Bifurcation from codimension one relative homoclinic cycles

Homburg, Ale; Jukes, Alice; Knobloch, Jürgen; Lamb, Jeroen

doi:10.1090/S0002-9947-2011-05193-7

Bifurcation from codimension one relative homoclinic cycles
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by Ale Jan Homburg, Alice C. Jukes, Jürgen Knobloch and Jeroen S.W. Lamb

Trans. Amer. Math. Soc. 363 (2011), 5663-5701

DOI: https://doi.org/10.1090/S0002-9947-2011-05193-7

Published electronically: June 20, 2011

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Abstract:

We study bifurcations of relative homoclinic cycles in flows that are equivariant under the action of a finite group. The relative homoclinic cycles we consider are not robust, but have codimension one. We assume real leading eigenvalues and connecting trajectories that approach the equilibria along leading directions. We show how suspensions of subshifts of finite type generically appear in the unfolding. Descriptions of the suspended subshifts in terms of the geometry and symmetry of the connecting trajectories are provided.

References

Manuela A. D. Aguiar, Sofia B. S. D. Castro, and Isabel S. Labouriau, Dynamics near a heteroclinic network, Nonlinearity 18 (2005), no. 1, 391–414. With multimedia enhancement available from the abstract page in the online journal. MR 2109482, DOI 10.1088/0951-7715/18/1/019
Peter Ashwin and Michael Field, Heteroclinic networks in coupled cell systems, Arch. Ration. Mech. Anal. 148 (1999), no. 2, 107–143. MR 1716564, DOI 10.1007/s002050050158
S. Bochner, Compact groups of differentiable transformations, Ann. of Math. (2) 46 (1945), 372–381. MR 13161, DOI 10.2307/1969157
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
Pascal Chossat and Reiner Lauterbach, Methods in equivariant bifurcations and dynamical systems, Advanced Series in Nonlinear Dynamics, vol. 15, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1831950, DOI 10.1142/4062
Shui-Nee Chow, Bo Deng, and Bernold Fiedler, Homoclinic bifurcation at resonant eigenvalues, J. Dynam. Differential Equations 2 (1990), no. 2, 177–244. MR 1050642, DOI 10.1007/BF01057418
Bo Deng, The Šil′nikov problem, exponential expansion, strong $\lambda$-lemma, $C^1$-linearization, and homoclinic bifurcation, J. Differential Equations 79 (1989), no. 2, 189–231. MR 1000687, DOI 10.1016/0022-0396(89)90100-9
G. L. dos Reis, Structural stability of equivariant vector fields on two-manifolds, Trans. Amer. Math. Soc. 283 (1984), no. 2, 633–643. MR 737889, DOI 10.1090/S0002-9947-1984-0737889-8
M. J. Field, Equivariant dynamical systems, Trans. Amer. Math. Soc. 259 (1980), no. 1, 185–205. MR 561832, DOI 10.1090/S0002-9947-1980-0561832-4
Michael Field, Lectures on bifurcations, dynamics and symmetry, Pitman Research Notes in Mathematics Series, vol. 356, Longman, Harlow, 1996. MR 1425388
Michael J. Field, Dynamics and symmetry, ICP Advanced Texts in Mathematics, vol. 3, Imperial College Press, London, 2007. MR 2358722, DOI 10.1142/9781860948541
J.-M. Gambaudo, P. Glendinning, and C. Tresser, The gluing bifurcation. I. Symbolic dynamics of the closed curves, Nonlinearity 1 (1988), no. 1, 203–214. MR 928953
Martin Golubitsky, Ian Stewart, and David G. Schaeffer, Singularities and groups in bifurcation theory. Vol. II, Applied Mathematical Sciences, vol. 69, Springer-Verlag, New York, 1988. MR 950168, DOI 10.1007/978-1-4612-4574-2
M. Golubitsky, J. W. Swift, and E. Knobloch, Symmetries and pattern selection in Rayleigh-Bénard convection, Phys. D 10 (1984), no. 3, 249–276. MR 763472, DOI 10.1016/0167-2789(84)90179-9
John Guckenheimer and Philip Holmes, Structurally stable heteroclinic cycles, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 1, 189–192. MR 913462, DOI 10.1017/S0305004100064732
Ale Jan Homburg, Global aspects of homoclinic bifurcations of vector fields, Mem. Amer. Math. Soc. 121 (1996), no. 578, viii+128. MR 1327210, DOI 10.1090/memo/0578
Ale Jan Homburg, Alice C. Jukes, Jürgen Knobloch, and Jeroen S. W. Lamb, Saddle-nodes and period-doublings of Smale horseshoes: a case study near resonant homoclinic bellows, Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 5, Dynamics in perturbations, 833–850. MR 2484136
Ale Jan Homburg and Jürgen Knobloch, Multiple homoclinic orbits in conservative and reversible systems, Trans. Amer. Math. Soc. 358 (2006), no. 4, 1715–1740. MR 2186994, DOI 10.1090/S0002-9947-05-03793-1
Ale Jan Homburg, Hiroshi Kokubu, and Martin Krupa, The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit, Ergodic Theory Dynam. Systems 14 (1994), no. 4, 667–693. MR 1304138, DOI 10.1017/S0143385700008117
A. C. Jukes. On homoclinic bifurcations with symmetry. Ph.D. thesis, Imperial College, 2006.
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
Vivien Kirk and Mary Silber, A competition between heteroclinic cycles, Nonlinearity 7 (1994), no. 6, 1605–1621. MR 1304441
Masashi Kisaka, Hiroshi Kokubu, and Hiroe Oka, Bifurcations to $N$-homoclinic orbits and $N$-periodic orbits in vector fields, J. Dynam. Differential Equations 5 (1993), no. 2, 305–357. MR 1223451, DOI 10.1007/BF01053164
J. Knobloch, Lin’s method for discrete dynamical systems, J. Differ. Equations Appl. 6 (2000), no. 5, 577–623. MR 1802448, DOI 10.1080/10236190008808247
J. Knobloch. Lin’s method for discrete and continuous dynamical systems and applications. Habilitation thesis, Technische Universität Ilmenau, 2004.
M. Krupa, Robust heteroclinic cycles, J. Nonlinear Sci. 7 (1997), no. 2, 129–176. MR 1437986, DOI 10.1007/BF02677976
Martin Krupa and Ian Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry, Ergodic Theory Dynam. Systems 15 (1995), no. 1, 121–147. MR 1314972, DOI 10.1017/S0143385700008270
Martin Krupa and Ian Melbourne, Nonasymptotically stable attractors in $\textrm {O}(2)$ mode interactions, Normal forms and homoclinic chaos (Waterloo, ON, 1992) Fields Inst. Commun., vol. 4, Amer. Math. Soc., Providence, RI, 1995, pp. 219–232. MR 1350551, DOI 10.1017/s0143385700008270
Martin Krupa and Ian Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry. II, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 6, 1177–1197. MR 2107489, DOI 10.1017/S0308210500003693
Xiao-Biao Lin, Using Mel′nikov’s method to solve Šilnikov’s problems, Proc. Roy. Soc. Edinburgh Sect. A 116 (1990), no. 3-4, 295–325. MR 1084736, DOI 10.1017/S0308210500031528
Karsten Matthies, A subshift of finite type in the Takens-Bogdanov bifurcation with $D_3$ symmetry, Doc. Math. 4 (1999), 463–485. MR 1713483
Ian Melbourne, An example of a nonasymptotically stable attractor, Nonlinearity 4 (1991), no. 3, 835–844. MR 1124335
Jacob Palis Jr. and Welington de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR 669541
B. Sandstede. Verzweigungstheorie homokliner Verdopplungen. Ph.D. thesis, Freie Universität Berlin, 1993.
L. P. Shil′nikov. On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type. Math. USSR Sbornik, 6:427–438, 1968.
Nicola Sottocornola, Simple homoclinic cycles in low-dimensional spaces, J. Differential Equations 210 (2005), no. 1, 135–154. MR 2114127, DOI 10.1016/j.jde.2004.10.023
D. V. Turaev, Bifurcations of a homoclinic “figure eight” of a multidimensional saddle, Uspekhi Mat. Nauk 43 (1988), no. 5(263), 223–224 (Russian); English transl., Russian Math. Surveys 43 (1988), no. 5, 264–265. MR 971495, DOI 10.1070/RM1988v043n05ABEH001952
André Vanderbauwhede and Bernold Fiedler, Homoclinic period blow-up in reversible and conservative systems, Z. Angew. Math. Phys. 43 (1992), no. 2, 292–318. MR 1162729, DOI 10.1007/BF00946632
Ferdinand Verhulst, Nonlinear differential equations and dynamical systems, 2nd ed., Universitext, Springer-Verlag, Berlin, 1996. Translated from the 1985 Dutch original. MR 1422255, DOI 10.1007/978-3-642-61453-8