Narrow Results

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.

The dynamics of a system defined by an endomorphism is essentially different from that of a system defined by a diffeomorphism due to interaction of invariant objects with the so-called critical locus. A planar endomorphism typically folds the phase space along curves J₀ where the Jacobian of the map is singular. The critical locus, denoted J₁, is the image of J₀. It is often only piecewise smooth due to the presence of isolated cusp points that are persistent under perturbation. We investigate what happens when the stable set W_s of a fixed point or periodic orbit interacts with J₁ near such a cusp point C₁. Our approach is in the spirit of bifurcation theory, and we classify the different unfoldings of the codimension-two singularity where the curve W_s is tangent to J₁ exactly at C₁. The analysis uses a local normal-form setup that identifies the possible local phase portraits. These local phase portraits give rise to different global manifestations of the behavior as organized by five different global bifurcation diagrams.

In this paper the one-seller/two-buyer problem with buyer externalities is investigated under the assumption that the two buyers have legal opportunities to cooperate. It is shown that the Competitive equilibrium and the Core are robust with respect to negligible externalities and that the range of market prices in the Core belongs to range of Competitive equilibrium prices. However, these concepts yield no prediction for relatively severe externalities. Therefore, in order to provide a prediction the Bargaining set and the Multilateral Nash (MN) solution are also investigated. Surprisingly, in case of an empty Core the Bargaining set predicts a unique tuple of payoffs which are independent of the externalities and each pair of participants is equally likely. Markets with market imperfections are captured by the MN solution concept. The MN solution yields the paradox that the seller's price can be higher under imperfect competition than under perfect competition.

The core and the stable set are possibly the two most crucially important solution concepts for cooperative games. The relation between the two has been investigated in the context of symmetric transferable utility games and this has been related to the notion of large core. In this paper the relation between the von-Neumann–Morgenstern stability of the core and the largeness of it is investigated in the case of non-transferable utility (NTU) games. The main findings are that under certain regularity conditions, if the core of an NTU game is large then it is a stable set and for symmetric NTU games the core is a stable set if and only if it is large.

Core elements (a la Aubin) of a fuzzy game can be associated with additive separable supporting functions of fuzzy games. Generalized cores whose elements consist of more general separable supporting functions of the game are introduced and studied. While the Aubin core of unanimity games can be empty, the generalized core of unanimity games is nonempty. Properties of the generalized cores and their relations to stable sets are studied. For convex fuzzy games interesting properties are found such as the fact that the generalized core is a unique generalized stable set.

This paper investigates core stability of cooperative (TU) games via a fuzzy extension of the totally balanced cover of a cooperative game. The stability of the core of the fuzzy extension of a game, the concave extension, is shown to reflect the core stability of the original game and vice versa. Stability of the core is then shown to be equivalent to the existence of an equilibrium of a certain correspondence.

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.