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.

scholarly journals Homoclinic and heteroclinic bifurcations in a two-dimensional endomorphism

2003 ◽  
Vol 16 (2) ◽  
pp. 273-283
Author(s):  
Ilham Djellit ◽  
Mohamed Ferchichi
Keyword(s):  
Saddle Point ◽  
Chaotic Attractor ◽  
Invariant Manifolds ◽  
Basin Of Attraction ◽  
Homoclinic Orbits ◽  
Specific Information ◽  
Two Dimensional ◽  
Global Bifurcations ◽  
Homoclinic Bifurcations ◽  
Noninvertible Maps

Our study concerns global bifurcations occurring in noninvertible maps, it consists to show that there exists a link between contact bifurcations of a chaotic attractor and homoclinic bifurcations of a saddle point or a saddle cycle being on the boundary of the chaotic attractor. We provide specific information about the intricate dynamics near such points. We study particularly a two-dimensional endomorphism of (Z\ - Z$ - Z\) type. We will show that points of contact, between boundary of the attractor and its basin of attraction, converge toward the saddle point or the saddle cycle. These points of contact are also points of intersection between the stable and unstable invariant manifolds. This gives rise to the birth of homoclinic orbits (homoclinic bifurcations).

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.

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.

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.

MULTIPARAMETRIC BIFURCATIONS IN AN ENZYME-CATALYZED REACTION MODEL

2005 ◽  
Vol 15 (03) ◽  
pp. 905-947 ◽  
Author(s):  
E. FREIRE ◽  
L. PIZARRO ◽  
A. J. RODRÍGUEZ-LUIS ◽  
F. FERNÁNDEZ-SÁNCHEZ
Keyword(s):  
Periodic Orbits ◽  
Dynamical Behavior ◽  
Homoclinic Orbits ◽  
Reaction Model ◽  
Numerical Continuation ◽  
Global Bifurcations ◽  
Codimension One ◽  
Local Bifurcations ◽  
Homoclinic Connections ◽  
Local And Global Bifurcations

An exhaustive analysis of local and global bifurcations in an enzyme-catalyzed reaction model is carried out. The model, given by a planar five-parameter system of autonomous ordinary differential equations, presents a great richness of bifurcations. This enzyme-catalyzed model has been considered previously by several authors, but they only detected a minimal part of the dynamical and bifurcation behavior exhibited by the system. First, we study local bifurcations of equilibria up to codimension-three (saddle-node, cusps, nondegenerate and degenerate Hopf bifurcations, and nondegenerate and degenerate Bogdanov–Takens bifurcations) by using analytical and numerical techniques. The numerical continuation of curves of global bifurcations allows to improve the results provided by the study of local bifurcations of equilibria and to detect new homoclinic connections of codimension-three. Our analysis shows that such a system exhibits up to sixteen different kinds of homoclinic orbits and thirty different configurations of equilibria and periodic orbits. The coexistence of up to five periodic orbits is also pointed out. Several bifurcation sets are sketched in order to show the dynamical behavior the system exhibits. The different codimension-one and -two bifurcations are organized around five codimension-three degeneracies.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document