Border collision bifurcations and power spectral density of chaotic signals generated by one-dimensional discontinuous piecewise linear maps

2014 ◽  
Vol 19 (8) ◽  
pp. 2771-2784 ◽  
Author(s):  
Kais Feltekh ◽  
Zouhair Ben Jemaa ◽  
Danièle Fournier-Prunaret ◽  
Safya Belghith
Keyword(s):  
Spectral Density ◽  
Power Spectral Density ◽  
Piecewise Linear ◽  
One Dimensional ◽  
Border Collision Bifurcations ◽  
Power Spectral ◽  
Linear Maps ◽  
Chaotic Signals ◽  
Piecewise Linear Maps

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 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.

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.

Sign in / Sign up

Export Citation Format

Share Document