One-Dimensional Semiconjugacy Revisited

2003 ◽  
Vol 13 (07) ◽  
pp. 1657-1663 ◽  
Author(s):  
J. F. Alves ◽  
J. Sousa Ramos
Keyword(s):  
Topological Entropy ◽  
Piecewise Linear ◽  
Linear Map ◽  
One Dimensional ◽  
Piecewise Linear Map ◽  
Piecewise Monotone ◽  
Positive Topological Entropy ◽  
Interval Map

Let f be a piecewise monotone interval map with positive topological entropy h(f)= log (s). Milnor and Thurston showed that f is topological semiconjugated to a piecewise linear map having slope s. Here we prove that these semiconjugacies are the eigenvectors of a certain linear endomorphism associated to f. Using this characterization, we prove a conjecture presented by those authors.

PIECEWISE LINEAR MODEL FOR TREE MAPS

2001 ◽  
Vol 11 (12) ◽  
pp. 3163-3169 ◽  
Author(s):  
MATHIEU BAILLIF ◽  
ANDRÉ DE CARVALHO
Keyword(s):  
Linear Model ◽  
Piecewise Linear ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Piecewise Monotone ◽  
Monotone Map ◽  
Piecewise Linear Model ◽  
Constant Slope

We generalize to tree maps the theorems of Parry and Milnor–Thurston about the semi-conjugacy of a continuous piecewise monotone map f to a continuous piecewise linear map with constant slope, equal to the exponential of the entropy of f.

CYCLES OF CHAOTIC INTERVALS IN A TIME-DELAYED CHUA'S CIRCUIT

1993 ◽  
Vol 03 (06) ◽  
pp. 1557-1572 ◽  
Author(s):  
Yu. L. MAISTRENKO ◽  
V. L. MAISTRENKO ◽  
L. O. CHUA
Keyword(s):  
Piecewise Linear ◽  
Period Doubling ◽  
Explicit Formulas ◽  
Stable Point ◽  
Linear Map ◽  
One Dimensional ◽  
Chua's Circuit ◽  
Piecewise Linear Map ◽  
Dense Trajectories ◽  
Interval Cycles

We study the bifurcations of attractors of a one-dimensional 2-segment piecewise-linear map. We prove that the parameter regions of existence of stable point cycles γ are separated by regions of existence of stable interval cycles Γ containing chaotic everywhere dense trajectories. Moreover, we show that the period-doubling phenomenon for cycles of chaotic intervals is characterized by two universal constants δ and α, whose values are calculated from explicit formulas.

Comparison of random and deterministic characteristics of chaotic signals issued from a one-dimensional piecewise linear map

2014 ◽  
Vol 1 ◽  
pp. 17-20 ◽  
Author(s):  
Kais Feltekh ◽  
Danièle Fournier-Prunaret ◽  
Safya Belghith ◽  
Zouhair Ben Jemaa
Keyword(s):  
Piecewise Linear ◽  
Linear Map ◽  
One Dimensional ◽  
Piecewise Linear Map ◽  
Chaotic Signals

Topological conjugation of Lorenz maps by β-transformations

1990 ◽  
Vol 107 (2) ◽  
pp. 401-413 ◽  
Author(s):  
Paul Glendinning
Keyword(s):  
Topological Entropy ◽  
Piecewise Linear ◽  
Sufficient Conditions ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Necessary And Sufficient ◽  
Constant Slope ◽  
Lorenz Maps ◽  
Basic Sets ◽  
Topological Conjugation

AbstractNecessary and sufficient conditions for a Lorenz map to be topologically conjugate to a piecewise linear map with constant slope (a β-transformation) are given, first in terms of kneading invariants of the maps and then in terms of the topological entropy restricted to basic sets. The dynamics of β-transformations is also described.

scholarly journals Coexistence of the Bandcount-Adding and Bandcount-Increment Scenarios

2011 ◽  
Vol 2011 ◽  
pp. 1-30 ◽  
Author(s):  
Viktor Avrutin ◽  
Michael Schanz ◽  
Björn Schenke
Keyword(s):  
Piecewise Linear ◽  
The Other ◽  
Coexisting Attractors ◽  
Linear Map ◽  
One Dimensional ◽  
Piecewise Linear Map ◽  
Periodic Domains

We investigate the structure of the chaotic domain of a specific one-dimensional piecewise linear map with one discontinuity. In this system, the region of ``robust" chaos is embedded between two periodic domains. One of them is organized by the period-adding scenario whereas the other one by the period-increment scenario with coexisting attractors. In the chaotic domain, the influence of both adjacent periodic domains leads to the coexistence of the recently discovered bandcount adding and bandcount-increment scenarios. In this work, we focus on the explanation of the overall structure of the chaotic domain and a description of the bandcount adding and bandcount increment scenarios.

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 The Chaotic Saddle in the Lozi Map, Autonomous and Nonautonomous Versions

2015 ◽  
Vol 25 (13) ◽  
pp. 1550184 ◽  
Author(s):  
Carlos Lopesino ◽  
Francisco Balibrea-Iniesta ◽  
Stephen Wiggins ◽  
Ana M. Mancho
Keyword(s):  
Piecewise Linear ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
The Third ◽  
Autonomous Version ◽  
Area Preserving ◽  
Chaotic Saddle

In this paper, we prove the existence of a chaotic saddle for a piecewise-linear map of the plane, referred to as the Lozi map. We study the Lozi map in its orientation and area preserving version. First, we consider the autonomous version of the Lozi map to which we apply the Conley–Moser conditions to obtain the proof of a chaotic saddle. Then we generalize the Lozi map on a nonautonomous version and we prove that the first and the third Conley–Moser conditions are satisfied, which imply the existence of a chaotic saddle. Finally, we numerically demonstrate how the structure of this nonautonomous chaotic saddle varies as parameters are varied.

DIGITAL CONTROL AS SOURCE OF CHAOTIC BEHAVIOR

2010 ◽  
Vol 20 (05) ◽  
pp. 1365-1378 ◽  
Author(s):  
GÁBOR CSERNÁK ◽  
GÁBOR STÉPÁN
Keyword(s):  
Mechanical System ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Linear Control ◽  
Control Loop ◽  
Digital Implementation ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Chaotic Vibrations ◽  
Processing Delay

In the present paper, we introduce and analyze a mechanical system, in which the digital implementation of a linear control loop may lead to chaotic behavior. The amplitude of such oscillations is usually very small, this is why these are called micro-chaotic vibrations. As a consequence of the digital effects, i.e. the sampling, the processing delay and the round-off error, the behavior of the system can be described by a piecewise linear map, the micro-chaos map. We examine a 2D version of the micro-chaos map and prove that the map is chaotic.

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