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dr hab. Paweł Pilarczyk
Employment
- Associate professor at Institute of Applied Mathematics
- 2018 - present at Faculty of Applied Physics and Mathematics
Publications
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total: 32
Catalog Publications
Year 2024
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An absorbing set for the Chialvo map
PublicationThe 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...
Year 2023
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Differentiating patients with obstructive sleep apnea from healthy controls based on heart rate-blood pressure coupling quantified by entropy-based indices
PublicationWe introduce an entropy-based classification method for pairs of sequences (ECPS) for quantifying mutual dependencies in heart rate and beat-to-beat blood pressure recordings. The purpose of the method is to build a classifier for data in which each item consists of two intertwined data series taken for each subject. The method is based on ordinal patterns and uses entropy-like indices. Machine learning is used to select a subset...
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Topological-numerical analysis of a two-dimensional discrete neuron model
PublicationWe 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...
Year 2021
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Nilpotent singularities and chaos: Tritrophic food chains
PublicationLocal bifurcation theory is used to prove the existence of chaotic dynamics in two well-known models of tritrophic food chains. To the best of our knowledge, the simplest technique to guarantee the emergence of strange attractors in a given family of vector fields consists of finding a 3-dimensional nilpotent singularity of codimension 3 and verifying some generic algebraic conditions. We provide the essential background regarding...
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Persistent homology as a new method of the assessment of heart rate variability
PublicationHeart 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...
Year 2020
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A space-efficient algorithm for computing the minimum cycle mean in a directed graph
PublicationAn algorithm is introduced for computing the minimum cycle mean in a strongly connected directed graph with n vertices and m arcs that requires O(n) working space. This is a considerable improvement for sparse graphs in comparison to the classical algorithms that require O(n^2) working space. The time complexity of the algorithm is still O(nm). An implementation in C++ is made publicly available at http://www.pawelpilarczyk.com/cymealg/.
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Rigorous numerics for critical orbits in the quadratic family
PublicationWe develop algorithms and techniques to compute rigorous bounds for finite pieces of orbits of the critical points, for intervals of parameter values, in the quadratic family of one-dimensional maps fa(x)=a−x2. We illustrate the effectiveness of our approach by constructing a dynamically defined partition P of the parameter interval Ω=[1.4,2] into almost 4 million subintervals, for each of which we compute to high precision the...
Year 2016
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A study of rigorous ODE integrators for multi-scale set-oriented computations
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Computation of Cubical Steenrod Squares
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Uniform Expansivity Outside a Critical Neighborhood in the Quadratic Family
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Year 2015
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An algorithmic approach to estimating the minimal number of periodic points for smooth self-maps of simply-connected manifolds
PublicationFor a given self-map f of M, a closed smooth connected and simply-connected manifold of dimension m 4, we provide an algorithm for estimating the values of the topological invariant D^m_r [f], which equals the minimal number of r-periodic points in the smooth homotopy class of f. Our results are based on the combinatorial scheme for computing D^m_r [f] introduced by G. Graff and J. Jezierski [J. Fixed Point Theory Appl. 13 (2013),...
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Computation of cubical homology, cohomology, and (co)homological operations via chain contraction
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Inducing a map on homology from a correspondence
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Rich Bifurcation Structure in a Two-Patch Vaccination Model
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Year 2014
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Shadowing is generic---a continuous map case
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Year 2012
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A dynamical model for plasma confinement transitions
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Combinatorial-topological framework for the analysis of global dynamics
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Global dynamics in a stage-structured discrete-time population model with harvesting
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Year 2011
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Finite Resolution Dynamics
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Year 2010
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Parallelization Method for a Continuous Property
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Year 2009
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A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems
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Decomposing Cavities in Digital Volumes into Products of Cycles
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Recent development in rigorous computational methods in dynamical systems
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Year 2008
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Excision-preserving cubical approach to the algorithmic computation of the discrete Conley index
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Homology algorithm based on acyclic subspace
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Quantitative hyperbolicity estimates in one-dimensional dynamics
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Year 2007
Year 2005
Year 2002
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Analysis of blood vessel topology by cubical homology
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The Conley Index and Rigorous Numerics for Attracting Periodic Orbits
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Year 2000
Year 1999
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