Saddle-nodes and period-doublings of Smale horseshoes: A case study near resonant homoclinic bellows

Bulletin of the Belgian Mathematical Society - Simon Stevin

Volumn 15, Issue 5, 2008, Pages 833-850

  Homburg, Ale Jan ^a   Jukes, Alice C ^b   Knobloch, Jürgen ^c   Lamb, Jeroen S W ^b  

^a UNIVERSITY OF AMSTERDAM   (Netherlands)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

^c ILMENAU UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Bifurcation; Homoclinic loop; Horseshoe

Indexed keywords

EID: 58249103073     PISSN: 13701444     EISSN: None     Source Type: Journal    
DOI: 10.36045/bbms/1228486411     Document Type: Article

References (21)

1. S.-N. Chow, B. Deng, and B. Fiedler. Homoclinic bifurcation at resonant eigenvalues. J. Dynam. Differential Equations, 2(2): 177-244, 1990.
2. S.-N. Chow and J. K. Hale. Methods of bifurcation theory, volume 251 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science]. Springer-Verlag, New York, 1982.
3. R. W. Ghrist. Resonant gluing bifurcations. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10(9):2141-2160, 2000.
4. J. Guckenheimer and P. J. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, volume 42 of Applied Mathematical Sciences. Springer-Verlag, New York, 1990. Revised and corrected reprint of the 1983 original.
5. A. J. Homburg. Global aspects of homoclinic bifurcations of vector fields. Mem. Amer. Math. Soc., 121(578):viii+128, 1996.
6. A. J. Homburg and J. Knobloch. Multiple homoclinic orbits in conservative and reversible systems. Trans. Amer. Math. Soc., 358(4):1715-1740, 2006.
7. A. J. Homburg, H. Kokubu, and M. Krupa. The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit. Ergodic Theory Dynam. Systems, 14(4):667-693, 1994.
8. A. C. Jukes. On homoclinic bifurcations with symmetry. PhD thesis, Imperial College, 2006.
9. M. Kisaka, H. Kokubu, and H. Oka. Bifurcations to N-homoclinic orbits and N-periodic orbits in vector fields. J. Dynam. Differential Equations, 5(2):305-357, 1993.
10. J. Knobloch. Lin's method for discrete and continuous dynamical systems and applications. 2004.
11. Yu. A. Kuznetsov. Elements of applied bifurcation theory, volume 112 of Applied Mathematical Sciences. Springer-Verlag, New York, third edition, 2004.
12. X.-B. Lin. Using Melnikovs method to solve Shil'nikovs problems. Proc. Roy. Soc. Edinburgh, 116A:295 - 325, 1990.
13. J. Palis and F. Takens. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, volume 35 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1993. Fractal dimensions and infinitely many attractors.
14. R. C. Robinson. Homoclinic bifurcation to a transitive attractor of Lorenz type. Nonlinearity, 2:495-518, 1989.
15. R. C. Robinson. Homoclinic bifurcation to a transitive attractor of Lorenz type. II. SIAM J. Math. Anal., 23:1255 - 1268, 1992.
16. B. Sandstede. Verzweigungstheorie homokliner Verdopplungen. PhD thesis, Free University Berlin, 1993.
17. B. Sandstede. Center manifolds for homoclinic solutions. J. Dynam. Differential Equations, 12(3):449-510, 2000.
18. J. Sotomayor. Generic bifurcations of dynamical systems. In Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pages 561-582. Academic Press, New York, 1973.
19. D. V. Turaev. Bifurcations of a homoclinic "figure eight" of a multidimensional saddle. Russian Math. Surveys, 43:264 - 265, 1988.
20. D. V. Turaev. On dimension of non-local bifurcational problems. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6(5):919-948, 1996.
21. A. Vanderbauwhede and B. Fiedler. Homoclinic period blow-up in reversible and conservative systems. Z. angew. Math. Phys., 43:292 - 318, 1992.