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Attractors and orbit-flip homoclinic orbits for star flows


Author: C. A. Morales
Journal: Proc. Amer. Math. Soc. 141 (2013), 2783-2791
MSC (2010): Primary 37D20; Secondary 37C10
DOI: https://doi.org/10.1090/S0002-9939-2013-11535-2
Published electronically: April 12, 2013
MathSciNet review: 3056568
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Abstract: We study star flows on closed $ 3$-manifolds and prove that they either have a finite number of attractors or can be $ C^1$ approximated by vector fields with orbit-flip homoclinic orbits.


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  • 1. Afraimovich, V. S., Bykov, V. V., Shilnikov, L. P., On attracting structurally unstable limit sets of Lorenz attractor type (Russian), Trudy Moskov. Mat. Obshch. 44 (1982), 150-212. MR 656286 (84a:58058)
  • 2. Aoki, N., The set of Axiom A diffeomorphisms with no cycles, Bol. Soc. Brasil. Mat. (N.S.) 23 (1992), no. 1-2, 21-65. MR 1203172 (94d:58080)
  • 3. Champneys, A. R., Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics, Time-reversal symmetry in dynamical systems (Coventry, 1996). Phys. D 112 (1998), no. 1-2, 158-186. MR 1605836 (99b:58169)
  • 4. Doering, C. I., Persistently transitive vector fields on three-dimensional manifolds, Dynamical systems and bifurcation theory (Rio de Janeiro, 1985), 59-89, Pitman Res. Notes Math. Ser., 160, Longman Sci. Tech., Harlow, 1987. MR 907891 (89c:58111)
  • 5. Gan, S., Wen, L., Nonsingular star flows satisfy Axiom A and the no-cycle condition, Invent. Math. 164 (2006), no. 2, 279-315. MR 2218778 (2007j:37045)
  • 6. Gan, S., Li, M., Wen, L., Robustly transitive singular sets via approach of an extended linear Poincaré flow, Discrete Contin. Dyn. Syst. 13 (2005), no. 2, 239-269. MR 2152388 (2006b:37056)
  • 7. Guckenheimer, J., A strange, strange attractor, The Hopf bifurcation and its applications, Applied Mathematical Series, 19, Springer-Verlag, 1976. MR 0494309 (58:13209)
  • 8. Guckenheimer, J., Williams, R., Structural stability of Lorenz attractors, Publ. Math. IHES 50 (1979), 59-72. MR 556582 (82b:58055a)
  • 9. Hasselblatt, B., Katok, A., Introduction to the modern theory of dynamical systems. With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. MR 1326374 (96c:58055)
  • 10. Hayashi, S., Connecting invariant manifolds and the solution of the $ C^1$ stability and $ \Omega $-stability conjectures for flows, Ann. of Math. (2) 145 (1997), no. 1, 81-137. MR 1432037 (98b:58096)
  • 11. Hayashi, S., Diffeomorphisms in $ \mathcal {F}^1(M)$ satisfy Axiom A, Ergodic Theory Dynam. Systems 12 (1992), no. 2, 233-253. MR 1176621 (94d:58081)
  • 12. Hirsch, M., Pugh, C., Shub, M., Invariant manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977. MR 0501173 (58:18595)
  • 13. Homburg, A. J., Krauskopf, B., Resonant homoclinic flip bifurcations, J. Dynam. Differential Equations 12 (2000), no. 4, 807-850. MR 1826963 (2002h:37089)
  • 14. Kokubu, H., Komuro, M., Oka, H., Multiple homoclinic bifurcations from orbit-flip. I. Successive homoclinic doublings, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 6 (1996), no. 5, 833-850. MR 1404121 (97e:58168)
  • 15. Liao, S., Qualitative theory of differentiable dynamical systems. Translated from the Chinese. With a preface by Min-de Cheng. Science Press, Beijing; distributed by American Mathematical Society, Providence, RI, 1996. MR 1449640 (98g:58041)
  • 16. Lopes, A. O., Structural stability and hyperbolic attractors, Trans. Amer. Math. Soc. 252 (1979), 205-219. MR 534118 (80j:58046)
  • 17. Mañé, R., A proof of the $ C^1$ stability conjecture, Inst. Hautes Études Sci. Publ. Math., No. 66 (1988), 161-210. MR 932138 (89e:58090)
  • 18. Mañé, R., Contributions to the stability conjecture, Topology 17 (1978), no. 4, 383-396. MR 516217 (84b:58061)
  • 19. Morales, C. A., The explosion of singular-hyperbolic attractors, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 577-591. MR 2054194 (2005d:37071)
  • 20. Morales, C. A., Pacifico, M. J., A dichotomy for three-dimensional vector fields, Ergodic Theory Dynam. Systems 23 (2003), no. 5, 1575-1600. MR 2018613 (2005a:37030)
  • 21. Morales, C. A., Pacifico, M. J., Lyapunov stability of $ \omega $-limit sets, Discrete Contin. Dyn. Syst. 8 (2002), no. 3, 671-674. MR 1897874 (2003b:37024)
  • 22. Morales, C. A., Pacifico, M. J., Inclination-flip homoclinic orbits arising from orbit-flip, Nonlinearity 14 (2001), no. 2, 379-393. MR 1819803 (2001m:37103)
  • 23. Naudot, V., A strange attractor in the unfolding of an orbit-flip homoclinic orbit, Dyn. Syst. 17 (2002), no. 1, 45-63. MR 1888697 (2002m:37032)
  • 24. Palis, J., Open questions leading to a global perspective in dynamics, Nonlinearity 21 (2008), no. 4, T37-T43. MR 2399817 (2009i:37003)
  • 25. Pliss, V. A., A hypothesis due to Smale, Differencial'nye Uravnenija 8 (1972), 268-282. MR 0299909 (45:8957)
  • 26. Pugh, C., An improved closing lemma and a general density theorem, Amer. J. Math. 89 (1967), 1010-1021. MR 0226670 (37:2257)
  • 27. Ruelle, D., Takens, F., On the nature of turbulence, Comm. Math. Phys. 20 (1971), 167-192. MR 0284067 (44:1297)
  • 28. Shilnikov, L. P., Shilnikov, A. L., Turaev, D., Chua, L., Methods of qualitative theory in nonlinear dynamics. Part II, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 5. World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR 1884710 (2003d:37002)
  • 29. Turaev, D., Multi-pulse homoclinic loops in systems with a smooth first integral, Ergodic theory, analysis, and efficient simulation of dynamical systems, 691-716, Springer, Berlin, 2001. MR 1850326 (2002f:37094)
  • 30. Wen, L., On the preperiodic set, Discrete Contin. Dynam. Systems 6 (2000), no. 1, 237-241. MR 1739926 (2001g:37035)
  • 31. Wen, L., On the $ C^1$ stability conjecture for flows, J. Differential Equations 129 (1996), no. 2, 334-357. MR 1404387 (97j:58082)

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Additional Information

C. A. Morales
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970 Rio de Janeiro, Brazil
Email: morales@impa.br

DOI: https://doi.org/10.1090/S0002-9939-2013-11535-2
Keywords: Star flow, attractor, orbit-flip homoclinic orbit
Received by editor(s): July 28, 2001
Received by editor(s) in revised form: November 3, 2011
Published electronically: April 12, 2013
Additional Notes: The author was partially supported by CNPq, FAPERJ and PRONEX-Brazil.
Communicated by: Bryna Kra
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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