# Definition and applications of the ascent-probability distribution in one-dimensional maps

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## Abstract

This paper introduces the discrete distribution of ascent probabilities {Pn}n=0, generalizing the concept of rotation number, already being defined in one-dimensional unimodal maps. The map domain is partitioned into subintervals {In}n=0, each one containing orbits with their route characterized by a specific number of n successive ascents before a descent occurs. Then, Pn is defined as the probability to have an orbit performing n successive ascents, and equals the portion of the invariant measure within In. The rotation number is found to be equal to = P0 = P1, that is the portion of invariant measure within I0 or I1. Some significant applications of this relation concern (i) the rotation number dependence on the nonlinear parameter p, (ii) the analytical derivation of the rotation number, given the invariant density explicit expression, (iii) an easy computation of the rotation number that characterizes a periodic window. Moreover, the dependence of the ascent probabilities on the nonlinear parameter p is examined. Emphasis is placed on the discrete distribution of the ascent probabilities {Pn}n=0 within the chaotic zone. A particular set of nonlinear parameter values {p(n)}n=3 is affiliated to the concept of ascent probabilities: For each probability Pn, n = 0, 1, 2, there is a lower limit of the nonlinear parameter values, p(n+1), so that, Pn = 0 ∀ p ≤ p(n+1). The set {p(n)} n=3 is analytically determined, and its specific arrangement in the chaotic zone is studied. Finally, the "u-S-P equivalence" between the triplet of the set of the fixed point and its preimages, {un} n=0, of the invariant density S, and of the set of the ascents probabilities, {Pn}n=0, is formulated. In particular, we show that each component of this triplet can be estimated whenever the other two components are given. Applications in the case of Logistic map are thoroughly examined.

Original language English (US) 3567-3591 25 International Journal of Bifurcation and Chaos 19 11 https://doi.org/10.1142/S0218127409025018 Published - Nov 2009 Yes

## All Science Journal Classification (ASJC) codes

• Modeling and Simulation
• Engineering (miscellaneous)
• General
• Applied Mathematics

## Keywords

• Eventually fixed points
• Invariant density
• Lyapunov characteristic number
• Map reconstruction
• Periodic windows
• Rotation number
• Unimodal maps

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