On Some Properties of Invariant Sets of Two-Dimensional Noninvertible Maps

1997 ◽  
Vol 07 (06) ◽  
pp. 1167-1194 ◽  
Author(s):  
Christos E. Frouzakis ◽  
Laura Gardini ◽  
Ioannis G. Kevrekidis ◽  
Gilles Millerioux ◽  
Christian Mira
Keyword(s):  
Periodic Points ◽  
Invariant Sets ◽  
Two Dimensional ◽  
S Locus ◽  
One Dimensional ◽  
Critical Curves ◽  
Invariant Circles ◽  
Saddle Type ◽  
Unstable Manifolds ◽  
Noninvertible Maps

We study the nature and dependence on parameters of certain invariant sets of noninvertible maps of the plane. The invariant sets we consider are unstable manifolds of saddle-type fixed and periodic points, as well as attracting invariant circles. Since for such maps a point may have more than one first-rank preimages, the geometry, transitions, and general properties of these sets are more complicated than the corresponding sets for diffeomorphisms. The critical curve(s) (locus of points having at least two coincident preimages) as well as its antecedent(s), the curve(s) where the map is singular (or "curve of merging preimages") play a fundamental role in such studies. We focus on phenomena arising from the interaction of one-dimensional invariant sets with these critical curves, and present some illustrative examples.

A METHOD TO CALCULATE BASIN BIFURCATION SETS FOR A TWO-DIMENSIONAL NONINVERTIBLE MAP

2000 ◽  
Vol 10 (08) ◽  
pp. 2001-2014 ◽  
Author(s):  
H. KITAJIMA ◽  
H. KAWAKAMI ◽  
C. MIRA
Keyword(s):  
Numerical Method ◽  
Periodic Points ◽  
Critical Curve ◽  
Stable Set ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Basin Boundary ◽  
Rank One ◽  
Saddle Type ◽  
Noninvertible Maps

For models in the form of noninvertible maps we propose a numerical method to calculate a class of basin bifurcation sets in a parameter space. It is known that basin bifurcations may result from the contact of a basin boundary with the critical curve (locus of points having two coincident rank-one preimages) segment. Therefore, when the map is smooth, we propose the method to obtain the tangent points of a basin boundary (stable set of saddle type periodic points) and a critical curve. Numerical examples for a two-dimensional quadratic noninvertible map are illustrated and new results of basin bifurcations are shown.

BASIN BIFURCATIONS OF TWO-DIMENSIONAL NONINVERTIBLE MAPS: FRACTALIZATION OF BASINS

1994 ◽  
Vol 04 (02) ◽  
pp. 343-381 ◽  
Author(s):  
C. MIRA ◽  
D. FOURNIER-PRUNARET ◽  
L. GARDINI ◽  
H. KAWAKAMI ◽  
J.C. CATHALA
Keyword(s):  
Phase Plane ◽  
The Other ◽  
Two Dimensional ◽  
Basin Boundary ◽  
Critical Curves ◽  
Noninvertible Maps

Properties of the basins of noninvertible maps of a plane are studied using the method of critical curves. Different kinds of basin bifurcation, some of them leading to basin boundary fractalization are described. More particularly the paper considers the simplest class of maps that of a phase plane which is made up of two regions, one with two preimages, the other with no preimage.

Thermodynamic formalism for certain nonhyperbolic maps

1999 ◽  
Vol 19 (5) ◽  
pp. 1365-1378 ◽  
Author(s):  
MICHIKO YURI
Keyword(s):  
Number Theory ◽  
Gibbs Measure ◽  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Periodic Points ◽  
Two Dimensional ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Conformal Measures ◽  
One Dimensional Maps

We establish a generalized thermodynamic formalism for certain nonhyperbolic maps with countably many preimages. We study existence and uniqueness of conformal measures and statistical properties of the equilibrium states absolutely continuous with respect to the conformal measures. We will see that such measures are not Gibbs but satisfy a version of Gibbs property (weak Gibbs measure). We apply our results to a one-parameter family of one-dimensional maps and a two-dimensional nonconformal map related to number theory. Both of them admit indifferent periodic points.

FRACTAL AGGREGATION OF BASIN ISLANDS IN TWO-DIMENSIONAL QUADRATIC NONINVERTIBLE MAPS

1995 ◽  
Vol 05 (04) ◽  
pp. 991-1019 ◽  
Author(s):  
C. MIRA ◽  
C. RAUZY
Keyword(s):  
Phase Plane ◽  
Parameter Variation ◽  
The Other ◽  
Two Dimensional ◽  
Quadratic Maps ◽  
Attracting Set ◽  
Critical Curves ◽  
Noninvertible Maps

Properties of basins of noninvertible maps of the plane are studied by using the method of critical curves. The paper considers the simplest class of quadratic maps, that having a phase plane made up of two regions, one with two first rank preimages, the other with no preimage, in situations different from those described in a previous publication. More specifically, the considered quadratic maps give rise to a basin made up of infinitely many nonconnected regions, a parameter variation leading to an aggregation of these regions, which occur in a fractal way. The nonconnected regions, different from that containing an attracting set, are called "islands".

Homoclinic and Heteroclinic Situations Specific to Two-Dimensional Noninvertible Maps

1997 ◽  
Vol 07 (01) ◽  
pp. 39-70 ◽  
Author(s):  
Gilles Millerioux ◽  
Christian Mira
Keyword(s):  
Qualitative Change ◽  
Closed Curve ◽  
Stable Set ◽  
Complex Structures ◽  
Two Dimensional ◽  
Parameter Region ◽  
Critical Curves ◽  
Rank One ◽  
Stable And Unstable Sets ◽  
Noninvertible Maps

These situations are put in evidence from two examples of (Z0 - Z2) maps. It is recalled that such maps (the simplest type of non-invertible ones) are related to the separation of the plane into a region without preimage, and a region each point of which has two rank-one preimages. With respect to diffeomorphisms, non-invertibility introduces more complex structures of the stable and unstable sets defining the homoclinic and heteroclinic situations, and the corresponding bifurcations. Critical curves permit the analysis of this question. More particularly, a basic global contact bifurcation (contact of the map critical curve with a non-connected saddle stable set Ws) plays a fundamental role for explaining the qualitative change of Ws, which occurs between two boundary homoclinic bifurcations limiting a parameter region related to the disappearing of an attracting invariant closed curve.

scholarly journals On the notion of the dimension with respect to a dynamical system

1984 ◽  
Vol 4 (3) ◽  
pp. 405-420 ◽  
Author(s):  
Ya. B. Pesin
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Invariant Sets ◽  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Common Features ◽  
Dynamical Properties ◽  
The One ◽  
Hyperbolic Sets

AbstractFor the invariant sets of dynamical systems a new notion of dimension-the so-called dimension with respect to a dynamical system-is introduced. It has some common features with the general topological notion of the dimension, but it also reflects the dynamical properties of the system. In the one-dimensional case it coincides with the Hausdorff dimension. For multi-dimensional hyperbolic sets formulae for the calculation of our dimension are obtained. These results are generalizations of Manning's results obtained by him for the Hausdorff dimension in the two-dimensional case.

FRACTALIZATION OF BASIN BOUNDARY IN TWO-DIMENSIONAL NONINVERTIBLE MAPS

1999 ◽  
Vol 09 (10) ◽  
pp. 1995-2025 ◽  
Author(s):  
J. C. CATHALA
Keyword(s):  
Critical Curve ◽  
The Other ◽  
Two Dimensional ◽  
Basin Boundary ◽  
Critical Curves ◽  
Noninvertible Maps

Properties of the basins of a two-dimensional noninvertible degenerated map is studied using the method of critical curves. More precisely, this paper considers a cubic map that belongs to the class of maps having the plane subdivided by the branches of the critical curve in three regions, two nonconnected with one preimage, the other with three distinct preimages. The fractalization of the basin of such an endomorphism is described.

ON BEHAVIORS OF TWO-DIMENSIONAL ENDOMORPHISMS: ROLE OF THE CRITICAL CURVES

1993 ◽  
Vol 03 (01) ◽  
pp. 187-194 ◽  
Author(s):  
CHRISTIAN MIRA ◽  
TONY NARAYANINSAMY
Keyword(s):  
Critical Points ◽  
Two Dimensional ◽  
One Dimensional ◽  
Critical Curves ◽  
Dynamical Properties

Critical curves are the natural two-dimensional extension of the notion of critical points in one-dimensional endomorphisms. They play a fundamental role in determining the dynamical properties and their bifurcations. This letter demonstrates such a role for two new behaviors.

scholarly journals Asset price dynamics in a financial market with fundamentalists and chartists

2001 ◽  
Vol 6 (2) ◽  
pp. 69-99 ◽  
Author(s):  
Carl Chairella ◽  
Roberto Dieci ◽  
Laura Gardini
Keyword(s):  
Financial Market ◽  
Asset Price ◽  
Homoclinic Orbits ◽  
Basins Of Attraction ◽  
Two Dimensional ◽  
Global Bifurcations ◽  
Critical Curves ◽  
Qualitative Structure ◽  
Local And Global Bifurcations ◽  
Noninvertible Maps

In this paper we consider a model of the dynamics of speculative markets involving the interaction of fundamentalists and chartists. The dynamics of the model are driven by a two-dimensional map that in the space of the parameters displays regions of invertibility and noninvertibility. The paper focuses on a study of local and global bifurcations which drastically change the qualitative structure of the basins of attraction of several, often coexistent, attracting sets. We make use of the theory of critical curves associated with noninvertible maps, as well as of homoclinic bifurcations and homoclinic orbits of saddles in regimes of invertibility.

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