scholarly journals On the mixing properties of piecewise expanding maps under composition with permutations, II: Maps of non-constant orientation

2016 ◽  
Vol 16 (03) ◽  
pp. 1660013 ◽  
Author(s):  
Nigel P. Byott ◽  
Congping Lin ◽  
Yiwei Zhang
Keyword(s):  
Piecewise Linear ◽  
Unit Interval ◽  
Interval Exchange ◽  
Expanding Maps ◽  
Mixing Properties ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Fold Map ◽  
Constant Slope ◽  
Piecewise Expanding Maps

For an integer [Formula: see text], let [Formula: see text] be the partition of the unit interval [Formula: see text] into [Formula: see text] equal subintervals, and let [Formula: see text] be the class of piecewise linear maps on [Formula: see text] with constant slope [Formula: see text] on each element of [Formula: see text]. We investigate the effect on mixing properties when [Formula: see text] is composed with the interval exchange map given by a permutation [Formula: see text] interchanging the [Formula: see text] subintervals of [Formula: see text]. This extends the work in a previous paper [N. P. Byott, M. Holland and Y. Zhang, DCDS 33 (2013) 3365–3390], where we considered only the “stretch-and-fold” map [Formula: see text].

Invariant densities for piecewise linear maps of the unit interval

2009 ◽  
Vol 29 (5) ◽  
pp. 1549-1583 ◽  
Author(s):  
PAWEŁ GÓRA
Keyword(s):  
Piecewise Linear ◽  
Unit Interval ◽  
Invariant Density ◽  
Piecewise Linear Map ◽  
Full Support ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
The Matrix ◽  
Definition Of ◽  
Invariant Densities

AbstractWe find an explicit formula for the invariant densityhof an arbitrary eventually expanding piecewise linear mapτof an interval [0,1]. We do not assume that the slopes of the branches are the same and we allow arbitrary number of shorter branches touching zero or touching one or hanging in between. The construction involves the matrixSwhich is defined in a way somewhat similar to the definition of the kneading matrix of a continuous piecewise monotonic map. Under some additional assumptions, we prove that if 1 is not an eigenvalue ofS, then the dynamical system (τ,h⋅m) is ergodic with full support.

scholarly journals Family of piecewise expanding maps having singular measure as a limit of ACIMs

2011 ◽  
Vol 33 (1) ◽  
pp. 158-167 ◽  
Author(s):  
ZHENYANG LI ◽  
PAWEŁ GÓRA ◽  
ABRAHAM BOYARSKY ◽  
HARALD PROPPE ◽  
PEYMAN ESLAMI
Keyword(s):  
Piecewise Linear ◽  
Dynamical Behavior ◽  
Singular Measure ◽  
Expanding Maps ◽  
Linear Maps ◽  
Absolutely Continuous Invariant Measure ◽  
Piecewise Linear Maps ◽  
Point Measure ◽  
Invariant Densities ◽  
Absolutely Continuous Invariant

AbstractKeller [Stochastic stability in some chaotic dynamical systems. Monatsh. Math.94(4) (1982), 313–333] introduced families of W-shaped maps that can have a great variety of behaviors. As a family approaches a limit W map, he observed behavior that was either described by a probability density function (PDF) or by a singular point measure. Based on this, Keller conjectured that instability of the absolutely continuous invariant measure (ACIM) can result only from the existence of small invariant neighborhoods of the fixed critical point of the limit map. In this paper, we show that the conjecture is not true. We construct a very simple family of W-maps with ACIMs supported on the whole interval, whose limiting dynamical behavior is captured by a singular measure. Key to the analysis is the use of a general formula for invariant densities of piecewise linear and expanding maps [P. Góra. Invariant densities for piecewise linear maps of interval. Ergod. Th. & Dynam. Sys. 29(5) (2009), 1549–1583].

ON THE EVENTUAL PERIODICITY OF PIECEWISE LINEAR CHAOTIC MAPS

2017 ◽  
Vol 95 (3) ◽  
pp. 467-475 ◽  
Author(s):  
M. ALI KHAN ◽  
ASHVIN V. RAJAN
Keyword(s):  
Periodic Point ◽  
Chaotic Maps ◽  
Piecewise Linear ◽  
Unit Interval ◽  
Linear Maps ◽  
The Family ◽  
Piecewise Linear Maps

We present a family of continuous piecewise linear maps of the unit interval into itself that are all chaotic in the sense of Li and Yorke [‘Period three implies chaos’, Amer. Math. Monthly82 (1975), 985–992] and for which almost every point (in the sense of Lebesgue) in the unit interval is an eventually periodic point of period $p,p\geq 3$, for a member of the family.

scholarly journals Parrondo’s Paradox for Tent Maps

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 85
Author(s):  
Jose S. Cánovas
Keyword(s):  
Piecewise Linear ◽  
Simple Complex ◽  
Tent Map ◽  
Linear Homeomorphism ◽  
Linear Maps ◽  
Parrondo’S Paradox ◽  
Tent Maps ◽  
Piecewise Linear Maps ◽  
Constant Slope

In this paper, we study the dynamic Parrondo’s paradox for the well-known family of tent maps. We prove that this paradox is impossible when we consider piecewise linear maps with constant slope. In addition, we analyze the paradox “simple + simple = complex” when a tent map with constant slope and a piecewise linear homeomorphism with two different slopes are considered.

scholarly journals How Many Inflections are There in the Lyapunov Spectrum?

2021 ◽  
Author(s):  
O. Jenkinson ◽  
M. Pollicott ◽  
P. Vytnova
Keyword(s):  
Piecewise Linear ◽  
Lyapunov Spectrum ◽  
Stat Phys ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Expanding Map ◽  
Lyapunov Spectra ◽  
Linear Maps ◽  
Inflection Points ◽  
Piecewise Linear Maps

AbstractIommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

Noninvertible Piecewise Linear Maps Applied to Chaos Synchronization and Secure Communications

1997 ◽  
Vol 07 (07) ◽  
pp. 1617-1634 ◽  
Author(s):  
G. Millerioux ◽  
C. Mira
Keyword(s):  
Dynamical Systems ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Piecewise Linear ◽  
A Priori ◽  
Secure Communications ◽  
Technological Parameters ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Necessary And Sufficient

Recently, it was demonstrated that two chaotic dynamical systems can synchronize each other, leading to interesting applications as secure communications. We propose in this paper a special class of dynamical systems, noninvertible discrete piecewise linear, emphasizing on interesting advantages they present compared with continuous and differentiable nonlinear ones. The generic aspect of such systems, the simplicity of numerical implementation, and the robustness to mismatch of technological parameters make them good candidates. The classical concept of controllability in the control theory is presented and used in order to choose and predict the number of appropriate variables to be transmitted for synchronization. A necessary and sufficient condition of chaotic synchronization is established without computing numerical quantities, introducing a state affinity structure of chaotic systems which provides an a priori establishment of synchronization.

Estimates of the topological degree of a class of piecewise linear maps with applications

2021 ◽  
pp. 2150073
Author(s):  
Laura Poggiolini ◽  
Marco Spadini
Keyword(s):  
Topological Degree ◽  
Piecewise Linear ◽  
Nonlinear Problems ◽  
Text Structure ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Classical Integral ◽  
Computation Formula

We provide some new estimates for the topological degree of a class of continuous and piecewise linear maps based on a classical integral computation formula. We provide applications to some nonlinear problems that exhibit a local [Formula: see text] structure.

Statistics of a family of piecewise linear maps

2021 ◽  
Vol 427 ◽  
pp. 133019
Author(s):  
J.J.P. Veerman ◽  
P.J. Oberly ◽  
L.S. Fox
Keyword(s):  
Piecewise Linear ◽  
Linear Maps ◽  
Piecewise Linear Maps
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