Cascade of sinks

Author:
Clark Robinson

Journal:
Trans. Amer. Math. Soc. **288** (1985), 841-849

MSC:
Primary 58F12; Secondary 34D30

DOI:
https://doi.org/10.1090/S0002-9947-1985-0776408-8

MathSciNet review:
776408

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper it is proved that if a one-parameter family of dissipative maps in dimension two creates a new homoclinic intersection for a fixed point when the parameter , then there is a cascade of quasi-sinks, i.e., there are parameter values converging to such that, for , has a quasi-sink with each point in having period . A quasi-sink for a map is a closed set such that each point in is a periodic point and is a quasi-attracting set (à la Conley), i.e., is the intersection of attracting sets , where each has a neighborhood such that . Thus, the quasi-sinks are almost attracting sets made up entirely of points of period . Gavrilov and Silnikov, and later Newhouse, proved this result when the new homoclinic intersection is created nondegenerately. In this case the sets are single, isolated (differential) sinks. In an earlier paper we proved the degenerate case when the homoclinic intersections are of finite order tangency (or the family is real analytic), again getting a cascade of sinks, not just quasi-sinks.

**[1]**K. Alligood and J. Yorke,*Cascades of period-doubling bifurcations*:*a prerequisite for horseshoes*, Bull. Amer. Math. Soc. (N. S.)**9**(1983), 319-322. MR**714994 (85b:58089)****[2]**D. G. Aronson, M. A. Chory, G. R. Hall and R. P. McGehee,*Bifurcation from an invariant circle for two-parameter families of the plane*, Comm. Math. Phys.**83**(1982), 303-354. MR**649808 (83j:58078)****[3]**J. Carr,*Applications of centre manifold theory*, Springer-Verlag, 1981. MR**635782 (83g:34039)****[4]**C. Conley,*Isolated invariant sets and the Morse index*, CBMS Regional Conf. Ser. in Math., no. 38, Amer. Math. Soc., Providence, R.I., 1978. MR**511133 (80c:58009)****[5]**J. Franks,*Homology and dynamical systems*, CBMS Regional Conf. Ser. in Math., no. 49, Amer. Math. Soc., Providence, R.I., 1982. MR**669378 (84f:58067)****[6]**N. K. Gavrilov and L. P. Silnikov,*On the three dimensional dynamical system close to a system with a structurally unstable homoclinic curve*. I, II, Math. USSR Sb.**17**(1972), 467-485; ibid.**19**(1973), 139-156. MR**0334280 (48:12599)****[7]**V. Guillemin and A. Pollack,*Differential topology*, Prentice-Hall, Englewood Cliffs, N.J., 1974. MR**0348781 (50:1276)****[8]**M. Hirsch, C. Pugh and M. Shub,*Invariant manifolds*, Lecture Notes in Math., vol. 583, Springer-Verlag, 1977. MR**0501173 (58:18595)****[9]**M. Hurley,*Attractors*:*persistence and density of their basins*, Trans. Amer. Math. Soc.**269**(1982), 247-271. MR**637037 (83c:58049)****[10]**S. Newhouse,*The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms*, Inst. Hautes. Études Sci. Publ. Math.**50**(1979), 101-151. MR**556584 (82e:58067)****[11]**-,*Lectures on dynamical systems*, Progress in Math. vol. 8, Birkhäuser, 1980, pp. 1-114.**[12]**C. Robinson,*Bifurcation to infinitely many sinks*, Comm. Math. Phys.**90**(1983), 433-459. MR**719300 (84k:58169)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58F12,
34D30

Retrieve articles in all journals with MSC: 58F12, 34D30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0776408-8

Article copyright:
© Copyright 1985
American Mathematical Society