Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

On the Omega Limit Sets for Analytic Flows

  • Choy, Jaeyoo (Department of Mathematics, Kyungpook National University) ;
  • Chu, Hahng-Yun (Department of Mathematics, Chungnam National University)
  • Received : 2014.02.13
  • Accepted : 2014.05.22
  • Published : 2014.06.23
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we describe the characterizations of omega limit sets (= ${\omega}$-limit set) on $\mathbb{R}^2$ in detail. For a local real analytic flow ${\Phi}$ by z' = f(z) on $\mathbb{R}^2$, we prove the ${\omega}$-limit set from the basin of a given attractor is in the boundary of the attractor. Using the result of Jim$\acute{e}$nez-L$\acute{o}$pez and Llibre [9], we can completely understand how both the attractors and the ${\omega}$-limit sets from the basin.

Keywords

References

  1. A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Qualitative theory of second-order dynamic systems, Translated from the Russian by D. Louvish. Halsted Press (A division of John Wiley & Sons), New York-Toronto, Ont., Jerusalem-London, 1973.
  2. V. I. Arnol'd and Y. S. Il'yashenko, Ordinary differential equations, [Current problems in mathematics. Fundamental directions, Vol. 1, 7-149, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985; MR0823489 (87e:34049)]. Translated from the Russian by E. R. Dawson and D. O'Shea. Encyclopaedia Math. Sci., 1, Dynamical systems, I, 1-148, Springer, Berlin, 1988.
  3. J. Choy and H.-Y. Chu, Attractors on Riemann spheres, submitted.
  4. J. Choy and H.-Y. Chu, On the Envelopes of Homotopies, Kyungpook Math. J., 49(3)(2009), 573-582 https://doi.org/10.5666/KMJ.2009.49.3.573
  5. C. M. Carballo and C. A. Morales, Omega-limit sets close to singular-hyperbolic at-tractors, Illinois J. Math., 48(2004), 645-663.
  6. C. Conley, Isolated invariant sets and the morse index, C. B. M. S. Regional Lect., 38, A. M. S., 1978
  7. F. Dumortier, Singularities of vector fields on the plane, J. Differential Equations, 23(1977), 53-106. https://doi.org/10.1016/0022-0396(77)90136-X
  8. F. Rodriguez Hertz and J. Rodriguez Hertz, Expansive attractors on surfaces, Ergodic Theory Dynam. Systems, 26(2006), 291-302.
  9. V. Jimenez Lopez and J. Llibre, A topological characterization of the !-limit sets for analytic flows on the plane, the sphere and the projective plane, Adv. Math., 216(2007), 677-710. https://doi.org/10.1016/j.aim.2007.06.007
  10. V. Jimenez Lopez and D. Peralta-Salas, Global attractors of analytic plane flows, To appear in Ergodic Theory Dynam. Systems.
  11. C. A. Morales, M. J. Pacifico and E. R. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. of Math., 160(2004), 375-432. https://doi.org/10.4007/annals.2004.160.375
  12. J. Milnor, On the concept of attractor, Commun. Math. Phy., 99(1978), 177-195.
  13. J. C. Robinson and O. M. Tearne, Boundaries of attractors of omega limit sets, Stoch. Dyn., 5(2005), 97-109. https://doi.org/10.1142/S0219493705001304
  14. A. Seidenberg, Reduction of singularities of the differential equation Ady = Bdx. Amer. J. Math., 90(1968), 248-269. https://doi.org/10.2307/2373435

Cited by

  1. A TOPOLOGICAL CHARACTERIZATION OF ­Ω-LIMIT SETS ON DYNAMICAL SYSTEMS vol.27, pp.3, 2014, https://doi.org/10.14403/jcms.2014.27.3.523