scholarly journals Iteration of piecewise linear maps on an interval

1981 ◽  
Vol 24 (3) ◽  
pp. 433-451 ◽  
Author(s):  
James B. McGuire ◽  
Colin J. Thompson
Keyword(s):  
Invariant Measures ◽  
Piecewise Linear ◽  
Point Of View ◽  
Complete Analysis ◽  
Functional Composition ◽  
One Dimensional ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Parameter Values ◽  
Dense Set

A complete analysis is given of the iterative properties of two piece-piecewise linear maps on an interval, from the point of view of a doubling transformation obtained by functional composition and rescaling. We show how invariant measures may be constructed for such maps and that parameter values where this may be done form a dense set in a one-dimensional subset of parameter space.

ORDER AND CHAOS FOR A CLASS OF PIECEWISE LINEAR MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1379-1394 ◽  
Author(s):  
VÍCTOR JIMÉNEZ LÓPEZ
Keyword(s):  
Positive Measure ◽  
Invariant Measures ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Absolutely Continuous ◽  
Linear Maps ◽  
Sensitive Dependence ◽  
Piecewise Linear Maps ◽  
Almost All ◽  
Absolutely Continuous Invariant

For a class of piecewise linear maps f: I → I from a compact interval I into itself, we describe the asymptotic behavior of the sequence [Formula: see text] for almost all x ∈ I. We also study in this setting the relations among sensitive dependence on initial conditions, existence of scrambled sets of positive measure and existence of absolutely continuous invariant measures.

scholarly journals TOPOLOGICAL ENTROPY IN THE SYNCHRONIZATION OF PIECEWISE LINEAR AND MONOTONE MAPS: COUPLED DUFFING OSCILLATORS

2009 ◽  
Vol 19 (11) ◽  
pp. 3855-3868 ◽  
Author(s):  
ACILINA CANECO ◽  
J. LEONEL ROCHA ◽  
CLARA GRÁCIO
Keyword(s):  
Topological Entropy ◽  
Cross Sections ◽  
Coupling Parameter ◽  
Piecewise Linear ◽  
Point Of View ◽  
Return Maps ◽  
Linear Maps ◽  
Monotone Maps ◽  
Piecewise Linear Maps ◽  
Coupled Duffing Oscillators

In this paper is presented a relationship between the synchronization and the topological entropy. We obtain the values for the coupling parameter, in terms of the topological entropy, to achieve synchronization of two unidirectional and bidirectional coupled piecewise linear maps. In addition, we prove a result that relates the synchronizability of two m-modal maps with the synchronizability of two conjugated piecewise linear maps. An application to the unidirectional and bidirectional coupled identical chaotic Duffing equations is given. We discuss the complete synchronization of two identical double-well Duffing oscillators, from the point of view of symbolic dynamics. Working with Poincaré cross-sections and the return maps associated, the synchronization of the two oscillators, in terms of the coupling strength, is characterized.

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.

DESIGN OF ONE-DIMENSIONAL CHAOTIC MAPS WITH PRESCRIBED STATISTICAL PROPERTIES

1995 ◽  
Vol 05 (06) ◽  
pp. 1585-1598 ◽  
Author(s):  
A. BARANOVSKY ◽  
D. DAEMS
Keyword(s):  
Inverse Problem ◽  
Correlation Function ◽  
Invariant Measure ◽  
Wide Class ◽  
Chaotic Maps ◽  
Piecewise Linear ◽  
Statistical Properties ◽  
One Dimensional ◽  
Linear Maps ◽  
Piecewise Linear Maps

The statistical properties of a wide class of 1D piecewise linear Markov maps are compiled. The method used enables one to address analytically the inverse problem of designing a map with a prescribed correlation function. This class of piecewise linear maps is then used as a system of reference to analyze non-Markov piecewise linear maps and to design maps with given invariant measure and correlation function.

Strange Nonchaotic Attractors From a Family of Quasiperiodically Forced Piecewise Linear Maps

2021 ◽  
Vol 31 (07) ◽  
pp. 2150111
Author(s):  
Denghui Li ◽  
Zhenbang Cao ◽  
Xiaoming Zhang ◽  
Celso Grebogi ◽  
Jianhua Xie
Keyword(s):  
Continuous Function ◽  
Lyapunov Exponents ◽  
Piecewise Linear ◽  
Almost Everywhere ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Strange Nonchaotic Attractors ◽  
Almost All ◽  
Parameter Values ◽  
Theoretical Results

In this paper, a family of quasiperiodically forced piecewise linear maps is considered. It is proved that there exists a unique strange nonchaotic attractor for some set of parameter values. It is the graph of an upper semi-continuous function, which is invariant, discontinuous almost everywhere and attracts almost all orbits. Moreover, both Lyapunov exponents on the attractor is nonpositive. Finally, to demonstrate and validate our theoretical results, numerical simulations are presented to exhibit the corresponding phase portrait and Lyapunov exponents portrait.

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