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.

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.

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.

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 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.

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.

BORDER-COLLISION BIFURCATIONS IN 1D PIECEWISE-LINEAR MAPS AND LEONOV'S APPROACH

2010 ◽  
Vol 20 (10) ◽  
pp. 3085-3104 ◽  
Author(s):  
LAURA GARDINI ◽  
FABIO TRAMONTANA ◽  
VIKTOR AVRUTIN ◽  
MICHAEL SCHANZ
Keyword(s):  
Coordinate Transformation ◽  
Analytical Study ◽  
Piecewise Linear ◽  
Analytical Calculation ◽  
Bifurcation Structure ◽  
Border Collision Bifurcation ◽  
Border Collision Bifurcations ◽  
Linear Maps ◽  
Piecewise Linear Maps

50 years ago (1959) in a series of publications by Leonov, a detailed analytical study of the nested period adding bifurcation structure occurring in piecewise-linear discontinuous 1D maps was presented. The results obtained by Leonov are barely known, although they allow the analytical calculation of border-collision bifurcation subspaces in an elegant and much more efficient way than it is usually done. In this work we recall Leonov's approach and explain why it works. Furthermore, we slightly improve the approach by avoiding an unnecessary coordinate transformation, and also demonstrate that the approach can be used not only for the calculation of border-collision bifurcation curves.

scholarly journals Unfolding Codimension-Two Subsumed Homoclinic Connections in Two-Dimensional Piecewise-Linear Maps

2020 ◽  
Vol 30 (03) ◽  
pp. 2030006 ◽  
Author(s):  
David J. W. Simpson
Keyword(s):  
Periodic Solution ◽  
Piecewise Linear ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Codimension Two ◽  
Codimension One ◽  
Border Collision Bifurcations ◽  
Homoclinic Connections ◽  
Linear Maps ◽  
Piecewise Linear Maps

For piecewise-linear maps, the phenomenon that a branch of a one-dimensional unstable manifold of a periodic solution is completely contained in its stable manifold is codimension-two. Unlike codimension-one homoclinic corners, such “subsumed” homoclinic connections can be associated with stable periodic solutions. The purpose of this paper is to determine the dynamics near a generic subsumed homoclinic connection in two dimensions. Assuming the eigenvalues associated with the periodic solution satisfy [Formula: see text], in a two-parameter unfolding there exists an infinite sequence of roughly triangular regions within which the map has a stable single-round periodic solution. The result applies to both discontinuous and continuous maps, although these cases admit different characterizations for the border-collision bifurcations that correspond to boundaries of the regions. The result is illustrated with a discontinuous map of Mira and the two-dimensional border-collision normal form.

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