Laura Gardini

Chaotic dynamics in two-dimensional noninvertible maps

Part 1: Generalities and definition one-dimensional endomorphisms the simplest type of two-dimensional endomorphisms. Part 2: Properties of the phase plane simplest type of two-dimensional endomorphisms. Part 3: Bifurcations more complex types of two-dimensional endomorphisms generalizations to higher dimensions.

Speculative behaviour and complex asset price dynamics: a global analysis

This paper analyses the dynamics of a model of a share market consisting of two groups of traders: fundamentalists, who base their trading decisions on the expectation of a return to the fundamental value of the asset, and chartists, who base their trading decisions on an analysis of past price trends. The model is reduced to a two-dimensional map whose global dynamic behaviour is analysed in detail. The dynamics are affected by parameters measuring the strength of fundamentalist demand and the speed with which chartists adjust their estimate of the trend to past price changes. The parameter space is characterized according to the local stability/instability of the equilibrium point as well as the non-invertibility of the map. The method of critical curves of non-invertible maps is used to understand and describe the range of global bifurcations that can occur. It is also shown how the knowledge of deterministic dynamics uncovered here can aid in understanding the behaviour of stochastic versions of the model.

Multistability and cyclic attractors in duopoly games

A dynamic Cournot duopoly game, whose time evolution is modeled by the iteration of a map T :Ox; yU!O r 1O yU ; r 2O xUU, is considered. Results on the existence of cycles and more complex attractors are given, based on the study of the one-dimensional map FOxUaO r 1 r 2UOxU. The property of multistability, i.e. the existence of many coexisting attractors (that may be cycles or cyclic chaotic sets), is proved to be a characteristic property of such games. The problem of the delimitation of the attractors and of their basins is studied. These general results are applied to the study of a particular duopoly game, proposed in M. Kopel [Chaos, Solitons & Fractals, 7 (12) (1996) 2031‐2048] as a model of an economic system, in which the reaction functions r1 and r2 are logistic maps. ” 2000 Elsevier Science Ltd. All rights reserved.

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

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.

Synchronization, intermittency and critical curves in a duopoly game

The phenomenon of synchronization of a two-dimensional discrete dynamical system is studied for the model of an economic duopoly game, whose time evolution is obtained by the iteration of a noninvertible map of the plane. In the case of identical players the map has a symmetry property that implies the invariance of the diagonal x1=x2, so that synchronized dynamics is possible. The basic question is whether an attractor of the one-dimensional restriction of the map to the diagonal is also an attractor for the two-dimensional map, and in which sense. In this paper, a particular dynamic duopoly game is considered for which the local study of the transverse stability, in a neighborhood of the invariant submanifold in which synchronized dynamics takes place, is combined with a study of the global behavior of the map. When measure theoretic, but not topological, attractors are present on the invariant diagonal, intermittency phenomena are observed. The global behavior of the noninvertible map is investigated by studying of the critical manifolds of the map, by which a two-dimensional region is defined that gives an upper bound to the amplitude of intermittent trajectories. Global bifurcations of the basins of attraction are evidenced through contacts between critical curves and basin boundaries.

DEGENERATE BIFURCATIONS AND BORDER COLLISIONS IN PIECEWISE SMOOTH 1D AND 2D MAPS

We recall three well-known theorems related to the simplest codimension-one bifurcations occurring in discrete time dynamical systems, such as the fold, flip and Neimark–Sacker bifurcations, and analyze these bifurcations in presence of certain degeneracy conditions, when the above mentioned theorems are not applied. The occurrence of such degenerate bifurcations is particularly important in piecewise smooth maps, for which it is not possible to specify in general the result of the bifurcation, as it strongly depends on the global properties of the map. In fact, the degenerate bifurcations mainly occur in piecewise smooth maps defined in some subspace of the phase space by a linear or linear-fractional function, although not necessarily only by such functions. We also discuss the relation between degenerate bifurcations and border-collision bifurcations.

Analysis of global bifurcations in a market share attraction model

In this paper we demonstrate how the global dynamics of an economic model can be analyzed. In particular, as an application, we consider a market share attraction model widely used in the analysis of interbrand competition in marketing theory. We analyze the local and global dynamic properties of the resulting two-dimensional noninvertible dynamical system in discrete time. The main result of this paper is given by the study of some global bifurcations that change the structure of the attractors and their basins. These bifurcations are investigated by the use of critical curves, a powerful tool for the analysis of the global properties of noninvertible two-dimensional maps. ( 2000 Elsevier Science B.V. All rights reserved. JEL classixcation: E32; M30

The dynamics of a triopoly Cournot game

This paper reconsiders the Cournot oligopoly (noncooperative) game with iso-elastic demand and constant marginal costs, one of the rare cases where the reaction functions can be derived in closed form. It focuses the case of three competitors, and so also extends the critical line method for non-invertible maps to the study of critical surfaces in 3D. By this method the various bifurcations of the attractors and their basins are studied. As a special case the restriction of the map to an invariant plane when two of the three firms are identical is focused.

Homoclinic bifurcations in n -dimensional endomorphisms, due to expanding periodic points

The study of bifurcation mechanisms which determine the transition from regular to chaotic behaviour (or “route to chaos”) in endomorphisms (maps with a nonunique inverse) has reached a good level of theoretical knowledge for one-dimensional maps. We refer to the book by Mira [l] as a review of the main results in this research area. The situation is quite different as regards endomorphisms of dimensions higher than one. A few results have been worked out until now [2-81, mainly in the simplest cases of two-dimensional endomorphisms. However, many mathematical models of interest in the applications are ultimately described by maps with a nonunique inverse. Clearly, in the plane IR2, or in higher-dimensional spaces, we can no longer take advantage of the Koenigs-Lemeray graphical construction, and of the qualitative behaviour of the graphs of powers of t(x) (t’(x), t3(x), . ..). which are powerful tools to identify the local and global bifurcations of one-dimensional endomorphisms, x’ = t(x), x E IR’. However, to study bifurcation mechanisms in IR”, n 2 2, we make use of properties which are the natural generalizations of those used in the one-dimensional case. Indeed, a basic tool for studying the dynamics and bifurcation mechanisms of endomorphisms is provided by (the analysis of) the critical manifolds, first introduced by Gumowski and Mira in 1965 (see [2] and references therein). The critical manifolds are the natural generalization to IR”, n 2 2, of the local extrema (also called critical points of rank-l) of one-dimensional endomorphisms. In this work, we deal with homoclinic bifurcations in endomorphisms of IR” (n 1 l), due to expanding periodic points. As in one-dimensional endomorphisms, we will see that critical manifolds play a pre-eminent role. Our purpose is to contribute to the understanding of the bifurcation mechanisms related to homoclinic orbits occurring in n-dimensional endomorphisms, mechanisms strictly related to the nonuniqueness of the inverse function, that is, typical of endomorphisms, not present in invertible maps, although similar to those occurring in these maps. In particular, this work concerns the global bifurcation (homoclinic bifurcations) occurring when a repelling fixed point becomes a “snap-back repeller”t (SBR henceforth), and in general when there exists a critical orbit homoclinic to an expanding

A DOUBLE LOGISTIC MAP

Several endomorphisms of a plane have been constructed by coupling two logistic maps. Here we study the dynamics occurring in one of them, a twisted version due to J. Dorband, which (like the other models) is rich in global bifurcations. By use of critical curves, absorbing and invariant areas are determined, inside which global bifurcations of the attracting sets (fixed points, closed invariant curves, cycles or chaotic attractors) take place. The basins of attraction of the absorbing areas are determined together with their bifurcations.