid instability; Hopf bifurcation; Symmetry

Indexed keywords

ARTICLE; DYNAMICS; MATHEMATICAL ANALYSIS; MATHEMATICAL COMPUTING; ROTATION; SIMULATION;

EID: 1542407680     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2002.1090     Document Type: Article

References (46)

1. Arnédo, A., Coullet, P. H. & Spiegel, E. 1983 Cascade of period doublings of tori. Phys. Lett. A 94, 1-6.
2. Arnold, V. I. 1965 Sur la topologie des écoulements stationnaires des fluides parfaits. C. R. Acad. Sci. Paris 261, 17-20.
3. Arnold, V. I. 1984 The evolution of a magnetic field under the influence of transport and diffusion. Some questions of present-day analysis (ed. V. M. Tikhomirov). Moscow University. (In Russian.)
4. Arnold, V. I. & Korkina, E. I. 1983 The growth of a magnetic field in a three-dimensional incompressible flow. Vestn. Moscow State Univ. Ser. Math. 3, 43-46. (In Russian.)
5. Ashwin, P. 1999 Chaotic intermittency of patterns in symmetric systems. In Proc. IMA Workshop on Pattern Formation in Coupled and Continuous Systems (ed. M. Golubitsky, D. Luss & S. H. Strogatz). IMA Volumes in Mathematics and Its Applications, vol. 115. Springer.
6. Ashwin, P. & Chossat, P. 1998 Attractors for robust heteroclinic cycles with continua of connections. J. Nonlin. Sci. 8, 103-129.
7. Ashwin, P. & Montaldi, J. 2002 Group theoretic conditions for existence of robust relative homoclinic trajectories. Math. Proc. Camb. Phil. Soc. 133, 125-141.
8. Back, A., Guckenheimer, J., Myers, M. R., Wicklin, F. J. & Worfolk, P. A. 1992 Dstool: computer-assisted exploration of dynamical systems. Not. Am. Math. Soc. 39, 303-309.
9. Callahan, T. K. & Knobloch, E. 1997 Symmetry breaking bifurcations on cubic lattices. Nonlinearity 10, 1179-1216.
10. Chenciner, A. & Iooss, G. 1979 Bifurcation de Tores invariants. Arch. Ration. Mech. Analysis 69, 109-198.
11. Childress, S. 1970 New solutions of the kinematic dynamo problem. J. Math. Phys. 11, 3063-3967.
12. Childress, S. & Gilbert, A. D. 1995 Stretch, twist, fold: the fast dynamo. Springer.
13. Childress, S. & Soward, A. M. 1985 On the rapid generation of magnetic field. In Chaos in astrophysics (ed. J. R. Buchler), pp. 233-244. Dordrecht: Kluwer.
14. Chossat, P. & Golubitsky, M. 1988 Symmetry increasing bifurcation of chaotic attractors. Physica D 32, 423-436.
15. Dias, A. P. & Stewart, I. N. 1999 Hopf bifurcation on a simple cubic lattice. Dynam. Stab. Syst. 14, 3-55.
16. Dionne, B., Golubitsky, M. & Stewart, I. 1996 Coupled cells with internal symmetry. II. Direct products. Nonlinearity 9, 575-599.
17. Dombre, T., Frisch, U., Greene, J. M., Hénon, M., Mehr, A. & Soward, A. 1986 Chaotic streamlines in the ABC flows. J. Fluid Mech. 167, 353-391.
18. Ermentrout, G. B. 2000 xppaut (dynamical systems software). (Available from http:// www.math.pitt.edu/~bard/bardware/.)
19. Galanti, B., Sulem, P. L. & Pouquet, A. 1992 Linear and non-linear dynamos associated with ABC flows. Geophys. Astrophys. Fluid Dynam. 66, 183-208.
20. Galanti, B., Sulem, P. L. & Pouquet, A. 1993 Influence of the period of an ABC flow on its dynamo action. In Solar and Planetary Dynamos, Proc. NATO Advanced Study Institute Cambridge, September 1992 (ed. M. R. E. Proctor, P. C. Matthews & A. M. Rucklidge), pp. 99-103. Cambridge University Press.
21. Galloway, D. J. & Frisch, U. 1984 A numerical investigation of magnetic field generation in a flow with chaotic streamlines. Geophys. Astrophys. Fluid Dynam. 29, 13-18.
22. Galloway, D. J. & Frisch, U. 1986 Dynamo action in a family of flows with chaotic streamlines. Geophys. Astrophys. Fluid Dynam. 36, 53-83.
23. Galloway, D. J. & Frisch, U. 1987 A note on the stability of a family of space-periodic Beltrami flows. J. Fluid Mech. 180, 557-564.
24. Gilbert, A. D. 1991 Fast dynamo action in a steady chaotic flow. Nature 350, 483-485.
25. Gilbert, A. D. 1992 Magnetic field evolution in steady chaotic flows. Phil. Trans. R. Soc. Lond. 339, 627-656.
26. Golubitsky, M. & Stewart, I. N. 1985 Hopf bifurcation in the presence of symmetry, Arch. Ration. Mech. Analysis 87, 107-165.
27. Golubitsky, M., Stewart, I. N. & Schaeffer, D. 1988 Groups and singularities in bifurcation theory 2. Applied Mathematical Science, vol. 69. Springer.
28. Guckenheimer, J. & Holmes, P. 1988 Structurally stable heteroclinic cycles. Math. Proc. Camb. Phil. Soc. 103, 189-192.
29. Hénon, M. 1966 Sur la topologie des lignes de courant dans un cas particulier. C. R. Acad. Sci. Paris 262, 312-314. (In French.)
30. Iooss, G. & Los, J. E. 1988 Quasi-genericity of bifurcations to high dimensional invariant tori for maps. Commun. Math. Phys. 119, 453-500.
31. Klapper, I. & Young, L. S. 1995 Rigorous bounds on the fast dynamo growth rate involving topological entropy. Commun. Math. Phys. 173, 623-646.
32. Libin, A. & Sivashinsky, G. 1990 Long wavelength instability of the ABC flow. Q. Appl. Math. 48, 611-623.
33. Libin, A., Sivashinsky, G. & Levich, E. 1987 Long wavelength instability of periodic flows at large Reynolds numbers. Phys. Fluids 30, 2984-2986.
34. 3 and doubling bifurcation. Nonlinearity 2, 149-174.
35. Melbourne, I. S. 1986 A singularity theory analysis of bifurcation problems with octahedral symmetry. Dynam. Stab. Syst. 4, 293-321.
36. Moffatt, H. K. 1986 Magnetostatic equilibria and analogous Euler flows of arbitrary complex topology 2. Stability considerations. J. Fluid Mech. 166, 359-378.
37. Podvigina, O. M. 1999a Spatially-periodic steady solutions to the three-dimensional Navier-Stokes equation with the ABC-force. Physica D 128, 250-272.
38. Podvigina, O. M. 1999b Evolutionary and steady solutions to the Navier-Stokes equation with the ABC forcing. Moscow: Institute of Mechanics, Lomonosov Moscow State University. (In Russian.)
39. Podvigina, O. M. 2003a A route to magnetic field reversals: an example of an ABC-forced non-linear dynamo. Geophys. Astrophys. Fluid Dynam. (Available at http://arxiv.org/abs/ physics/0107029.) (Submitted.)
40. Podvigina, O. 2003b Center manifold reduction for a Hopf bifurcation with cubic symmetry. Geophys. Astrophys. Fluid Dynam. (In the press.)
41. Podvigina, O. & Pouquet, A. 1994 On the non-linear stability of the 1:1:1 ABC flow. Physica D 75, 471-508.
42. Silber, M. & Knobloch, E. 1991 Hopf bifurcation on a square lattice. Nonlinearity 4, 1063-1106.
43. Swift, J. W. 1988 Hopf bifurcation with the symmetry of the square. Nonlinearity 1, 333-377.
44. Swift, J. W. & Barany, E. 1991 Chaos in the Hopf bifurcation with tetrahedral symmetry; convection in a rotating fluid with low Prandtl number. Eur. J. Mech. B 102, 99-104.
45. Vishik, M. M. 1988 On magnetic field generation by a three-dimensional steady flow of conducting fluid at high magnetic Reynolds number. Izv. Akad. Sci. USSR 3, 3-12. (In Russian.)
46. Vishik, M. M. 1989 Magnetic field generation by the motion of a highly conducting fluid. Geophys. Astrophys. Fluid Dynam. 48, 151-167.