Asymptotic Periodicity in One-Dimensional Maps

The purpose of this paper is to present a weighted kneading theory for one-dimensional maps with a hole. We consider extensions of the kneading theory of Milnor and Thurston to expanding discontinuous maps with a hole and introduce weights in the formal power series. This method allows us to derive techniques to compute explicitly the topological entropy, the Hausdorff dimension, and the escape rate.

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The Stability of One-Dimensional Maps

Discrete Chaos ◽

10.1201/9781420011043-6 ◽

2007 ◽

pp. 21-80

Keyword(s):

One Dimensional ◽

The Stability ◽

One Dimensional Maps

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A Note on Non-hyperbolic Fixed Points of One-Dimensional Maps

Progress on Difference Equations and Discrete Dynamical Systems - Springer Proceedings in Mathematics & Statistics ◽

10.1007/978-3-030-60107-2_12 ◽

2020 ◽

pp. 257-267

Author(s):

Sinan Kapçak

Keyword(s):

Fixed Points ◽

One Dimensional ◽

Hyperbolic Fixed Points ◽

One Dimensional Maps

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STABILIZATION OF CHAOTIC DYNAMICS OF ONE-DIMENSIONAL MAPS BY A CYCLIC PARAMETRIC TRANSFORMATION

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127496000382 ◽

1996 ◽

Vol 06 (04) ◽

pp. 725-735 ◽

Cited By ~ 4

Author(s):

ALEXANDER Yu. LOSKUTOV ◽

VALERY M. TERESHKO ◽

KONSTANTIN A. VASILIEV

Keyword(s):

Periodic Orbits ◽

Chaotic Dynamics ◽

Chaotic Maps ◽

Logistic Map ◽

Exponential Map ◽

One Dimensional ◽

Stable Periodic ◽

One Dimensional Maps ◽

Parameter Values ◽

Parametric Transformation

We consider one-dimensional maps, the logistic map and an exponential map, in those sets of parameter values which correspond to their chaotic dynamics. It is proven that such dynamics may be stabilized by a certain cyclic parametric transformation operating strictly within the chaotic set. The stabilization is a result of the creation of stable periodic orbits in the initially chaotic maps. The period of these stable orbits is a multiple of the period of the cyclic transformation. It is shown that stabilized behavior cannot be destroyed by a weak noise smearing of the required parameter values. The regions where the behavior stabilization takes place are numerically estimated. Periods of the created stabile periodic orbits are calculated.

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