Rok 2024

• Algebraic periods and minimal number of periodic points for smooth self-maps of 1-connected 4-manifolds with definite intersection forms

  Publikacja

  • H. Duan
  • G. Graff
  • J. Jezierski
  • A. Myszkowski

  - Journal of Fixed Point Theory and Applications - Rok 2024

  Let M be a closed 1-connected smooth 4-manifolds, and let r be a non-negative integer. We study the problem of finding minimal number of r-periodic points in the smooth homotopy class of a given map f: M-->M. This task is related to determining a topological invariant D^4_r[f], defined in Graff and Jezierski (Forum Math 21(3):491–509, 2009), expressed in terms of Lefschetz numbers of iterations and local fixed point indices of...

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• An absorbing set for the Chialvo map

  Publikacja

  • P. Pilarczyk
  • G. Graff

  - Communications in Nonlinear Science and Numerical Simulation - Rok 2024

  The classical Chialvo model, introduced in 1995, is one of the most important models that describe single neuron dynamics. In order to conduct effective numerical analysis of this model, it is necessary to obtain a rigorous estimate for the maximal bounded invariant set. We discuss this problem, and we correct and improve the results obtained by Courbage and Nekorkin (2010). In particular, we provide an explicit formula for an...

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Rok 2023

• Attractors of dissipative homeomorphisms of the infinite surface homeomorphic to a punctured sphere

  Publikacja

  • G. Graff
  • R. Ortega
  • A. Ruiz-Herrera

  - COMMUNICATIONS IN CONTEMPORARY MATHEMATICS - Rok 2023

  A class of dissipative orientation preserving homeomorphisms of the infinite annulus,pairs of pants, or generally any infinite surface homeomorphic to a punctured sphere isconsidered. We prove that in some isotopy classes the local behavior of such homeomor-phisms at a fixed point, namely the existence of so-called inverse saddle, impacts thetopology of the attractor — it cannot be arcwise connected

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

  Publikacja

  • P. Pilarczyk
  • J. Signerska-Rynkowska
  • G. Graff

  - CHAOS - Rok 2023

  We conduct computer-assisted analysis of a two-dimensional model of a neuron introduced by Chialvo in 1995 [Chaos, Solitons Fractals 5, 461–479]. We apply the method of 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 afterward. Additionally, we introduce a new algorithm to analyze the return times...

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Rok 2021

• Dold sequences, periodic points, and dynamics

  Publikacja

  • J. Byszewski
  • G. Graff
  • T. Ward

  - BULLETIN OF THE LONDON MATHEMATICAL SOCIETY - Rok 2021

  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.

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

  Publikacja

  • G. Graff
  • B. Graff
  • P. Pilarczyk
  • G. Jabłoński
  • D. Gąsecki
  • K. Narkiewicz

  - PLOS ONE - Rok 2021

  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...

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Rok 2020

• Computations of the least number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds

  Publikacja

  • G. Graff
  • J. Jezierski
  • A. Myszkowski

  - Topological Methods in Nonlinear Analysis - Rok 2020

  Let $r$ be an odd natural number, $M$ a compact simply-connected smooth manifold, $\dim M\geq 4$, such that its boundary $\partial M$ is also simply-connected. We consider $f$, a $C^1$ self-maps of $M$, preserving $\partial M$. In [G. Graff and J. Jezierski, Geom. Dedicata 187 (2017), 241-258] the smooth Nielsen type periodic number $D_r(f;M,\partial M)$ was defined and proved to be equal to the minimal number of $r$-periodic points...

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