Narrow Results

This paper is the second part of an earlier work devoted to the properties specific to maps of the plane characterized by the presence of a vanishing denominator, which gives rise to the generation of new types of singularities, called set of nondefinition, focal points and prefocal curves. A prefocal curve is a set of points which are mapped (or "focalized") into a single point, called focal point, by the inverse map when it is invertible, or by at least one of the inverses when it is noninvertible. In the case of noninvertible maps, the previous text dealt with the simplest geometrical situation, which is nongeneric. To be more precise this situation occurs when several focal points are associated with a given prefocal curve. The present paper defines the generic case for which only one focal point is associated with a given prefocal curve. This is due to the fact that only one inverse of the map has the property of focalization, but with properties different from those of invertible maps. Then the noninvertible maps of the previous Part I appear as resulting from a bifurcation leading to the merging of two prefocal curves, without merging of two focal points.

The present work describes a family of polynomial noninvertible maps of the plane shared within two open regions: (i) (denoted by Z₀) each point having no real preimage, and (ii) (denoted by Z₂) each point having two real preimages. The regions Z₀, Z₂ are separated by the critical curve LC, locus of points having two coincident preimages. Z₂ is made up of two sheets (giving rise to a plane foliation) joining on LC, each being associated with a well-defined inverse of the map. The considered maps family is structurally unstable. For a wide choice of the parameter space it generates a singular foliation in the sense that the region Z₂ is separated into two zones, Z'₂ and , inside which the two preimages do not have the same qualitative behavior. Moreover, the boundary between Z'₂ and is made up of points having only one real preimage at finite distance, the second one being at infinity. This situation gives rise to a nonclassical homoclinic bifurcation. The maps of the family have another important feature: their inverses present a denominator which vanish along a line of the plane. This has a great consequence on a chaotic attractor structure, when it exists. The imbedding of the map into a wider structurally stable family, generating regions Z_k (k=1,3 being the number of real preimages), permits to understand the foliation nature when the imbedding parameter cancels leading to the structurally unstable map.

This paper continues the study of the global dynamic properties specific to maps of the plane characterized by the presence of a denominator that vanishes in a one-dimensional submanifold. After two previous papers by the same authors, where the definitions of new kinds of singularities, called focal points and prefocal sets, are given, as well as the particular structures of the basins and the global bifurcations related to the presence of such singularities, this third paper is devoted to the analysis of nonsimple focal points, and the bifurcations associated with them. We prove the existence of a one-to-one relation between the points of a prefocal curve and arcs through the focal point having all the same tangent but different curvatures. In the case of nonsimple focal points, such a relation replaces the one-to-one correspondence between the slopes of arcs through a focal point and the points along the associated prefocal curve that have been proved and extensively discussed in the previous papers. Moreover, when dealing with noninvertible maps, other kinds of relations can be obtained in the presence of nonsimple focal points or prefocal curves, and some of them are associated with qualitative changes of the critical sets, i.e. with the structure of the Riemann foliation of the plane.

The present paper focuses on the two time scale dynamics generated by 2D polynomial noninvertible maps T of (Z₀ - Z₂) and (Z₁ - Z₃ - Z₁) types. This symbolism, specific to noninvertible maps, means that the phase plane is partitioned into zones Z_k, where each point possesses the k real rank-one preimages. Of special interest here is the structure of slow and fast motion sets of such maps. The formation mechanism of a stable invariant close curve through the interaction of fast and slow dynamics, as well as its transformation into a canard are studied. A few among the plethora of chaotic attractors and chaotic transients produced by such maps are described as well.

Two-dimensional (Z₁–Z₃–Z₁) maps are such that the plane is divided into three unbounded open regions: a region Z₃, whose points generate three real rank-one preimages, bordered by two regions Z₁, whose points generate only one real rank-one preimage. This paper is essentially devoted to the study of the structures, and the global bifurcations, of the basins of attraction generated by such maps. In particular, the cases of fractal structure of such basins are considered. For the class of maps considered in this paper, a large variety of dynamic situations is shown, and the bifurcations leading to their occurrence are explained.

We develop an asset pricing model based on the interaction of heterogeneous trading groups. In addition to the two main trader groups, fundamentalists and trend-chasing chartists, we include a third significant group known as contrarian chartists. We model the case of opportunistic contrarian behavior, where the contrarian group disagrees with the trend-chasing chartists only when the return differential is high. We also consider absolute contrarian behavior, in which the contrarians consistently disagree with trend-chasers. The models are nonlinear planar maps, exhibiting period doubling, Neimark–Sacker and global bifurcations leading to local chaotic behavior. Absolute contrarian behavior is found to have a moderating effect on price change, while opportunistic contrarian behavior is found to further complicate the price cycles present in other models.

We study the dynamics of three planar, noninvertible maps which rotate and fold the plane. Two maps are inspired by real-world applications whereas the third map is constructed to serve as a toy model for the other two maps. The dynamics of the three maps are remarkably similar. A stable fixed point bifurcates through a Hopf–Neĭmark–Sacker which leads to a countably infinite set of resonance tongues in the parameter plane of the map. Within a resonance tongue a periodic point can bifurcate through a period-doubling cascade. At the end of the cascade we detect Hénon-like attractors which are conjectured to be the closure of the unstable manifold of a saddle periodic point. These attractors have a folded structure which can be explained by means of the concept of critical lines. We also detect snap-back repellers which can either coexist with Hénon-like attractors or which can be formed when the saddle-point of a Hénon-like attractor becomes a source.

In this paper we consider a two species population model based on the discretisation of the original Lotka-Volterra competition equations. We analyse the global dynamic properties of the resulting two-dimensional noninvertible dynamical system in the case where the interspecific competition is considered to be “mixed”. The main results of this paper are derived from the study of some global bifurcations that change the structure of the attractors and their basins. These bifurcations are investigated using the method of Critical Curves.