PapersNo Access

EXAMPLES OF FORCED SYMMETRY-BREAKING TO HETEROCLINIC CYCLES AND NETWORKS IN THREE-DIMENSIONAL EUCLIDEAN-INVARIANT SYSTEMS

M. J. PARKER

School of Computing and Information Technology, University of Wolverhampton,Wolverhampton WV1 1SB, UK

Author for correspondence.

Search for more papers by this author

M. G. M. GOMES

Instituto Gulbenkian de Ciencia, Rua da Quinta Grande 6, 2780-156 Oeiras, Portugal

Search for more papers by this author

, and

I. N. STEWART

Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

Search for more papers by this author

https://doi.org/10.1142/S0218127409023767Cited by:0 (Source: Crossref)

Abstract

In [Parker et al., 2008a] group theory was employed to prove the existence of homoclinic cycles in forced symmetry-breaking of simple (SC), face-centered (FCC), and body-centered (BCC) cubic planforms. In this paper we extend this classification demonstrating that more elaborate heteroclinic cycles and networks can arise through the same process. Our methods naturally generate graphs that represent possible heteroclinic cycles and networks. The results do not depend on the representation of the symmetry group and are thus quite general.

This study is motivated by pattern formation in three dimensions which occur in reaction–diffusion systems, certain nonlinear optical systems and the polyacrylamide methylene blue oxygen reaction. This work extends previous work by Parker et al. [2006, 2008a, 2008b] and Hou and Golubitsky [1997].

Keywords:

References

P. Ashwin and M. Field, Arch. Ration. Mech. Anal. 148, 107 (1999), DOI: 10.1007/s002050050158. Crossref, Web of Science, Google Scholar
W. Brannath, Nonlinearity 7, 1367 (1994), DOI: 10.1088/0951-7715/7/5/006. Crossref, Web of Science, Google Scholar
P. C. Bresloffet al., Phil. Trans. Roy. Soc. London Ser. B 356, 299 (2001). Crossref, Web of Science, Google Scholar
E. Buzano and M. Golubitsky, Philos. Trans. Roy. Soc. London Ser. A 308, 617 (1983). Crossref, Web of Science, Google Scholar
T. K. Callahan and E. Knobloch, Nonlinearity 10, 1179 (1997), DOI: 10.1088/0951-7715/10/5/009. Crossref, Web of Science, Google Scholar
T. K. Callahan and E. Knobloch, Phys. D 132, 339 (1999), DOI: 10.1016/S0167-2789(99)00041-X. Crossref, Web of Science, Google Scholar
J. Carr , Applications of Centre Manifold Theory , Applied Mathematical Sciences 35 ( Springer-Verlag , NY , 1981 ) . Crossref, Google Scholar
V. Castetset al., Phys. Rev. Lett. 64, 2953 (1990), DOI: 10.1103/PhysRevLett.64.2953. Crossref, Web of Science, Google Scholar
G. Cicogna, Lett. Nuovo Cimento (2) 31, 600 (1981), DOI: 10.1007/BF02777979. Crossref, Web of Science, Google Scholar
A. De Witet al., Phys. D 61, 289 (1992). Crossref, Web of Science, Google Scholar
E. V. Degtiarev and M. A. Vorontsov, J. Mod. Opt. 43, 93 (1996). Crossref, Web of Science, Google Scholar
B. Dionne and M. Golubitsky, Z. Angew. Math. Phys. 43, 36 (1992), DOI: 10.1007/BF00944740. Crossref, Web of Science, Google Scholar
B. Dionne, Z. Angew. Math. Phys. 44, 673 (1993), DOI: 10.1007/BF00948482. Crossref, Web of Science, Google Scholar
B. Dionne, M. Silber and A. C. Skeldon, Nonlinearity 10, 321 (1997), DOI: 10.1088/0951-7715/10/2/002. Crossref, Web of Science, Google Scholar
W. S. Edwards and S. Fauve, Phys. Rev. E 47, 788 (1993), DOI: 10.1103/PhysRevE.47.R788. Crossref, Web of Science, Google Scholar
W. S. Edwards and S. Fauve, J. Fluid Mech. 278, 123 (1994), DOI: 10.1017/S0022112094003642. Crossref, Web of Science, Google Scholar
G. B. Ermentrout and J. D. Cowan, Biol. Cybern. 34, 137 (1979), DOI: 10.1007/BF00336965. Crossref, Web of Science, Google Scholar
M. J. Field, Trans. Amer. Math. Soc. 259, 185 (1980), DOI: 10.2307/1998153. Crossref, Web of Science, Google Scholar
M. Golubitsky and D. G. Schaeffer , Singularities and Groups in Bifurcation Theory. Vol. I , Applied Mathematical Sciences 51 ( Springer-Verlag , NY , 1985 ) . Crossref, Google Scholar
M. Golubitsky , I. Stewart and D. G. Schaeffer , Singularities and Groups in Bifurcation Theory. Vol. II , Applied Mathematical Sciences 69 ( Springer-Verlag , NY , 1988 ) . Crossref, Google Scholar
M. Golubitsky and I. Stewart , The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space , Progress in Mathematics 200 ( Birkhäuser Verlag , Basel , 2002 ) . Crossref, Google Scholar
M. G. M. Gomes and I. N. Stewart, Nonlinearity 7, 253 (1994), DOI: 10.1088/0951-7715/7/1/012. Crossref, Web of Science, Google Scholar
M. G. M. Gomes and I. Stewart, Int. J. Bifurcation and Chaos 7, 147 (1997). Web of Science, Google Scholar
M. G. M. Gomes, Phys. Rev. E (3) 60(4), 3741 (1999), DOI: 10.1103/PhysRevE.60.3741. Crossref, Web of Science, Google Scholar
M. G. M. Gomes, I. S. Labouriau and E. M. Pinho, Pattern Formation in Continuous and Coupled Systems (Minneapolis, MN, 1998), IMA Vol. Math. Appl. 115 (Springer, NY, 1999) pp. 83–99. Crossref, Google Scholar
G. H. Gunaratne, Q. Ouyang and H. L. Swinney, Phys. Rev. E (3) 50, 2802 (1994), DOI: 10.1103/PhysRevE.50.2802. Crossref, Web of Science, Google Scholar
F. Guyard and R. Lauterbach, Nonlinearity 10, 291 (1997), DOI: 10.1088/0951-7715/10/1/019. Crossref, Web of Science, Google Scholar
C. Hou and M. Golubitsky, J. Diff. Eqs. 133, 30 (1997), DOI: 10.1006/jdeq.1996.3201. Crossref, Web of Science, Google Scholar
V. Kirk and M. Silber, Nonlinearity 7, 1605 (1994), DOI: 10.1088/0951-7715/7/6/005. Crossref, Web of Science, Google Scholar
N. L. Komarovaet al., Phys. Rev. A 56, 803 (1997), DOI: 10.1103/PhysRevA.56.803. Crossref, Web of Science, Google Scholar
M. Krupa and I. Melbourne, Erg. Th. Dyn. Syst. 15, 121 (1995). Crossref, Web of Science, Google Scholar
M. Krupa, J. Nonlin. Sci. 7, 129 (1997), DOI: 10.1007/BF02677976. Crossref, Web of Science, Google Scholar
K. Kurin-Csörgeiet al., Chem. Rev. Lett. 295, 70 (1998). Google Scholar
R. Lauterbach and M. Roberts, J. Diff. Eqs. 100, 22 (1992), DOI: 10.1016/0022-0396(92)90124-6. Crossref, Web of Science, Google Scholar
R. Lauterbach, Nonlinear Dynamical Systems and Chaos (Groningen, 1995), Progr. Nonlinear Differential Equations Appl. 19 (Birkhäuser, Basel, 1996) pp. 121–143. Crossref, Google Scholar
R. Lauterbach, S. Maier-Paape and E. Reissner, Proc. Roy. Soc. Edinburgh Sect. A 126, 885 (1996). Crossref, Web of Science, Google Scholar
S. Maier-Paape and R. Lauterbach, Trans. Amer. Math. Soc. 352, 2937 (2000), DOI: 10.1090/S0002-9947-00-02311-4. Crossref, Web of Science, Google Scholar
I. Melbourne, Nonlinearity 4, 835 (1991), DOI: 10.1088/0951-7715/4/3/010. Crossref, Web of Science, Google Scholar
E. Mosekilde, O. G. D. Jensen and P. Borckmans, SAMS 19, 45 (1995). Web of Science, Google Scholar
A. F. Münster, M. Watzl and F. W. Schneider, Phys. Scr. T 67, 58 (1996). Google Scholar
M. Orbánet al., J. Phys. Chem. 103, 36 (1999). Crossref, Web of Science, Google Scholar
Q. Ouyang and H. L. Swinney, Nature 352, 610 (1991), DOI: 10.1038/352610a0. Crossref, Web of Science, Google Scholar
Parker, M. J. [2003] "Forced symmetry-breaking of Euclidean-equivariant partial differential equations, pattern formation and Turing instabilities," PhD thesis, University of Warwick . Google Scholar
M. J. Parker, I. N. Stewart and M. G. M. Gomes, J. Dyn. Diff. Eqs. 18, 223 (2006), DOI: 10.1007/s10884-005-9004-z. Crossref, Web of Science, Google Scholar
M. J. Parker, I. N. Stewart and M. G. M. Gomes, Int. J. Bifurcation and Chaos 18, 83 (2008). Link, Web of Science, Google Scholar
M. J. Parker, I. N. Stewart and M. G. M. Gomes, Dyn. Syst. 23, 137 (2008), DOI: 10.1080/14689360601070771. Crossref, Web of Science, Google Scholar
C. M. Postlethwaite and J. H. P. Dawes, Nonlinearity 18, 1477 (2005), DOI: 10.1088/0951-7715/18/4/004. Crossref, Web of Science, Google Scholar
G. W. Schwarz, Inst. Hautes Études Sci. Publ. Math. 37 (1980), DOI: 10.1007/BF02684776. Google Scholar
K. Staliunas, G. Slekys and C. O. Weiss, Phys. Rev. Lett. 79, 2658 (1997), DOI: 10.1103/PhysRevLett.79.2658. Crossref, Web of Science, Google Scholar
K. Staliunas, Phys. Rev. Lett. 81, 81 (1998), DOI: 10.1103/PhysRevLett.81.81. Crossref, Web of Science, Google Scholar
K. Staliunas and J. V. Sánchez-Morcillo, Opt. Commun. 177, 389 (2000), DOI: 10.1016/S0030-4018(00)00561-7. Crossref, Web of Science, Google Scholar
O. Steinbock, E. Kasper and S. Müller, J. Phys. Chem. A 103, 3442 (1999), DOI: 10.1021/jp990657a. Crossref, Web of Science, Google Scholar
A. M. Turing, Phil. Trans. R. Soc. London B 327, 37 (1952). Google Scholar
A. Vanderbauwhede , Local Bifurcation and Symmetry , Research Notes in Mathematics 75 ( Pitman , Boston, MA , 1982 ) . Google Scholar
M. A. Vorontsov and W. J. Firth, Phys. Rev. A 49, 2891 (1994), DOI: 10.1103/PhysRevA.49.2891. Crossref, Web of Science, Google Scholar
M. A. Vorontsov and A. Y. Karpov, J. Opt. Soc. Am. B 14, 34 (1997), DOI: 10.1364/JOSAB.14.000034. Crossref, Web of Science, Google Scholar
M. A. Vorontsov and B. A. Samson, Phys. Rev. A 57, 3040 (1998), DOI: 10.1103/PhysRevA.57.3040. Crossref, Web of Science, Google Scholar
D. Walgraef, G. Dewel and P. Borckmans, Adv. Chem. Phys. 49, 311 (1982), DOI: 10.1002/9780470142691.ch5. Crossref, Web of Science, Google Scholar
W. Zhang and J. Winñals, Phys. Rev. E 53, 4283 (1996), DOI: 10.1103/PhysRevE.53.R4283. Crossref, Web of Science, Google Scholar