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.

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.

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 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.

On the Fractal Structure of Basin Boundaries in Two-Dimensional Noninvertible Maps

2003 ◽  
Vol 13 (07) ◽  
pp. 1767-1785 ◽  
Author(s):  
A. Agliari ◽  
L. Gardini ◽  
C. Mira
Keyword(s):  
Fractal Structure ◽  
Real Plane ◽  
Two Dimensional ◽  
Numerical Examples ◽  
The Real ◽  
Basin Boundary ◽  
Fractal Basin Boundary ◽  
Noninvertible Maps ◽  
Analytical Tools ◽  
Basin Boundaries

In this paper we give an example of transition to fractal basin boundary in a two-dimensional map coming from the applicative context, in which the hard-fractal structure can be rigorously proved. That is, not only via numerical examples, although theoretically guided, as often occurs in maps coming from the applications, but also via analytical tools. The proposed example connects the two-dimensional maps of the real plane to the well-known complex map.

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.

ON CRITICAL CURVES IN TWO-DIMENSIONAL ENDOMORPHISMS

2001 ◽  
Vol 11 (03) ◽  
pp. 821-839 ◽  
Author(s):  
J. C. CATHALA
Keyword(s):  
Critical Curve ◽  
Two Dimensional ◽  
Curve Segment ◽  
Fundamental Properties ◽  
Critical Curves ◽  
Parameter Variations ◽  
Noninvertible Maps

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

1998 ◽  
Vol 08 (11) ◽  
pp. 2147-2189 ◽  
Author(s):  
J. C. Cathala
Keyword(s):  
Phase Plane ◽  
Critical Curve ◽  
The Other ◽  
Two Dimensional ◽  
Multiply Connected ◽  
Critical Curves ◽  
Noninvertible Maps

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.

PLANE FOLIATION OF TWO-DIMENSIONAL NONINVERTIBLE MAPS

1996 ◽  
Vol 06 (08) ◽  
pp. 1439-1462 ◽  
Author(s):  
MIRA CHRISTIAN ◽  
CARCASSES JEAN-PIERRE ◽  
MILLÉRIOUX GILLES ◽  
GARDINI LAURA
Keyword(s):  
Chaotic Attractors ◽  
Two Dimensional ◽  
Critical Curves ◽  
Rank One ◽  
Noninvertible Maps

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.

scholarly journals BIFURCATIONS OF STABLE SETS IN NONINVERTIBLE PLANAR MAPS

2005 ◽  
Vol 15 (03) ◽  
pp. 891-904 ◽  
Author(s):  
J. P. ENGLAND ◽  
B. KRAUSKOPF ◽  
H. M. OSINGA
Keyword(s):  
Basin Of Attraction ◽  
Critical Curve ◽  
Point Of View ◽  
Stable Set ◽  
Stable Sets ◽  
Codimension One ◽  
Critical Curves ◽  
Theory Point ◽  
Planar Maps ◽  
Noninvertible Maps

Many applications give rise to systems that can be described by maps that do not have a unique inverse. We consider here the case of a planar noninvertible map. Such a map folds the phase plane, so that there are regions with different numbers of preimages. The locus, where the number of preimages changes, is made up of so-called critical curves, that are defined as the images of the locus where the Jacobian is singular. A typical critical curve corresponds to a fold under the map, so that the number of preimages changes by two.We consider the question of how the stable set of a hyperbolic saddle of a planar noninvertible map changes when a parameter is varied. The stable set is the generalization of the stable manifold for the case of an invertible map. Owing to the changing number of preimages, the stable set of a noninvertible map may consist of finitely or even infinitely many disjoint branches. It is now possible to compute stable sets with the Search Circle algorithm that we developed recently.We take a bifurcation theory point of view and consider the two basic codimension-one interactions of the stable set with a critical curve, which we call the outer-fold and the inner-fold bifurcations. By taking into account how the stable set is organized globally, these two bifurcations allow one to classify the different possible changes to the structure of a basin of attraction that are reported in the literature. The fundamental difference between the stable set and the unstable manifold is discussed. The results are motivated and illustrated with a single example of a two-parameter family of planar noninvertible maps.

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