In the present paper we introduce and study a generalization of the Hukuhara di¤erence and also generalizations of the Hukuhara di¤erentiability to the case of interval valued functions. We consider several possible de…nitions for the derivative of an interval valued function and we study connections between them and their properties. Using these concepts we study interval di¤erential equations. Local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval di¤erential equation by ODE systems and by di¤erential algebraic equations. We also show some connection with di¤erential inclusions. The thoretical results are turned into practical algorithms to solve interval di¤erential equations.
In the present paper, using novel generalizations of the Hukuhara di¤erence for fuzzy sets, we introduce and study new generalized di¤erentiability concepts for fuzzy valued functions. Several properties of the new concepts are investigated and they are compared to similar fuzzy di¤erentiabilities …nding connections between them. Characterization and relatively simple expressions are provided for the new derivatives.
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 x 1 x 2 , 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. # 1998 IMACS/Elsevier Science B.V.
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