This paper was extensively circulated in manuscript form beginning in the Summer of 1989. It is being published here for the first time in its original form except for minor corrections, updated references and some concluding comments.
We use one-dimensional techniques to characterize the Devil’s staircase route to chaos in a relaxation oscillator of the van der Pol type with periodic forcing term. In particular, by using symbolic dynamics, we give the behaviour for certain range of parameter values of a Cantor set of solutions having a certain rotation set associated to a rational number. Finally, we explain the phenomena observed experimentally in the system by Kennedy, Krieg and Chua (in [10]) related with the appearance of...
Let f: X→ X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m ∈ ℕ, is transitive, (ii) for each m ∈ ℕ, there exists x ∈ X such that the diagonal m-tuple (x,x,...,x) has a dense orbit in under the action of . We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying (ii)....
In 2008 Juhász and Szentmiklóssy established that for every compact space there exists a discrete with . We generalize this result in two directions: the first one is to prove that the same holds for any Lindelöf -space and hence is -separable. We give an example of a countably compact space such that is not -separable. On the other hand, we show that for any Lindelöf -space there exists a discrete subset such that ; in particular, the diagonal is a retract of and the projection...
There is a general conjecture, the dichotomy (C) about Borel equivalence relations E: (i) E is Borel reducible to the equivalence relation where X is a Polish space, and a Polish group acting continuously on X; or (ii) a canonical relation is Borel reducible to E. (C) is only proved for special cases as in [So]. In this paper we make a contribution to the study of (C): a stronger conjecture is true for hereditary subspaces of the Polish space of real sequences, i.e., subspaces such that ...
This paper is concerned with distributional chaos of time-varying discrete systems in metric spaces. Some basic concepts are introduced for general time-varying systems, including sequentially distributive chaos, weak mixing, and mixing. We give an example of sequentially distributive chaos of finite-dimensional linear time-varying dynamical systems, which is not distributively chaotic of type i (DCi for short, i = 1, 2). We also prove that two uniformly topological equiconjugate time-varying systems...