<p style='text-indent:20px;'>We present a mechanism for the emergence of strange attractors in one-parameter family differential equations defined on 3-dimensional sphere. When parameter is zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between saddles-foci with different Morse indices. Afte...

This paper concerns with the reducibility loss of (periodic) invariant curves of quasi-periodically forced one dimensional maps and its relationship with the renormalization operator. Let gα be a one-parametric family of one dimensional maps with a cascade of period doubling bifurcations. Between each of these bifurcations, there exists a parameter value αn such that gαn has a superstable perio...

We assume that M is a stable homogeneous model of large cardinality. We prove a nonstructure theorem for (slightly saturated) elementary submodels of M, assuming M has dop. We do not assume that th(M) is stable. In this paper we study elementary submodels of a stable homogeneous L-structure M. We use M as a monster model used in “classical” stability theory and so as in [HS], we assume that |M|...

We give a model theoretic proof, replacing admisssible set theory by the LopezEscobar theorem, of Makkai’s theorem: Every counterexample to Vaught’s conjecture has an uncountable model which realizes only countably many Lω1,ω-types. The following two results are new. Theorem I. If a first order theory is a counterexample to the Vaught conjecture then it has 2א1 models of cardinality א1. Theorem...

Abstract We uncover a connection between the model-theoretic notion of superstability and that noetherian rings pure-semisimple rings. characterize via class left modules with embeddings. Theorem 0.1 For ring R following are equivalent. (1) is noetherian. (2) The R-modules embeddings superstable. (3) every ? ? | + ? 0 , there ? such has uniqueness limit models cardinality ?. (4) Every model in ...

We obtain a characterization of left perfect rings via superstability the class flat modules with pure embeddings. $\mathbf{Theorem.}$ For ring $R$ following are equivalent. - is perfect. The $R$-modules embeddings superstable. There exists $\lambda \geq (|R| + \aleph_0)^+$ such that has uniqueness limit models cardinality $\lambda$. Every model in $\Sigma$-cotorsion. A key step our argum...

Motivated by the notion of chip-firing on dual graph a planar graph, we consider ‘integral flow chip-firing’ an arbitrary G. The rule is governed $${\mathcal {L}}^*(G)$$ , Laplacian G determined choosing basis for lattice integral flows We show that any admits such so M-matrix, leading to firing these elements avalanche finite. This follows from more general result bases lattices may be indepen...