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The relation between the complexity of a time-switched dynamics and the complexity of its control sequence depends critically on the concept of a non-autonomous pullback attractor. For instance, the switched dynamics associated with scalar dissipative affine maps has a pullback attractor consisting of singleton component sets. This entails that the complexity of the control sequence and switched dynamics, as quantified by the topological entropy, coincide. In this paper we extend the previous framework to pullback attractors with nontrivial components sets in order to gain further insights in that relation. This calls, in particular, for distinguishing two distinct contributions to the complexity of the switched dynamics. One proceeds from trajectory segments connecting different component sets of the attractor; the other contribution proceeds from trajectory segments within the component sets. We call them “macroscopic” and “microscopic” complexity, respectively, because only the first one can be measured by our analytical tools. As a result of this picture, we obtain sufficient conditions for a switching system to be more complex than its unswitched subsystems, i.e., a complexity analogue of Parrondo’s paradox.
Cyrioctea (Araneae, Zodariidae) in Africa: temperate Gondwanaland relict, recent radiation, or both?
(2013)
Two new species of the zodariid genus Cyrioctea Simon, 1889 are described: C. sawadee sp. nov. and C. lotzi sp. nov., both only known from males. The genus now contains seven Afrotropical species and this abundance is discussed in the context of its basal situation in the family and its apparent temperate Gondwanaland distribution, which implies a much greater age of the Zodariidae than presently accepted. Unlike most taxa with a temperate Gondwana distribution, Cyrioctea boasts a high number of species with small distribution areas. This points in the direction of a recent radiation initiated after a long period of stasis.