With Andrzej Lasota There and Back Again

Ryszard Rudnicki

doi:10.2478/amsil-2024-0018

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With Andrzej Lasota There and Back Again

Annales Mathematicae Silesianae

Volume 38 (2024): Issue 2 (September 2024)

By: Ryszard Rudnicki

Open Access

|Jul 2024

Abstract

The paper below is a written version of the 17th Andrzej Lasota Lecture presented on January 12th, 2024 in Katowice. During the lecture we tried to show the impact of Andrzej Lasota’s results on the author’s research concerning various fields of mathematics, including chaos and ergodicity of dynamical systems, Markov operators and semigroups and partial differential equations.

References

J. Auslander and J.A. Yorke, Interval maps, factors of maps and chaos, Tohoku Math. J. (2) 32 (1980), 177–188.
Search in Google Scholar Back to article
A. Bobrowski, T. Lipniacki, K. Pichór, and R. Rudnicki, Asymptotic behavior of distributions of mRNA and protein levels in a model of stochastic gene expression, J. Math. Anal. Appl. 333 (2007), 753–769.
Search in Google Scholar Back to article
A. Bobrowski and R. Rudnicki, On convergence and asymptotic behaviour of semi-groups of operators, Philos. Trans. Roy. Soc. A 378 (2020), 20190613, 18 pp.
Search in Google Scholar Back to article
F. Comets, S. Popov, G.M. Schütz, and M. Vachkovskaia, Billiards in a general domain with random reflections, Arch. Ration. Mech. Anal. 191 (2009), 497–537.
Search in Google Scholar Back to article
M.H.A. Davis, Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B 46 (1984), 353–388.
Search in Google Scholar Back to article
R.L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley Stud. Nonlinearity, Addison-Wesley Publishing Company, Redwood City, CA, 1989.
Search in Google Scholar Back to article
H.M. Hilden and L.J. Wallen, Some cyclic and non-cyclic vectors of certain operators, Indiana Univ. Math. J. 23 (1974), 557–565.
Search in Google Scholar Back to article
A. Lasota, Invariant measures and a linear model of turbulence, Rend. Sem. Mat. Univ. Padova 61 (1979), 40–48.
Search in Google Scholar Back to article
A. Lasota, Stable and chaotic solutions of a first order partial differential equation, Nonlinear Anal. 5 (1981), 1181–1193.
Search in Google Scholar Back to article
A. Lasota, Asymptotic stability of some nonlinear Boltzmann-type equations, J. Math. Anal. Appl. 268 (2002), 291–309.
Search in Google Scholar Back to article
A. Lasota and M.C. Mackey, Chaos, Fractals and Noise. Stochastic Aspects of Dynamics, Appl. Math. Sci., 97, Springer-Verlag, New York, 1994.
Search in Google Scholar Back to article
A. Lasota and R. Rudnicki, Asymptotic behaviour of semigroups of positive operators on C(X), Bull. Polish Acad. Sci. Math. 36 (1988), 151–159.
Search in Google Scholar Back to article
A. Lasota and J. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481–488.
Search in Google Scholar Back to article
A. Lasota and J. Yorke, On the existence of invariant measures for transformations with strictly turbulent trajectories, Bull. Polish Acad. Sci. Math. 25 (1977), 233–238.
Search in Google Scholar Back to article
A. Lasota and J. Yorke, Exact dynamical systems and the Frobenius–Perron operator, Trans. Amer. Math. Soc. 273 (1982), 375–384.
Search in Google Scholar Back to article
A. Lasota and J. Yorke, When the long time behavior is independent of the initial density, SIAM J. Math. Anal. 27 (1996), 221–240.
Search in Google Scholar Back to article
B. Lods, M. Mokhtar-Kharroubi, and R. Rudnicki, Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators, Ann. Inst. H. Poincaré C Anal. Non Linéaire 37 (2020), 877–923.
Search in Google Scholar Back to article
M.C. Mackey and R. Rudnicki, Asymptotic similarity and Malthusian growth in autonomous and nonautonomous populations, J. Math. Anal. Appl. 187 (1994), 548–566.
Search in Google Scholar Back to article
M.C. Mackey and R. Rudnicki, A new criterion for global stability of cell simultaneous cell replication and maturation processes, J. Math. Biol. 38 (1999), 195–219.
Search in Google Scholar Back to article
M. Mokhtar-Kharroubi and R. Rudnicki, On asymptotic stability and sweeping of collisionless kinetic equations, Acta Appl. Math. 147 (2017), 19–38.
Search in Google Scholar Back to article
G. Pianigiani and J.A. Yorke, Expanding maps on sets which are almost invariant: decay and chaos, Trans. Amer. Math. Soc. 252 (1979), 351–366.
Search in Google Scholar Back to article
K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl. 249 (2000), 668–685.
Search in Google Scholar Back to article
K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic operators and semigroups, J. Math. Anal. Appl. 436 (2016), 305–321.
Search in Google Scholar Back to article
K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic semigroups and applications, Stoch. Dyn. 18 (2018), 1850001, 18 pp.
Search in Google Scholar Back to article
K. Pichór and R. Rudnicki, Stability of stochastic semigroups and applications to Stein’s neuronal model, Discrete Contin. Dyn. Syst. Ser. B 23 (2018), 377–385.
Search in Google Scholar Back to article
K. Pichór and R. Rudnicki, Applications of stochastic semigroups to cell cycle models, Discrete Contin. Dyn. Syst. Ser. B 24 (2019), 2365–2381.
Search in Google Scholar Back to article
K. Pichór and R. Rudnicki, Dynamics of antibody levels: asymptotic properties, Math. Methods Appl. Sci. 43 (2020), 10490–10499.
Search in Google Scholar Back to article
K. Pichór and R. Rudnicki, Cell cycle length and long-time behavior of an age-size model, Math. Methods Appl. Sci. 45 (2022), 5797–5820.
Search in Google Scholar Back to article
K. Pichór and R. Rudnicki, Asymptotic properties of a general model of immune status, SIAM J. Appl. Math. 83 (2023), 172–193.
Search in Google Scholar Back to article
R. Rudnicki, Invariant measures for the flow of a first order partial differential equation, Ergodic Theory Dynam. Systems 5 (1985), 437–443.
Search in Google Scholar Back to article
R. Rudnicki, Asymptotic properties of the iterates of positive operators on C(X), Bull. Polish Acad. Sci. Math. 34 (1986), 181–187.
Search in Google Scholar Back to article
R. Rudnicki, Strong ergodic properties of a first-order partial differential equation, J. Math. Anal. Appl. 133 (1988), 14–26.
Search in Google Scholar Back to article
R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math. 43 (1995), 245–262.
Search in Google Scholar Back to article
R. Rudnicki, Chaos for some infinite-dimensional dynamical systems, Math. Methods Appl. Sci. 27 (2004), 723–738.
Search in Google Scholar Back to article
R. Rudnicki, Chaoticity of the blood cell production system, Chaos 19 (2009), 043112, 6 pp.
Search in Google Scholar Back to article
R. Rudnicki, Chaoticity and invariant measures for a cell population model, J. Math. Anal. Appl. 393 (2012), 151–165.
Search in Google Scholar Back to article
R. Rudnicki, An ergodic theory approach to chaos, Discrete Contin. Dyn. Syst. 35 (2015), 757–770.
Search in Google Scholar Back to article
R. Rudnicki, Models and Methods Mathematical Biology. Part II: Probabilistic Models, (in Polish), Księgozbiór Matematyczny 4, IMPAN, Warszawa, 2022. (Modele i Metody Biologii Matematycznej. Część II: Modele Probabilistyczne.)
Search in Google Scholar Back to article
R. Rudnicki, Ergodic properties of a semilinear partial differential equation, J. Differential Equations 372 (2023), 235–253.
Search in Google Scholar Back to article
R. Rudnicki and A. Tomski, On a stochastic gene expression with pre-mRNA, mRNA and protein contribution, J. Theoret. Biol. 387 (2015), 54–67.
Search in Google Scholar Back to article
R. Rudnicki and M. Tyran-Kamińska, Piecewise Deterministic Processes in Biological Models, SpringerBriefs Appl. Sci. Technol., Math. Methods, Springer, Cham, 2017.
Search in Google Scholar Back to article
R. Rudnicki and P. Zwoleński, Model of phenotypic evolution in hermaphroditic populations, J. Math. Biol. 70 (2015), 1295–1321.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/amsil-2024-0018 | Journal eISSN: 2391-4238 (formerly 0860-2107) | Journal ISSN: 0860-2107

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Language: English

Page range: 134 - 154

Submitted on: Apr 14, 2024

Published on: Jul 15, 2024

Published by: University of Silesia in Katowice, Institute of Mathematics

In partnership with: Paradigm Publishing Services

Keywords:

chaos,

invariant measure,

partial differential equation,

Markov operator,

semigroup of operators,

asymptotic stability,

piecewise deterministic Markov process,

application to biological models

MSC:

35F25,

37A05,

37L40,

47D06,

60J76,

92D25

Related subjects:

Mathematics,

General mathematics

© 2024 Ryszard Rudnicki, published by University of Silesia in Katowice, Institute of Mathematics
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 38 (2024): Issue 2 (September 2024)