Global structural stability of a saddle node bifurcation

Author:
Clark Robinson

Journal:
Trans. Amer. Math. Soc. **236** (1978), 155-171

MSC:
Primary 58F10

DOI:
https://doi.org/10.1090/S0002-9947-1978-0467832-8

MathSciNet review:
0467832

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: S. Newhouse, J. Palis, and F. Takens have recently proved the global structural stability of a one parameter unfolding of a saddle node when the nonwandering set is finite and transversality conditions are satisfied. (The diffeomorphism is Morse-Smale except for the saddle node.) Using their local unfolding of a saddle node and our method of compatible families of unstable disks (instead of the more restrictive method of compatible systems of unstable tubular families), we are able to extend one of their results to the case where the nonwandering set is infinite. We assume that a saddle node is introduced away from the rest of the nonwandering set which is hyperbolic (Axiom A), and that a (strong) transversality condition is satisfied.

**[1]**P. Brunovsky,*On one-parameter families of diffeomorphisms*, Comment. Math. Univ. Carolinae**11**(1970), 559-582. MR**43**#5548. MR**0279827 (43:5548)****[2]**N. Fenichel,*Asymptotic stability with rate conditions*, Indiana Univ. Math. J.**23**(1973/74), 1109-1137. MR**49**#4036. MR**0339276 (49:4036)****[3]**M. Hirsch, C. Pugh and M. Shub,*Invariant manifolds*, Lecture Notes in Math., Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR**0501173 (58:18595)****[4]**S. Newhouse and J. Palis,*Bifurcations of Morse-Smale dynamical systems*, Dynamical Systems, (Proc. Sympos., Univ. of Bahia, Salvador, 1971), Academic Press, New York, 1973, pp. 303-366. MR**48**#12600. MR**0334281 (48:12600)****[5]**-,*Cycles and bifurcation theory*, Asterisque, Soc. Math. France**31**(1976), 43-140. MR**0516408 (58:24366)****[6]**S. Newhouse, J. Palis and F. Takens,*Stable arcs of diffeomorphisms*, Bull. Amer. Math. Soc.**82**(1976), 499-502. MR**0402826 (53:6640)****[7]**Z. Nitecki,*Differentiable dynamics*, M.I.T. Press, Cambridge, Mass., 1971. MR**0649788 (58:31210)****[8]**J. Palis,*On Morse-Smale dynamical systems*, Topology**8**(1969), 385-404. MR**39**#7620. MR**0246316 (39:7620)****[9]**J. Palis and S. Smale,*Structural stability theorems*, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R.I., 1970, pp. 223-232. MR**42**#2505. MR**0267603 (42:2505)****[10]**J. Robbin,*Topological conjugacy and structural stability for discrete dynamical systems*, Bull. Amer. Math. Soc.**78**(1972), 923-952. MR**47**#1086. MR**0312529 (47:1086)****[11]**C. Robinson,*Structural stability of**diffeomorphisms*, J. Differential Equations**22**(1976), 28-73. MR**0474411 (57:14051)****[12]**-,*The geometry of the structural stability proof using unstable disks*, Bol. Soc. Brasil. (to appear).**[13]**S. Smale,*Differentiable dynamical systems*, Bull. Amer. Math. Soc.**73**(1967), 747-817. MR**37**#3598;**39**, p. 1593. MR**0228014 (37:3598)****[14]**J. Sotomayor,*Generic one-parameter families of vector fields on two-dimensional manifolds*, Inst. Hautes Études Sci. Publ. Math. No. 43 (1974), 5-46. MR**49**#4039. MR**0339279 (49:4039)****[15]**J. Hale,*Ordinary differential equations*, Interscience, New York, 1969. MR**0419901 (54:7918)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58F10

Retrieve articles in all journals with MSC: 58F10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0467832-8

Keywords:
Structural stability,
bifurcation,
saddle node

Article copyright:
© Copyright 1978
American Mathematical Society