The global superstability of a dynamical system is allowed when all bounded orbits of this system are superstable, .i.e. There exist a minus infinity Lyapunouv exponent. In this paper it is shown that any globally superstable 2-D quadratic map is conjugate to the 1D quadratic map. This result allow us to determine some forms of superstable 1-D quadratic maps.

We show, assuming a mild set-theoretic hypothesis, that if an abstract elementary class (AEC) has a superstable-like forking notion for models of cardinality λ and a superstable-like forking notion for models of cardinality λ+, then orbital types over models of cardinality λ+ are determined by their restrictions to submodels of cardinality λ. By a superstable-like forking notion, we mean here a...

We consider chip-firing dynamics defined by arbitrary M-matrices. M-matrices generalize graph Laplacians and were shown by Gabrielov to yield avalanche finite systems. Building on the work of Baker and Shokrieh, we extend the concept of energy minimizing chip configurations. Given an M-matrix, we show that there exists a unique energy minimizing configuration in each equivalence class defined b...

S. Shelah [20] has given a description of all countable complete theories T that permit a classification of their models. A necessary condition for T is superstability. A stable (superstable) group is a group interpreted in some stable (superstable) structure. It is equivalent to say that the groups investigated have additional relations and functions such that the whole structure is stable (su...

We prove that a non-abelian superstable CSA-group has an infinite definable simple subgroup all of whose proper definable subgroups are abelian. This imply in particular that the existence of non-abelian CSAgroup of finite Morley rank is equivalent to the existence of a simple bad group all whose definable proper subgroups are abelian. We give a new proof of a result of E. Mustafin and B. Poiza...

This paper studies rich superstable phenomena in a nonautonomous piecewise constant circuit including one impulsive switch. Since the vector field of circuit equation is piecewise constant, embedded return map is piecewise linear and can be described explicitly in principle. As parameters vary the map can have infinite extrema with one flat segment. Such maps can cause complicated periodic orbi...

We study superstable groups acting on trees. We prove that an action of an ω-stable group on a simplicial tree is trivial. This shows that an HNN-extension or a nontrivial free product with amalgamation is not ωstable. It is also shown that if G is a superstable group acting nontrivially on a Λ-tree, where Λ = Z or Λ = R, and if G is either α-connected and Λ = Z, or if the action is irreducible...