Resonance bifurcation from homoclinic cycles

Journal of Differential Equations

Volumn 246, Issue 7, 2009, Pages 2681-2705

  Driesse, Ramon ^a   Homburg, Ale Jan ^a  

^a UNIVERSITY OF AMSTERDAM   (Netherlands)

Author keywords

Homoclinic cycle; Resonant bifurcation

Indexed keywords

EID: 60349112995     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2009.01.034     Document Type: Article

References (19)

1. Berger M.S. Nonlinearity and Functional Analysis (1977), Academic Press
2. Chossat P., Krupa M., Melbourne I., and Scheel A. Transverse bifurcations of homoclinic cycles. Phys. D 100 1-2 (1997) 85-100
3. Chossat P., and Lauterbach R. Methods in Equivariant Bifurcations and Dynamical Systems. Adv. Ser. Nonlinear Dynam. vol. 15 (2000), World Scientific Publishing Co. Inc., River Edge, NJ
4. Chow S.-N., Deng B., and Fiedler B. Homoclinic bifurcation at resonant eigenvalues. J. Dynam. Differential Equations 2 2 (1990) 177-244
5. 1-linearization, and homoclinic bifurcation. J. Differential Equations 79 2 (1989) 189-231
6. R. Driesse, A.J. Homburg, Essentially asymptotically stable homoclinic networks, preprint
7. Field M.J. Lectures on Bifurcations, Dynamics and Symmetry. Pitman Res. Notes Math. Ser. vol. 356 (1996), Longman, Harlow
8. Field M.J. Dynamics and Symmetry. ICP Adv. Texts Math. vol. 3 (2007), World Scientific, London
9. A.J. Homburg, A. Jukes, J. Knobloch, J.S.W. Lamb, Bifurcations from codimension one homoclinic cycles in equivariant flows, preprint
10. Homburg A.J., Kokubu H., and Naudot V. Homoclinic-doubling cascades. Arch. Ration. Mech. Anal. 160 3 (2001) 195-243
11. Krupa M. Robust heteroclinic cycles. J. Nonlinear Sci. 7 2 (1997) 129-176
12. Krupa M., and Melbourne I. Asymptotic stability of heteroclinic cycles in systems with symmetry. Ergodic Theory Dynam. Systems 15 1 (1995) 121-147
13. Krupa M., and Melbourne I. Asymptotic stability of heteroclinic cycles in systems with symmetry. II. Proc. Roy. Soc. Edinburgh Sect. A 134 6 (2004) 1177-1197
14. Ovsyannikov I.M., and Shil'nikov L.P. On systems with saddle-focus homoclinic curve. Mat. USSR Sb. 58 (1987) 557-574
15. Postlethwaite C.M., and Dawes J.H.P. A codimension-two resonant bifurcation from a heteroclinic cycle with complex eigenvalues. Dyn. Syst. 21 3 (2006) 313-336
16. Scheel A., and Chossat P. Bifurcation d'orbites périodiques à partir d'un cycle homocline symétrique. C. R. Acad. Sci. Paris Sér. I Math. 314 1 (1992) 49-54
17. 4 in the case of a symmetry group G ⊂ SO (4). C. R. Math. Acad. Sci. Paris 334 10 (2002) 859-864
18. 4. Nonlinearity 16 1 (2003) 1-24
19. Sottocornola N. Simple homoclinic cycles in low-dimensional spaces. J. Differential Equations 210 1 (2005) 135-154