On the Omega Limit Sets for Analytic Flows

  • Choy, Jaeyoo ;
  • Chu, Hahng-Yun
  • Received : 2014.02.13
  • Accepted : 2014.05.22
  • Published : 2014.06.23


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.


attractors;${\omega}$-limit sets;analytic flows


  1. J. Milnor, On the concept of attractor, Commun. Math. Phy., 99(1978), 177-195.
  2. J. C. Robinson and O. M. Tearne, Boundaries of attractors of omega limit sets, Stoch. Dyn., 5(2005), 97-109.
  3. A. Seidenberg, Reduction of singularities of the differential equation Ady = Bdx. Amer. J. Math., 90(1968), 248-269.
  4. 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.
  5. 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.
  6. J. Choy and H.-Y. Chu, Attractors on Riemann spheres, submitted.
  7. J. Choy and H.-Y. Chu, On the Envelopes of Homotopies, Kyungpook Math. J., 49(3)(2009), 573-582
  8. C. M. Carballo and C. A. Morales, Omega-limit sets close to singular-hyperbolic at-tractors, Illinois J. Math., 48(2004), 645-663.
  9. C. Conley, Isolated invariant sets and the morse index, C. B. M. S. Regional Lect., 38, A. M. S., 1978
  10. F. Dumortier, Singularities of vector fields on the plane, J. Differential Equations, 23(1977), 53-106.
  11. F. Rodriguez Hertz and J. Rodriguez Hertz, Expansive attractors on surfaces, Ergodic Theory Dynam. Systems, 26(2006), 291-302.
  12. 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.
  13. V. Jimenez Lopez and D. Peralta-Salas, Global attractors of analytic plane flows, To appear in Ergodic Theory Dynam. Systems.
  14. 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.

Cited by



Supported by : National Research Foundation of Korea(NRF)