Hopf bifurcation on a simple cubic lattice

Dynamics and Stability of Systems

Volumn 14, Issue 1, 1999, Pages 3-55

  Dias, Ana Paula S ^a,b   Stewart, Ian ^b  

^a UNIVERSITY OF PORTO   (Portugal)

^b UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL GEOMETRY; CRYSTAL LATTICES; CRYSTAL SYMMETRY; ORDINARY DIFFERENTIAL EQUATIONS; PROBABILITY; THEOREM PROVING;

HOPF BIFURCATION; SYMMETRY-BREAKING BIFURCATION;

BIFURCATION (MATHEMATICS);

EID: 0033097564     PISSN: 02681110     EISSN: None     Source Type: Journal    
DOI: 10.1080/026811199282065     Document Type: Article

References (33)

1. Callahan, T. K. and Knobloch, E. (1997) Symmetry-breaking bifurcations on cubic lattices. Nonlinearity 10, 1179-1216.
2. Callahan, T. K. and Knobloch, E. (1998) Pattern formation in three-dimensional reaction-diffusion systems. Preprint.
3. Chandrasekhar, S. (1961) Hydrodynamic and Hydromagnetic Stability (Oxford University Press, Oxford).
4. De Wit, A., Borckmans, P. and Dewel, G. (1997) Twist grain boundaries in three-dimensional lamellar Turing structures. Proceedings of the National Academy of Sciences of the United States of America 94, 12765-12768.
5. De Wit, A., Dewel, G., Borckmans, P. and Walgraef, D. (1992) Three-dimensional dissipative structures in reaction-diffusion systems. Physica D 61, 289-296.
6. Dias, A. P. S. (1997) Bifurcations with wreath product symmetry. PhD thesis, Mathematics Institute, University of Warwick.
7. Dias, A. P. S. (1998) Hopf bifurcation for wreath products. Nonlinearity 11, 247-264.
8. Dias, A. P. S. and Stewart, I. (1999) Symmetry-breaking bifurcations of wreath product systems. Journal of Nonlinear Science (forthcoming).
9. Dionne, B. and Golubitsky, M. (1992) Planforms in two and three dimensions. Zeitschrift angewandte Mathematik Physik 43, 36-62.
10. Dionne, B., Golubitsky, M., Silber, M. and Stewart, I. N. (1995) Time-periodic spatially periodic planforms in Euclidean equivariant partial differential equations. Philosophical Transactions of the Royal Society of London A 352, 125-168.
11. Dionne, B., Golubitsky, M. and Stewart, I. N. (1996) Coupled cells with internal symmetry: I. Wreath products. Nonlinearity 9, 559-574.
12. Field, M. (1996) Lectures on Bifurcations, Dynamics and Symmetry, Research Notes in Mathematics 356 (Longman, London).
13. Gils, S. A. van (1984) Some studies in Dynamical Systems Theory. Thesis, Vrije Universitet, Amsterdam.
14. Golubitsky, M. and Stewart, I. N. (1985) Hopf bifurcation with dihedral group symmetry: coupled nonlinear oscillators. Multiparameter Bifurcation Theory (Arcata, Calif.), Contemporary Mathematics, American Mathematical Society, Providence, Rhode Island 56, 131-173.
15. Golubitsky, M., Stewart, I. N. and Dionne, B. (1994) Coupled cells: wreath products and direct products. Dynamics, Bifurcation and Symmetry, NATO ARW Series (eds P. Chossat and J. M. Gambaudo; Kluwer, Amsterdam), pp. 127-138.
16. Golubitsky, M., Stewart, I. N. and Schaeffer, D. G. (1988) Singularities and Groups in Bifurcation Theory, Vol. II, Applied Mathematical Sciences 69 (Springer, New York).
17. Gomes, M. G. M. and Stewart, I. N. (1994) Steady PDEs on generalized rectangles: a change of genericity in mode interactions. Nonlinearity 7, 253-272.
18. Guckenheimer, J. and Holmes, P. (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences 42 (Springer, New York).
19. Guckenheimer, J. and Holmes, P. (1988) Structurally stable heteroclinic cycles. Mathematical Proceedings of the Cambridge Philosophical Society 103, 189-192.
20. Hurle, D. T. J. and Jakeman, E. (1971) Soret-driven thermosolutal convection. Journal of Fluid Mechanics 47, 667-687.
21. Marsden, J. E. and McCracken, M. (1976) The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences 19 (Springer, New York).
22. Mathews, P. C., Proctor, M. R. E. and Weiss, N. O. (1995) Compressible magnetoconvection in three dimensions: planform selection and weakly nonlinear behaviour. Journal of Fluid Mechanics 305, 281-305.
23. Melbourne, I., Chossat, P. and Golubitsky, M. (1989) Heteroclinic cycles involving periodic solutions in mode interactions with O(2) symmetry. Proceedings of the Royal Mathematical Society of Edinburgh 113A, 315-345.
24. Miller, Jr W. (1972) Symmetry Groups and their Applications (Academic Press, New York), p. 55.
25. Renardy, Y. (1993) Pattern selection for the oscillatory onset in thermosolutal convection. Physics Fluids A 5, 1376-1389.
26. Renardy, M. and Renardy, Y. (1988) Bifurcation solutions at the onset of convection in the Bénard problem for two fluids. Physica D 32, 227-258.
27. Renardy, M. and Renardy, Y. (1992) Pattern selection in the Bénard problem for a viscoelastic fluid. Zeitschrift angewandte Mathematik Physik 43, 154-180.
28. Roberts, M., Swift, J. W. and Wagner, D. H. (1986) The Hopf bifurcation on a hexagonal lattice. Contemporary Mathematics 56, 283-318.
29. Scott, W. R. (1964) Group Theory (Prentice-Hall, Englewood Cliffs, New Jersey).
30. Silber, M. and Knobloch, E. (1991) Hopf bifurcation on a square lattice. Nonlinearity 4, 1063-1106.
31. Silber, M., Riecke, H. and Kramer, L. (1992) Symmetry-breaking Hopf bifurcation in anisotropic systems. Physica D 61, 260-278.
32. Swift, J. W. (1988) Hopf bifurcation with the symmetry of the square. Nonlinearity 1, 333-377.
33. Veronis, G. (1968) Effect of stabilizing gradient of solute on thermal convection. Journal of Fluid Mechanics 34, 315-336.