Bursts in oscillatory systems with broken D4 symmetry

Physica D: Nonlinear Phenomena

Volumn 135, Issue 3-4, 2000, Pages 263-304

  Moehlis, Jeff ^a   Knobloch, Edgar ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Bursts; Forced symmetry breaking; Global bifurcations

Indexed keywords

EID: 0347318516     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(99)00141-4     Document Type: Article

References (46)

1. R. Lauterbach, M. Roberts, Heteroclinic cycles in dynamical systems with broken spherical symmetry, J. Diff. Eq. 100 (1992) 22-48.
2. E. Knobloch, System symmetry breaking and Shil'nikov Dynamics, in: J. Chadam, M. Golubitsky, W. Langford, B. Wetton (Eds.), Pattern Formation: Symmetry Methods and Applications, American Mathematical Society, Providence, 1996, pp. 271-279.
3. P. Hirschberg, E. Knobloch, Complex dynamics in the Hopf bifurcation with broken translation symmetry, Physica D 92 (1996) 56-78.
4. J.D. Crawford, E. Knobloch, Symmetry and symmetry-breaking bifurcations in fluid mechanics, Ann. Rev. Fluid Mech. 23 (1991) 341-387.
5. M. Golubitsky, I. Stewart, D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, vol. II, Springer, New York, 1988.
6. T. Clune, E. Knobloch, Mean flow suppression by endwalls in oscillatory binary fluid convection, Physica D 61 (1992) 106-112.
7. A.D.D. Craik, I.M. Moroz, Temporal evolution of interaction waves in non-conservative systems: some exact solutions, Wave Motion 10 (1988)443-452.
8. E. Knobloch, J. Moehlis, Bursting mechanisms for hydrodynamical systems, in: M. Golubitsky, D. Luss, S.H. Strogatz (Eds.), Pattern Formation in Continuous and Coupled Systems, Springer, New York, 1999, pp. 157-174.
9. E. Knobloch, J. Moehlis, Burst mechanisms in hydrodynamics, in: L. Debnath, D. Riahi (Eds), Nonlinear Instability, Chaos, and Turbulence, vol. II, Computational Mechanics Publications, Southampton, in press.
10. J.W. Swift, Hopf bifurcation with the symmetry of the square, Nonlinearity 1 (1988) 333-377.
11. A.S. Landsberg, E. Knobloch, Oscillatory bifurcation with broken translation symmetry, Phys. Rev. E 53 (1996) 3579-3600.
12. J. Moehlis, E. Knobloch, Forced symmetry breaking as a mechanism for bursting, Phys. Rev. Lett. 80 (1998) 5329-5332.
13. G. Dangelmayr, E. Knobloch, Hopf bifurcation with broken circular symmetry, Nonlinearity 4 (1991) 399-27.
14. 1 symmetric systems, Nonlinearity 8 (1995) 615-628.
15. T. Clune, E. Knobloch, Pattern selection in three-dimensional magnetoconvection, Physica D 74 (1994) 151-176.
16. X.M. Gu, P.R. Sethna, Resonant surface waves and chaotic phenomena, J. Fluid Mech. 183 (1987) 543-565.
17. Z.C. Feng, P.R. Sethna, Symmetry-breaking bifurcations in resonant surface waves, J. Fluid Mech. 199 (1989) 495-518.
18. I. Melbourne, P Chossat, M. Golubitsky, Heteroclinic cycles involving periodic solutions in mode interactions with O(2) symmetry, Proc. Roy. Soc. Edinburgh A 113 (1989) 315-345.
19. C. Tresser, About some theorems by L.P. Shil'nikov, Ann. Inst. Henri Poincaré 40 (1984) 441-61.
20. S. Wiggins, Global Bifurcations and Chaos, Springer, New York, 1988.
21. P Hirschberg, E. Knobloch, Shil'nikov-Hopf bifurcation, Physica D 62 (1993) 202-216.
22. P Glendinning, C. Sparrow, Local and global behavior near homoclinic orbits, J. Stat. Phys. 35 (1984) 645-695.
23. J.W. Swift, K. Wiesenfeld, Suppression of period doubling in symmetric systems, Phys. Rev. Lett. 52 (1984) 705-708.
24. E. Knobloch, N.O. Weiss, Bifurcations in a model of double-diffusive convection, Phys. Lett. A 85 (1981) 127-130.
25. C.T. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer, New York, 1982.
26. A. Back, J. Guckenheimer, M.R. Myers, F.J. Wicklin, P.A. Worfolk, dstool: Computer assisted exploration of dynamical systems, Not. Am. Math. Soc. 39 (1992) 303-309.
27. E. Doedel, A. Champneys, T. Fairgrieve, Y Kuznetsov, B. Sandstede, X. Wang, AUTO 97: Continuation and bifurcation software for ordinary differential equations (available via FTP from directory pub/doedel/auto at ftp.cs.concordia.ca) (1997).
28. P Chossat, M. Golubitsky, Symmetry-increasing bifurcation of chaotic attractors, Physica D 32 (1988) 423-36.
29. J.D. Rodriguez, M. Schell, Global bifurcations into chaos in systems with SO(2) symmetry, Phys. Lett. A 146 (1990) 25-31.
30. A.C. Newell, D.A. Rand, D. Russell, Turbulent dissipation rates and the random occurrence of coherent events, Phys. Lett. A 132 (1988) 112-123.
31. A.C. Newell, D.A. Rand, D. Russell, Turbulent transport and the random occurrence of coherent events, Physica D 33 (1988) 281-303.
32. E. Knobloch, D.R. Moore, Minimal model of binary fluid convection, Phys. Rev. A 42 (1990) 4693-709.
33. E. Knobloch, D.R. Moore, Chaotic travelling wave convection, Eur. J. Mech. B/Fluids 10(2, suppl.) (1991) 37-2.
34. N. Platt, E.A. Spiegel, C. Tresser, On-off intermittency: a mechanism for bursting, Phys. Rev. Lett. 70 (1993) 279-282.
35. P Ashwin, E. Covas, R. Tavakol, Transverse instability for non-normal parameters, Nonlinearity 12 (1999) 563-577.
36. D. Armbruster, Codimension 2 bifurcation in binary convection with square symmetry, in: F.H. Busse, L. Kramer (Eds.), Nonlinear Evolution of Spatio-Temporal Structures in Dissipative Continuous Systems, Plenum Press, New York, 1990, pp. 385-398.
37. D. Armbruster, Square and almost square symmetry in binary convection, Eur. J. Mech. B/Fluids 10(2, suppl.) (1991) 7-12.
38. P Ashwin, Z. Mei, Normal form for Hopf bifurcation of partial differential equations on the square, Nonlinearity 8 (1995) 715-734.
39. M. Golubitsky, I.N. Stewart. Hopf bifurcation with dihedral group symmetry, in: Multiparameter Bifurcation Theory, Contemporary Maths, vol. 56, American Mathematical Society, Providence, 1986, pp. 131-173.
40. P Ashwin, P Stork, Permissible symmetries of coupled cell networks, Math. Proc. Camb. Phil. Soc. 116 (1994) 27-36.
41. 4 symmetry, Nonlinear Dyn. 8 (1995) 161-178.
42. T.S. Sullivan, G. Ahlers, Nonperiodic time dependence at the onset of convection in a binary liquid mixture, Phys. Rev. A 38 (1988) 3143-3146.
43. F. Simonelli, J.P. Gollub, Surface wave mode interactions: effects of symmetry and degeneracy, J. Fluid Mech. 199 (1989) 471-94.
44. E. Knobloch, A.S. Landsberg, A new model of the solar cycle, Mon. Not. Roy. Astr. Soc. 278 (1996) 294-302.
45. E. Knobloch, S.M. Tobias, N.O. Weiss, Modulation and symmetry changes in stellar dynamos, Mon. Not. Roy. Astr. Soc. 297 (1998) 1123-1138.
46. P Ashwin, G. Dangelmayr, I. Stewart, M. Wegelin, Oscillator networks with the symmetry of the unit quaternion group, in: P Chossat (Ed.), Dynamics, Bifurcation and Symmetry, Kluwer Academic Publishers, Dordrecht, 1994, pp. 35-48.