Bifurcation analysis of one-dimensional maps using the renormalization technique in a parameter space

1998 ◽  
Vol 81 (8) ◽  
pp. 41-51
Author(s):  
Ikuo Matsuba
Keyword(s):  
Parameter Space ◽  
Bifurcation Analysis ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Renormalization Technique

BORDER COLLISION BIFURCATIONS IN 1D PWL MAP WITH ONE DISCONTINUITY AND NEGATIVE JUMP: USE OF THE FIRST RETURN MAP

2010 ◽  
Vol 20 (11) ◽  
pp. 3529-3547 ◽  
Author(s):  
LAURA GARDINI ◽  
FABIO TRAMONTANA
Keyword(s):  
Parameter Space ◽  
Complete Characterization ◽  
One Dimensional ◽  
Return Map ◽  
Border Collision Bifurcations ◽  
One Dimensional Maps ◽  
Negative Jump

The aim of this work is to study discontinuous one-dimensional maps in the case of slopes and offsets having opposite signs. Such models represent the dynamics of applied systems in several disciplines. We analyze in particular attracting cycles, their border collision bifurcations and the properties of the periodicity regions in the parameter space. The peculiarity of this family is that we can make use of the technical instrument of the first return map. With this, we can rigorously prove properties which were known numerically, as well as prove new ones, giving a complete characterization of the overlapping periodicity regions.

A PICTURE BOOK OF TWO FAMILIES OF CUBIC MAPS

1993 ◽  
Vol 04 (03) ◽  
pp. 553-568 ◽  
Author(s):  
FERNANDO CABRAL ◽  
ALEXANDRE LAGO ◽  
JASON A. C. GALLAS
Keyword(s):  
Dynamical Systems ◽  
High Resolution ◽  
Parameter Space ◽  
Picture Book ◽  
Initial Conditions ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Two Parameters ◽  
The Way

This paper reports high-resolution isoperiodic diagrams for two model-families of dynamical systems characterised by one-dimensional maps depending on two parameters. We present a comparison of both diagrams, investigating the way in which initial conditions affect isoperiodic sets in the parameter space of both systems and the similarities between them. Although both models represent quite different dynamical systems, they are found to have many properties in common in their space of parameters.

scholarly journals The role of extreme orbits in the global organization of periodic regions in parameter space for one dimensional maps

2016 ◽  
Vol 380 (18-19) ◽  
pp. 1610-1614 ◽  
Author(s):  
Diogo Ricardo da Costa ◽  
Matheus Hansen ◽  
Gustavo Guarise ◽  
Rene O. Medrano-T ◽  
Edson D. Leonel
Keyword(s):  
Parameter Space ◽  
One Dimensional ◽  
Global Organization ◽  
One Dimensional Maps

scholarly journals On the finiteness of attractors for one-dimensional maps with discontinuities

2021 ◽  
Vol 389 ◽  
pp. 107891
Author(s):  
P. Brandão ◽  
J. Palis ◽  
V. Pinheiro
Keyword(s):  
One Dimensional ◽  
One Dimensional Maps

scholarly journals Weighted kneading theory of one-dimensional maps with a hole

2004 ◽  
Vol 2004 (38) ◽  
pp. 2019-2038 ◽  
Author(s):  
J. Leonel Rocha ◽  
J. Sousa Ramos
Keyword(s):  
Hausdorff Dimension ◽  
Power Series ◽  
Formal Power Series ◽  
Topological Entropy ◽  
Escape Rate ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Formal Power ◽  
Kneading Theory

The purpose of this paper is to present a weighted kneading theory for one-dimensional maps with a hole. We consider extensions of the kneading theory of Milnor and Thurston to expanding discontinuous maps with a hole and introduce weights in the formal power series. This method allows us to derive techniques to compute explicitly the topological entropy, the Hausdorff dimension, and the escape rate.

STABILIZATION OF CHAOTIC DYNAMICS OF ONE-DIMENSIONAL MAPS BY A CYCLIC PARAMETRIC TRANSFORMATION

1996 ◽  
Vol 06 (04) ◽  
pp. 725-735 ◽  
Author(s):  
ALEXANDER Yu. LOSKUTOV ◽  
VALERY M. TERESHKO ◽  
KONSTANTIN A. VASILIEV
Keyword(s):  
Periodic Orbits ◽  
Chaotic Dynamics ◽  
Chaotic Maps ◽  
Logistic Map ◽  
Exponential Map ◽  
One Dimensional ◽  
Stable Periodic ◽  
One Dimensional Maps ◽  
Parameter Values ◽  
Parametric Transformation

We consider one-dimensional maps, the logistic map and an exponential map, in those sets of parameter values which correspond to their chaotic dynamics. It is proven that such dynamics may be stabilized by a certain cyclic parametric transformation operating strictly within the chaotic set. The stabilization is a result of the creation of stable periodic orbits in the initially chaotic maps. The period of these stable orbits is a multiple of the period of the cyclic transformation. It is shown that stabilized behavior cannot be destroyed by a weak noise smearing of the required parameter values. The regions where the behavior stabilization takes place are numerically estimated. Periods of the created stabile periodic orbits are calculated.

Sign in / Sign up

Export Citation Format

Share Document