## Cascade of sinks

HTML articles powered by AMS MathViewer

- by Clark Robinson
- Trans. Amer. Math. Soc.
**288**(1985), 841-849 - DOI: https://doi.org/10.1090/S0002-9947-1985-0776408-8
- PDF | Request permission

## Abstract:

In this paper it is proved that if a one-parameter family $\{ {F_t}\}$ of ${C^1}$ dissipative maps in dimension two creates a new homoclinic intersection for a fixed point ${P_t}$ when the parameter $t = {t_0}$, then there is a cascade of quasi-sinks, i.e., there are parameter values ${t_n}$ converging to ${t_0}$ such that, for $t = {t_n}$, ${F_t}$ has a quasi-sink ${A_n}$ with each point $q$ in ${A_n}$ having period $n$. A quasi-sink ${A_n}$ for a map $F$ is a closed set such that each point $q$ in ${A_n}$ is a periodic point and ${A_n}$ is a quasi-attracting set (à la Conley), i.e., ${A_n}$ is the intersection of attracting sets $A_n^j, {A_n} = { \cap _j}A_n^j$, where each $A_n^j$ has a neighborhood $U_n^j$ such that $\cap \{ {F^k}(U_n^j):k \geqslant 0\} = A_n^j$. Thus, the quasi-sinks ${A_n}$ are almost attracting sets made up entirely of points of period $n$. Gavrilov and Silnikov, and later Newhouse, proved this result when the new homoclinic intersection is created nondegenerately. In this case the sets ${A_n}$ 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.## References

- James A. Yorke and Kathleen T. Alligood,
*Cascades of period-doubling bifurcations: a prerequisite for horseshoes*, Bull. Amer. Math. Soc. (N.S.)**9**(1983), no. 3, 319–322. MR**714994**, DOI 10.1090/S0273-0979-1983-15191-1 - D. G. Aronson, M. A. Chory, G. R. Hall, and R. P. McGehee,
*Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study*, Comm. Math. Phys.**83**(1982), no. 3, 303–354. MR**649808** - Jack Carr,
*Applications of centre manifold theory*, Applied Mathematical Sciences, vol. 35, Springer-Verlag, New York-Berlin, 1981. MR**635782** - Charles Conley,
*Isolated invariant sets and the Morse index*, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR**511133** - John M. Franks,
*Homology and dynamical systems*, CBMS Regional Conference Series in Mathematics, vol. 49, Published for the Conference Board of the Mathematical Sciences, Washington, D.C. by the American Mathematical Society, Providence, R.I., 1982. MR**669378** - N. K. Gavrilov and L. P. Šil′nikov,
*Three-dimensional dynamical systems that are close to systems with a structurally unstable homoclinic curve. II*, Mat. Sb. (N.S.)**90(132)**(1973), 139–156, 167 (Russian). MR**0334280** - Victor Guillemin and Alan Pollack,
*Differential topology*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR**0348781** - M. W. Hirsch, C. C. Pugh, and M. Shub,
*Invariant manifolds*, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR**0501173** - Mike Hurley,
*Attractors: persistence, and density of their basins*, Trans. Amer. Math. Soc.**269**(1982), no. 1, 247–271. MR**637037**, DOI 10.1090/S0002-9947-1982-0637037-7 - Sheldon E. 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**
—, - Clark Robinson,
*Bifurcation to infinitely many sinks*, Comm. Math. Phys.**90**(1983), no. 3, 433–459. MR**719300**

*Lectures on dynamical systems*, Progress in Math. vol. 8, Birkhäuser, 1980, pp. 1-114.

## Bibliographic Information

- © Copyright 1985 American Mathematical Society
- 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