Probability density functions of some skew tent maps

Random talk: Random walk and synchronizability in a moving neighborhood network

“…Here we show that suitably adapting our methods of proof in [25], and adding further substantial arguments, now allows us to rigorously address the problem where the fast switching is due to stochastically moving agents, similar to what we studied empirically in [24]. Furthermore, it is well known that for some classes of dynamical systems, the socalled "Markov maps" are dense in an appropriate space of ergodic transformations [28], such as skew tent maps [29]. Consequently, we believe the treatment via Markov chains in this paper provides a significant step toward the analysis of a broad class of moving neighborhood networks.…”

Comparing dynamical systems by a graph matching method

“…As example of this approximation via grid refinement, L. Billing and E. Bollt proved that the family of skew tent map is Markov for a dense set of parameters in the chaotic region [10]. In particular, in any given neighborhood of a non-Markov map in this chaotic region, there exists a Markov map that uniformly approximates the non-Markov one.…”

Feedback control with finite accuracy: more knowledge and better control for free

“…We analyze the time average asymptotic behavior of the controlled system by studying the invariant measures for maps from this family. The density function is easily and directly calculated in the case of rational β by the dominant eigenvector of a Markov transition matrix [5,6] and is well approximated by a pdf of a nearby map of rational slope, in the case of irrational β [7]. 4 If β is an integer, the invariant density is uniform.…”