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 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.

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].

Kendall’s Tau Based Correlation Analysis of Chaotic Sequences Generated by Piecewise Linear Maps

2015 ◽  
Vol 25 (13) ◽  
pp. 1550177
Author(s):  
Zouhair Ben Jemaa ◽  
Daniele Fournier-Prunaret ◽  
Safya Belghith
Keyword(s):  
Correlation Analysis ◽  
Chaotic Maps ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Chaotic Map ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Chaotic Sequences ◽  
Linear Maps ◽  
Piecewise Linear Maps

In many applications, sequences generated by chaotic maps have been considered as pseudo-random sequences. This paper deals with the correlation between chaotic sequences generated by a given piecewise linear map; we have based the measure of the correlation on the statistics of the Kendall tau, which is usually used in the field of statistics. We considered three piecewise linear maps to generate chaotic sequences and computed the statistics of the Kendall tau of couples of sequences obtained from randomly chosen couples of initial conditions. We essentially found that the results depend on the considered chaotic map and that it is possible to approach the uncorrelated case.

SENSITIVE PARAMETER DEPENDENCE OF AUTOCORRELATION FUNCTION IN PIECEWISE LINEAR MAPS

2007 ◽  
Vol 17 (04) ◽  
pp. 1185-1197 ◽  
Author(s):  
ALEXANDER L. BARANOVSKI ◽  
ANTHONY J. LAWRANCE
Keyword(s):  
Autocorrelation Function ◽  
Piecewise Linear ◽  
Chaotic Sequence ◽  
Parameter Dependence ◽  
Computational Studies ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Sensitive Parameter ◽  
Linear Maps ◽  
Piecewise Linear Maps

This paper is principally concerned with the effect of splitting and permuting, or shifting, the branches of an onto piecewise linear map, with particular regard to the effect on the autocorrelation of the associated chaotic sequence. This is shown to be a chaotic function of the shifting parameter of the map, and its sensitivity with respect to minute changes in this parameter is termed autocorrelation chaos. This paper presents both analytical and computational studies of the phenomenon.

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].

scholarly journals SYNTHESIS OF PIECEWISE-LINEAR CHAOTIC MAPS: INVARIANT DENSITIES, AUTOCORRELATIONS, AND SWITCHING

2008 ◽  
Vol 18 (08) ◽  
pp. 2169-2189 ◽  
Author(s):  
ALAN ROGERS ◽  
ROBERT SHORTEN ◽  
DANIEL M. HEFFERNAN ◽  
DAVID NAUGHTON
Keyword(s):  
Matrix Method ◽  
Chaotic Maps ◽  
Piecewise Linear ◽  
Image Generation ◽  
Two Dimensional ◽  
Invariant Density ◽  
Interesting Application ◽  
The Matrix ◽  
Simple Matrix ◽  
Invariant Densities

In this paper, we give a review of the Inverse Frobenius–Perron problem (IFPP): how to create chaotic maps with desired invariant densities. After describing some existing methods for solving the IFPP, we present a new and simple matrix method of doing this. We show how the invariant density and the autocorrelation properties of the maps can be controlled independently. We also give some fundamental results on switching between a number of different chaotic maps and the effect this has on the overall invariant density of the system. The invariant density of the switched system can be controlled by varying the probabilities of choosing each individual map. Finally, we present an interesting application of the matrix method to image generation, by synthesizing a two-dimensional map, which when iterated, generates a well-known image.

PERIOD ADDING IN PIECEWISE LINEAR MAPS WITH TWO DISCONTINUITIES

2012 ◽  
Vol 22 (03) ◽  
pp. 1250068 ◽  
Author(s):  
FABIO TRAMONTANA ◽  
LAURA GARDINI ◽  
VIKTOR AVRUTIN ◽  
MICHAEL SCHANZ
Keyword(s):  
Piecewise Linear ◽  
Discontinuity Point ◽  
Big Bang ◽  
One Dimensional ◽  
Piecewise Linear Map ◽  
Stable Regime ◽  
Bifurcation Points ◽  
Border Collision Bifurcations ◽  
Linear Maps ◽  
Piecewise Linear Maps

In this work we consider the border collision bifurcations occurring in a one-dimensional piecewise linear map with two discontinuity points. The map, motivated by an economic application, is written in a generic form and considered in the stable regime, with all slopes between zero and one. We prove that the period adding structures occur in maps with more than one discontinuity points and that the Leonov's method to calculate the bifurcation curves forming these structures is applicable also in this case. We demonstrate the existence of particular codimension-2 bifurcation (big-bang bifurcation) points in the parameter space, from which infinitely many bifurcation curves are issuing associated with cycles involving several partitions. We describe how the bifurcation structure of a map with one discontinuity is modified by the introduction of a second discontinuity point, which causes orbits to appear located on three partitions and organized again in a period-adding structure. We also describe particular codimension-2 bifurcation points which represent limit sets of doubly infinite sequences of bifurcation curves and appear due to the existence of two discontinuities.

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.

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.

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