Abstract
This paper shows that if a horseshoe is created in a natural manner as a parameter is varied, then the process of creation involves the appearance of attracting periodic orbits of all periods. Furthermore, each of these orbits will period double repeatedly, with those periods going to infinity.
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Alligood, K.T.: A canonical partition of the periodic orbits of chaotic maps. Trans. Am. Math. Soc. (in press)
Alligood, K.T., Mallet-Paret, J., Yorke, J.A.: An index for the global continuation of relatively isolated sets of periodic orbits. Geometric dynamics. Lecture Notes in Mathematics, Vol. 1007, pp. 1–21. Berlin, Heidelberg, New York: Springer 1983
Alligood, K.T., Yorke, J.A.: Families of periodic orbits: virtual periods and global continuability. J. Differential Equations55, 59–71 (1984)
Alligood, K.T., York, J.A.: Hopfbifurcation: the appearance of virtual periods in cases of resonance. J. Differential Equations (in press)
Brunovsky, P.: One parameter families of diffeomorphisms. Symposium on differential equations and dynamical systems, University of Warwick, 1969, Springer Lecture Notes206. Berlin, Heidelberg, New York: Springer 1969;
: On one-parameter families of diffeomorphisms. Comment. Math. Univ. Carol.11, 559–582 (1970)
Chirikov, B.V.: A universal instability of many-dimensional oscillator systems. Phys. Rep.52, 265–379 (1979)
Chow, S.N., Mallet-Paret, J., Yorke, J.: A bifurcation invariant: degenerate orbits treated as clusters of simple orbits. Geometric dynamic. Lecture Notes in Mathematics, Vol. 1007, pp. 109–131. Berlin, Heidelberg, New York: 1983
Devaney, R., Nitecki, Z.: Shift automorphisms in the Henon mapping. Commun. Math. Phys.67, 137–146 (1979)
Franks, J.: Period doubling and the Lefschetz formula. Trans. Am. Math. Soc.287, 275–283 (1985)
Guckenheimer, J., Holmes, P.J.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Berlin, Heidelberg, New York: Springer 1983
Holmes, P.J.: The dynamics of repeated impacts with a sinusoidally vibrating table. J. Sound Vib.84 (2), 173–189 (1982)
Myrberg, P.J.: Sur l'iteration des polynomes reels quadratiques. J. Math. Pures Appl.41, 339–351 (1962)
Mallet-Paret, J., Yorke, J.: Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation. J. Differential Equations43, 419–450 (1982)
Newhouse, S.: Lectures on dynamical systems. In: Dynamical systems. Progress in Mathematics, Vol. 8. Boston: Birkhäuser 1980
Peixoto, M.M.: On an approximation theorem of Kupka and Smale. J. Differential Equations3, 214–227 (1966)
Robinson, C.: Bifurcation to infinitely many sinks. Commun. Math. Phys.90, 433–459 (1983)
Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc.73, 747–817 (1967)
Short, T., Yorke, J.A.: Truncated development of chaotic attractors in a map when the Jacobian is not small. In: Chaos and statistical mechanics. Proc. of the 1983 Kyoto Summer Institute. Berlin, Heidelberg, New York: Springer 1983
Yorke, J.A., Alligood, K.T.: Cascades of period-doubling bifurcations: a prerequisite for horseshoes. Bull. Am. Math. Soc.9, 319–322 (November, 1983)
Zaslovsky, G.M.: The simplest case of a strange attractor. Phys. Lett.69 A (3), 145–147 (1978)
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Communicated by O. E. Lanford
In memory of Charles Conley
Research partially supported by the National Science Foundation
Research partially supported by the Air Force Office of Scientific Research
The paper was written while Dr. Alligood was visiting Michigan State University and the University of Maryland
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Yorke, J.A., Alligood, K.T. Period doubling cascades of attractors: A prerequisite for horseshoes. Commun.Math. Phys. 101, 305–321 (1985). https://doi.org/10.1007/BF01216092
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DOI: https://doi.org/10.1007/BF01216092