ON SOME GLOBAL BIFURCATIONS OF THE DOMAINS OF FEASIBLE TRAJECTORIES: AN ANALYSIS OF RECURRENCE EQUATIONS

2005 ◽  
Vol 15 (05) ◽  
pp. 1625-1639 ◽  
Author(s):  
EN-GUO GU ◽  
JIONG RUAN
Keyword(s):  
Fixed Point ◽  
Stable Manifold ◽  
The Other ◽  
Computer Assisted ◽  
Two Dimensional ◽  
Global Bifurcations ◽  
Basin Boundary ◽  
Recurrence Equations ◽  
Critical Curves ◽  
Global Properties

This paper is an attempt to give new results, by a computer-assisted study, on some global bifurcations that change the structure of the domain of feasible trajectories (bounded discrete trajectories having an ecological sense) which can be obtained by the union of all rank preimages of axes. Three two-dimensional recurrence equations (or maps) are analyzed. The two first maps are degenerated invertible maps (i.e. the inverses of them are well defined except a set of zero lebergue measure) for which the basins of attractor are obtained by the backward iteration of a stable manifold of a saddle fixed point belonging to the basin boundary, and the interior domains of feasible trajectories are given by the intersection between the basin of attractor and the first quadrant. The other is a noninvertible map which is investigated by the use of critical curves, a powerful tool for the analysis of global properties of two-dimensional maps.

GLOBAL BIFURCATIONS OF DOMAINS OF FEASIBLE TRAJECTORIES: AN ANALYSIS OF A DISCRETE PREDATOR–PREY MODEL

2006 ◽  
Vol 16 (09) ◽  
pp. 2601-2613 ◽  
Author(s):  
EN-GUO GU ◽  
YIBING HUANG
Keyword(s):  
Fixed Point ◽  
Domain Boundary ◽  
Computer Assisted ◽  
Two Dimensional ◽  
Global Bifurcations ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Global Properties ◽  
Singular Sets ◽  
Domain Boundaries

This paper aims to provide some new results, by a computer-assisted study, on some global bifurcations that change the domains of feasible trajectories (bounded discrete trajectories having an ecological sense). It is shown that the domain boundaries of two-dimensional maps can be generally obtained by the union of all rank preimages of two axes. A discrete predator–prey ecosystem is employed to demonstrate how the global properties of persistence in a biological model can be analyzed. The main results of this paper are from the study of some global bifurcations that change the stable manifold of a saddle fixed point belonging to the domain boundary. The basin fractalization is explained by using two types of nonclassical singular sets. The first one, critical curve, separates the plane into two regions having a different number of real inverses (here zero and two). The second one is a line of nondefinition for one of the two inverses of the map, i.e. this inverse has a vanishing denominator on this line.

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.

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.

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".

FROM THE BOX-WITHIN-A-BOX BIFURCATION STRUCTURE TO THE JULIA SET PART II: BIFURCATION ROUTES TO DIFFERENT JULIA SETS FROM AN INDIRECT EMBEDDING OF A QUADRATIC COMPLEX MAP

2009 ◽  
Vol 19 (10) ◽  
pp. 3235-3282 ◽  
Author(s):  
CHRISTIAN MIRA ◽  
ANNA AGLIARI ◽  
LAURA GARDINI
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Real Variable ◽  
Detailed Knowledge ◽  
Two Dimensional ◽  
Global Bifurcations ◽  
Bifurcation Structure ◽  
Basin Boundary ◽  
Fractal Organization ◽  
Quadratic Complex

Part I of this paper has been devoted to properties of the different Julia set configurations, generated by the complex map TZ: z′ = z2 - c, c being a real parameter, -1/4 < c < 2. These properties were revisited from a detailed knowledge of the fractal organization (called "box-within-a-box"), generated by the map x′ = x2 - c with x a real variable. Here, the second part deals with an embedding of TZ into the two-dimensional noninvertible map [Formula: see text]; y′ = γ y + 4x2y, γ ≥ 0. For [Formula: see text] is semiconjugate to TZ in the invariant half plane (y ≤ 0). With a given value of c, and with γ decreasing, the identification of the global bifurcations sequence when γ → 0, permits to explain a route toward the Julia sets, from a study of the basin boundary of the attractor located on y = 0.

scholarly journals Properties of RG interfaces for 2D boundary flows

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Anatoly Konechny
Keyword(s):  
Boundary Condition ◽  
Fixed Point ◽  
Variational Method ◽  
Boundary Conditions ◽  
The Other ◽  
Two Dimensional ◽  
Conformal Boundary ◽  
First Approximation ◽  
Local Operators

Abstract We consider RG interfaces for boundary RG flows in two-dimensional QFTs. Such interfaces are particular boundary condition changing operators linking the UV and IR conformal boundary conditions. We refer to them as RG operators. In this paper we study their general properties putting forward a number of conjectures. We conjecture that an RG operator is always a conformal primary such that the OPE of this operator with its conjugate must contain the perturbing UV operator when taken in one order and the leading irrelevant operator (when it exists) along which the flow enters the IR fixed point, when taken in the other order. We support our conjectures by perturbative calculations for flows between nearby fixed points, by a non-perturbative variational method inspired by the variational method proposed by J. Cardy for massive RG flows, and by numerical results obtained using boundary TCSA. The variational method has a merit of its own as it can be used as a first approximation in charting the global structure of the space of boundary RG flows. We also discuss the role of the RG operators in the transport of states and local operators. Some of our considerations can be generalised to two-dimensional bulk flows, clarifying some conceptual issues related to the RG interface put forward by D. Gaiotto for bulk 𝜙1,3 flows.

scholarly journals Two-dimensional Dynamics of Cubic Maps

2021 ◽  
Vol 45 (03) ◽  
pp. 427-438
Author(s):  
I. DJELLIT ◽  
W. SELMANI
Keyword(s):  
Bifurcation Analysis ◽  
Basins Of Attraction ◽  
Global Structure ◽  
Two Dimensional ◽  
Global Bifurcations ◽  
Global Properties ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis ◽  
The Creation

We investigate the global properties of two cubic maps on the plane, we try to explain the basic mechanisms of global bifurcations leading to the creation of nonconnected basins of attraction. It is shown that in some certain conditions the global structure of such systems can be simple. The main results here can be seen as an improvement of the results of stability and bifurcation analysis.

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.

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.

EXTENSIONS OF THE NOTION OF CHAOTIC AREA IN SECOND-ORDER ENDOMORPHISMS

1995 ◽  
Vol 05 (03) ◽  
pp. 751-777 ◽  
Author(s):  
A. BARUGOLA ◽  
J.C. CATHALA ◽  
C. MIRA
Keyword(s):  
Fixed Point ◽  
Finite Number ◽  
Periodic Point ◽  
Critical Points ◽  
Unstable Manifold ◽  
Two Dimensional ◽  
One Dimensional ◽  
Critical Curves ◽  
The One ◽  
Invariant Domains

Properties of chaotic areas (i.e. invariant domains of points positively stable in the Poisson’s sense) of non-invertible maps of the plane are studied by using the method of critical curves (two-dimensional extension of the notion of critical points in the one-dimensional case). The classical situation is that of a chaotic area bounded by a finite number of critical curves segments. This paper considers another class of chaotic areas bounded by the union of critical curves segments and segments of the unstable manifold of a saddle fixed point, or that of saddle cycle (periodic point). Different configurations are examined, as their bifurcations when a map parameter varies.

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