Anti-integrability for the Logistic Maps

Abstract

The embedding of the Bernoulli shift into the logistic map x → μx(1 − x) for μ > 4 is reinterpreted by the theory of anti-integrability: it is inherited from the anti-integrable limit μ → ∞.

Keywords

Logistic maps / Hyperbolicity / Symbolic dynamics / Anti-integrable limit / 37C15 / 37D05 / 37E05

References

[1] Aubry Phys. D, 1990, 43: 199

[2] Aulbach J. Difference Equ. Appl., 2004, 10: 1243

[3] Bolotin Nonlinearity, 1997, 10: 1015

[4] Chen Dyn. Syst., 2004, 19: 145

[5] Chen Regul. Chaotic Dyn., 2003, 8: 243

[6] Chen Discrete Contin. Dyn. Syst. Ser. B, 2005, 15: 587

[7] Devaney, R. L., An Introduction to Chaotic Dynamical Systems, Second Edition, Addison-Wesley, New York, 1989

[8] Easton Phys. D, 2001, 156: 201

[9] Feigenbaum J. Stat. Phys., 1978, 19: 25

[10] Glendinning Nonlinear Anal., 2001, 47: 3323

[11] Guckenheimer Comm. Math. Phys., 1979, 70: 133

[12] Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983

[13] Gulick, D., Encounters with Chaos, McGraw-Hill, New York, 1992

[14] Kraft Amer. Math. Monthly, 1999, 106: 400

[15] MacKay Nonlinearity, 1992, 5: 149

[16] May Nature, 1976, 261: 459

[17] de Melo, W. and van Strien, S., One-dimensional Dynamics, Springer-Verlag, New York, 1993

[18] Robinson, C., Dynamical Systems-Stability, Symbolic Dynamics, and Chaos, CRC Press, London, 1995

[19] Zeller Amer. Math. Monthly, 2001, 108: 155