From Morse-Smale to all knots and links

Nonlinearity

Volumn 11, Issue 4, 1998, Pages 1111-1125

  Ghrist, Robert ^a   Young, Todd ^b  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b NORTHWESTERN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0042722421     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/11/4/021     Document Type: Article

References (34)

1. Arnol'd V I, Afraimovich V S, Il'yashenko Y S and Shil'nikov L P 1986 The theory of bifurcations Modern Problems of Mathematics, Fundamental Directions. VINITI (Moscow: Nauka) (Engl. transl. 1993 Encyclopedia of Mathematical Sciences vol 5 (Berlin: Springer))
2. Birman J and Williams R 1983 Knotted periodic orbits in dynamical systems-I. Lorenz's equations Topology 22 47-82
3. Birman J and Williams R 1983 Knotted periodic orbits in dynamical systems-II. Knot holders for fibered knots Cont. Math. 20 1-60
4. Burchard A, Deng B and Lu K 1992 Smooth conjugacy of centre manifolds Proc. R. Soc. A 120 61-77
5. Burde G and Zieschang H 1985 Knots (Berlin: De Gruyter)
6. 3 Nonlinearity 10 1339-55
7. Casasayas J, Martinez Alfaro J and Nunes A 1993 Knotted periodic orbits and integrability Hamiltonian Systems and Celestial Mechanics (Guanajuato 1991) (Adv. Ser. Nonlinear Dynam. 4) (River Edge, NJ: World Science) 35-44
8. Champneys A, Harterich J and Sandstede B 1998 A nontransverse homoclinic orbit to a saddle-node equalibrium Preprint
9. Fenichel N 1971 Persistence and smoothness of invariant manifolds for flows Indiana Univ. Math. J. 21 193-226
10. Franks J and Williams R F 1985 Entropy and knots Trans. Am. Math. Soc. 291 241-53
11. 3 supporting all links as closed orbits Electron. Res. Ann. AMS 191-7
12. Ghrist R 1997 Branched two-manifolds supporting all links Topology 36 423-47
13. Ghrist R and Holmes P 1996 An ODE whose solutions contain all knots and links Int. J. Bif. Chaos 6 779-800
14. Ghrist R, Holmes: P and Sullivan M 1997 Knots and links in three-dimensional flows Springer Lecture Notes in Mathematics vol 1654 (Berlin: Springer)
15. Guckenheimer J and Holmes P 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (New York: Springer)
16. Holmes P 1986 Knotted periodic orbits in suspensions of Smale's horseshoe period mutiplying and cabled knots Physica D 21 7-41
17. Holmes P 1987 Knotted periodic orbits in suspensions of annulus maps Proc. R. Soc. A 411 351-78
18. Holmes P and R F Williams 1985 Knotted periodic orbits in suspensions of Smale's horseshoe torus knots and bifurcation sequences Arch. Ration. Mech. Anal. 90 115-93
19. Il'yashenko Yu S and Li C Z 1998 Preprint
20. Lorenz E 1963 Deterministic nonperiodic flow J. Atmos. Sci. 20- 130-41
21. Morgan J 1978 Nonsingular Morse-Smale flows on 3-dimensional manifolds Topology 18 41-54
22. Newhouse S, Palis J and Takens F 1983 Bifurcations and stability of families of diffeomorphisms Publ. Math. IHES 57 5-71
23. Palis J and Takens F 1993 Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations (Cambridge: Cambridge University Press)
24. Pomeau Y and Manneville P 1980 Intermittent transition to turbulence in dissipative dynamical systems Commun. Math. Phys. 74 189-97
25. Robinson C 1989 Homoclinic bifurcation to a transitive attractor of Lorenz type, I Nonlinearity 2 495-518
26. Robinson C 1992 Homoclinic bifurcation to a transitive attractor of Lorenz type, II SIAM J. Appl. Math. Anal. 23 1255-68
27. Robinson C 1995 Dynamical Systems Stability, Symbolic Dynamics, and Chaos (Ann Arbor, MI: Chemical Rubber Company)
28. Rolfsen D 1977 Knots and Links (Berkeley, CA: Publish or Perish)
29. Rychlik M 1989 Lorenz attractors through Silnikov-type bifurcation I Ergod. Theor. Dynam. Syst. 10 793-821
30. Shil'nikov L P 1969 A new type of bifurcation of many-dimensional dynamical systems Dokl. Akad. Nauk SSSR 189 59-62
31. Takens F 1971 Partially hyperbolic fixed points Topology 10 133-47
32. 3 J. Math. Soc. Japan 41 405-13
33. Williams R 1997 The universal templates of Ghrist Bull. AMS to be published
34. Young T 1995 Partially hyperbolic sets from a co-dimension one bifurcation Discrete Continuous Dynam. Syst. 1 253-75