Estimation of the minimal number of periodic points for smooth self-maps of odd dimensional real projective spaces

AbstractLet f be a smooth self-map of a closed connected manifold of dimension m⩾3. The authors introduced in [G. Graff, J. Jezierski, Minimizing the number of periodic points for smooth maps. Non-simply connected case, Topology Appl. 158 (3) (2011) 276–290] the topological invariant NJDr[f], where r is a fixed natural number, which is equal to the minimal number of r-periodic points in the smooth homotopy class of f. In this paper smooth self-maps of real projective space RPm, where m>3 is odd, are considered and the estimations from below and above for NJDr[f] are given

Persistent homology as a new method of the assessment of heart rate variability

Heart rate variability (hrv) is a physiological phenomenon of the variation in the length of the time interval between consecutive heartbeats. In many cases it could be an indicator of the development of pathological states. The classical approach to the analysis of hrv includes time domain methods and frequency domain methods. However, attempts are still being made to define new and more effective hrv assessment tools. Persistent homology is a novel data analysis tool developed in the recent decades that is rooted at algebraic topology. The Topological Data Analysis (TDA) approach focuses on examining the shape of the data in terms of connectedness and holes, and has recently proved to be very effective in various fields of research. In this paper we propose the use of persistent homology to the hrv analysis. We recall selected topological descriptors used in the literature and we introduce some new topological descriptors that reflect the specificity of hrv, and we discuss their relation to the standard hrv measures. In particular, we show that this novel approach provides a collection of indices that might be at least as useful as the classical parameters in differentiating between series of beat-to-beat intervals (RR-intervals) in healthy subjects and patients suffering from a stroke episode

Dold sequences, periodic points, and dynamics

In this survey we describe how the so-called Dold congruence arises in topology, and how it relates to periodic point counting in dynamical systems.Comment: 38 pages; surve

Periodic points of latitudinal maps of the m-dimensional sphere

Let f be a smooth self-map of the m-dimensional sphere Sm. Under the assumption that f preserves latitudinal foliations with the fibres S1, we estimate from below the number of fixed points of the iterates of f. The paper generalizes the results obtained by Pugh and Shub and by Misiurewicz

Shub’s conjecture for smooth longitudinal maps of Sm

Let f be a smooth map of the m-dimensional sphere Sm to itself, preserving the longitudinal foliation. We estimate from below the number of fixed points of the iterates of f, reduce Shub’s conjecture for longitudinal maps to a lower dimensional classical version, and prove the conjecture in case m=2 and in a weak form for m=3

Topological-numerical analysis of a two-dimensional discrete neuron model

We conduct computer-assisted analysis of the two-dimensional model of a neuron introduced by Chialvo in 1995 (Chaos, Solitons & Fractals 5, 461-479). We apply the method for rigorous analysis of global dynamics based on a set-oriented topological approach, introduced by Arai et al. in 2009 (SIAM J. Appl. Dyn. Syst. 8, 757-789) and improved and expanded afterwards. Additionally, we introduce a new algorithm to analyze the return times inside a chain recurrent set. Based on this analysis, together with the information on the size of the chain recurrent set, we develop a new method that allows one to determine subsets of parameters for which chaotic dynamics may appear. This approach can be applied to a variety of dynamical systems, and we discuss some of its practical aspects. The data and the software described in the paper are available at http://www.pawelpilarczyk.com/neuron/