TWO-DIMENSIONAL NONINVERTIBLE MAPS.: PROPERTIES OF CRITICAL CURVES

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.

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

10.1142/s0218127495000752 ◽

1995 ◽

Vol 05 (04) ◽

pp. 991-1019 ◽

Cited By ~ 9

Author(s):

C. MIRA ◽

C. RAUZY

Keyword(s):

Phase Plane ◽

Parameter Variation ◽

The Other ◽

Two Dimensional ◽

Quadratic Maps ◽

Attracting Set ◽

Critical Curves ◽

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

10.1142/s0218127497000042 ◽

1997 ◽

Vol 07 (01) ◽

pp. 39-70 ◽

Cited By ~ 5

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 ◽

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.

FRACTALIZATION OF BASIN BOUNDARY IN TWO-DIMENSIONAL NONINVERTIBLE MAPS

10.1142/s0218127499001450 ◽

1999 ◽

Vol 09 (10) ◽

pp. 1995-2025 ◽

Cited By ~ 3

Author(s):

J. C. CATHALA

Keyword(s):

Critical Curve ◽

The Other ◽

Two Dimensional ◽

Basin Boundary ◽

Critical Curves ◽

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.

Asset price dynamics in a financial market with fundamentalists and chartists

Discrete Dynamics in Nature and Society ◽

10.1155/s1026022601000103 ◽

2001 ◽

Vol 6 (2) ◽

pp. 69-99 ◽

Cited By ~ 13

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 ◽

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.

ON CRITICAL CURVES IN TWO-DIMENSIONAL ENDOMORPHISMS

10.1142/s021812740100247x ◽

2001 ◽

Vol 11 (03) ◽

pp. 821-839 ◽

Cited By ~ 1

Author(s):

J. C. CATHALA

Keyword(s):

Critical Curve ◽

Two Dimensional ◽

Curve Segment ◽

Fundamental Properties ◽

Critical Curves ◽

Parameter Variations ◽

Properties of the critical curves of noninvertible maps are studied using the representation of the plane in the form of sheets. In such a representation, every sheet is associated with a well-defined determination of the inverse map which leads to a foliation of the plane directly related to fundamental properties of the map. The paper describes the change of the plane foliation occurring in the presence of parameter variations, leading to a modification of the nature of the map by crossing through a foliation bifurcation. The degenerated map obtained at the foliation bifurcation is characterized by the junction of more than two sheets on a critical curve segment. Examples illustrating these situations are given.

Basin Properties in Two-Dimensional Noninvertible Maps

10.1142/s0218127498001777 ◽

1998 ◽

Vol 08 (11) ◽

pp. 2147-2189 ◽

Cited By ~ 6

Author(s):

J. C. Cathala

Keyword(s):

Phase Plane ◽

Critical Curve ◽

The Other ◽

Two Dimensional ◽

Multiply Connected ◽

Critical Curves ◽

Properties of the basins of noninvertible maps of the plane are studied using the method of critical curves. The paper considers the class of maps having a region of the phase plane where the number of first rank preimages is greater than two. More particularly, the paper gives the conditions of existence of a connected basin, a nonconnected basin and a multiply connected basin in maps where the segments of the critical curve separate the plane into regions, one with three first rank preimages and the other with only one preimage. Then, maps having a maximum number of first rank preimages equal to four are considered.

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

10.1142/s0218127497000972 ◽

1997 ◽

Vol 07 (06) ◽

pp. 1167-1194 ◽

Cited By ~ 44

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 ◽

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.

PLANE FOLIATION OF TWO-DIMENSIONAL NONINVERTIBLE MAPS

10.1142/s0218127496000850 ◽

1996 ◽

Vol 06 (08) ◽

pp. 1439-1462 ◽

Cited By ~ 14

Author(s):

MIRA CHRISTIAN ◽

CARCASSES JEAN-PIERRE ◽

MILLÉRIOUX GILLES ◽

GARDINI LAURA

Keyword(s):

Chaotic Attractors ◽

Two Dimensional ◽

Critical Curves ◽

Rank One ◽

The plane foliation consists in associating a sheet with each determination of the inverse map. Such sheets join by pairs at curves called “critical curves”, locus of points having two coincident rank-one preimages. The knowledge of the foliation, i.e., the sheets organization, is of first importance for understanding the properties of noninvertible maps, in particular those of chaotic attractors, basins, and their bifurcations.

Global properties of symmetric competition models with riddling and blowout phenomena

Discrete Dynamics in Nature and Society ◽

10.1155/s1026022600000492 ◽

2000 ◽

Vol 5 (3) ◽

pp. 149-160 ◽

Cited By ~ 22

Author(s):

Giant-italo Bischi ◽

Laura Gardini

Keyword(s):

Market Share ◽

Discrete Time ◽

Chaos Synchronization ◽

Two Dimensional ◽

Critical Curves ◽

Global Properties ◽

Competition Models ◽

Symmetric Competition ◽

Dynamical Properties ◽

In this paper the problem of chaos synchronization, and the related phenomena of riddling, blowout and on–off intermittency, are considered for discrete time competition models with identical competitors. The global properties which determine the different effects of riddling and blowout bifurcations are studied by the method of critical curves, a tool for the study of the global dynamical properties of two-dimensional noninvertible maps. These techniques are applied to the study of a dynamic market-share competition model.