Piotr Kowalczyk - Personal Homepage, Wroclaw University of Science and Technology, Department of Mathematics

                                                                   Politechnika Wrocławska, Wydział Matematyki

Piotr3.jpg Email: piotr.s.kowalczyk@pwr.edu.pl
Adres: Budynek C-19, pokój A.3.19,
           ul. Hoene-Wrońskiego 13C,
           50-376 Wrocław
Konsultacje: Poniedziałek 14:30-16:30
                    Piątek 10:00-12:00

Research interests 

Dynamics of piecewise-smooth and hybrid systems

1) Dynamical systems theory with an emphasis on the analysis and computations of bifurcations in non-smooth dynamical systems.

2) Engineering applications of dynamical systems. For instance, understanding the phenomena brought about by the presence of discontinuous nonlinearities in mechanical systems with dry-friction, preload, impacts, hysteresis, and saturation. Other applications of interest are power electronics and hybrid systems.

3) Robustness of discontinuity induced bifurcations and singularly perturbed piecewise-smooth systems. It is important to understand how small imperfections that are usually ignored by the modeler influence system dynamics. For instance, in the context of modelling DC/DC power converters there are small capacitances or inductances that are ignored. The question araises how and if these produce some significant effects on the system dynamics.

4) Dynamics of multiple scale systems with switchings and resets.


Biped dynamics, human balance and human locomotion

  • It has been reported that with aging the ability to maintain an upright position decreases and a rate of falling causing serious injuries and even death are quite frequent. With the aging society it is therefore important to understand and isolate the mechanisms that are critical for maintaining balance. Mathematical models of upright standing can be studied to address this question. Some of these models involve the presence of discontinuous nonlinearities and time delays.
  • Some of the reduced models of human running are based on the essential fact that human running consists of two phases, which are repeated. That is, the phase of contact with the ground (support phase) and the phase of no contact with the grund (flight phase). This situation natuarlly leads to a switched model system. One of the most elementary of such models is so-called SLIP model (Spring Loaded Inverted Pendulum model). In spite of its simplicity, the presence of switching and asymmetry captures some essential dynamics of running and the model has a rich bifurcation structure. 
  • Dynamics of piecewise-smooth systems with time delay

  • Dynamics of systems with relay feedback, hysteresis and time delay.

  • Kursy (2023/2024 - semestr letni)

    Diploma Seminar (materiały na ePortalu)

    Analiza matematyczna 2A (materiały na ePortalu)

    Polish scientist worth knowing that you may not know:


  • Books
  • M. di Bernardo, C. Budd, A. R. Champneys, P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications (2008) (see Springer).

  • Accepted/Submitted for publication/In revision/In preparation
  • [29] M. Desroches P. Kowalczyk, and S. Rodrigues, Discontinuity induced dynamics in Conductance-Based Adaptive Exponential Integrate-and-Fire Model, submitted for publication to Bulletin of Mathematical Biology, February 2024 

    [28] Zofia Wróblewska, P. Kowalczyk, and Łukasz PłociniczakNonexistence of asymmetric solutions of human running gaits in a switched inverted pendulum model, accepted for publication to Mathematica Applicanda, February 2024

    [27] Zofia Wróblewska, P. Kowalczyk, and Krzysztof Przednowek, Leg stiffness and energy minimization in human running gaits, Sports Engineering, accepted for publication, April 2024

    [26] Zofia Wróblewska, P. Kowalczyk, and Łukasz Płociniczak, Stability of fixed points in an approximate solution of the spring-mass running model, accepted for publication in the IMA Journal of Applied Mathematics, March 2023

  • Journal publications

  • [25] P. Kowalczyk, Łukasz Płociniczak and Zofia WróblewskaEnergy variations and periodic solutions in a switched inverted pendulum model of human running gaits, Physica D, 2022

    [24] M. Desroches P. Kowalczyk, and S. Rodrigues, Spike-adding and reset-induced canard cycles in adaptive integrate and fire models, Nonlinear Dynamics, 2021

    [23] P. Kowalczyk, The dynamics and event-collision bifurcations in switched control systems with delayed switching, Physica D, Vol. 406, 2020

    [22] P. Kowalczyk, A novel route to a Hopf-bifurcation scenario in switched systems with dead zone, Physica D, Vol. 348, pp. 60-66, 2017

    [21] S. Nema, P. Kowalczyk, I. Loram, Wavelet-frequency analysis for the detection of discontinuities in switched system models of human balance, Human Movement Science, 2016

    [20] P. Glendinning, P. Kowalczyk, and A. B. Nordmark, Multiple attractors in grazing-sliding bifurcations in Filippov type flows, the IMA Journal of Applied Mathematics,  2016

    [19] S. Nema, P. Kowalczyk, and I. Loram, Complexity and dynamics of switched human balance during quiet standing, Biological Cybernetics, 2015

    [18] S. Nema, P. Kowalczyk, Detecting abrupt changes in a noisy Van der Pol type oscillator, Differential Equations and Dynamical Systems, 2015

    [17] P. Kowalczyk, S. Nema, P. Glendinning, I. Loram and M. Brown, ARMA analysis of linear and discontinuous models of human balance during quiet standing, Chaos: An Interdisciplinary Journal of Nonlinear Science, June 2014

    [16] P. Glendinning, P. Kowalczyk, and A. B. Nordmark, Attractors near grazing-sliding bifurcations, Nonlinearity, Vol 25(6), pp. 1867- 1885, 2012

    [15] P. Kowalczyk, P. Glendinning, Martin Brown, Gustavo Medrano-Cerda, Houman Dallali, and Jonathan Shapiro Modelling human balance using switched systems with linear feedback control, In Print, Interdiscpilinary Journal of the Royal Society Interface, Available online 21 June 2011

    [14] P. Kowalczyk, P. Glendinning Boundary-equilibrium bifurcations in piecewise-smooth slow-fast systems, Chaos: An interdisciplinary Journal of Nonlinear Science, June 2011

    [13] P. Glendinning, P. Kowalczyk, Micro-chaotic dynamics due to digital sampling in hybrid systems of Filippov type, Physica D, 239(1-2), pp.58-71, (2010)

    [12] J. Sieber, P. Kowalczyk, Small-scale instabilities in dynamical systems with sliding, Physica D, 239(1-2), pp. 44-57, 2010

    [11] J. Sieber, P. Kowalczyk, S.J. Hogan, M. di Bernardo: Dynamics of symmetric dynamical systems with delayed switching. Special Issue of Journal of Vibration and Control on Dynamics and Control of Systems with Time-Delay, Vol. 16(7-8), 2010

    [10] P. Glendinning, P. Kowalczyk, Dynamics of a hybrid thermostat model with discrete sampling time control, Dynamical Systems, 24(3), pp. 343-360, 2009

    [9] M. di Bernardo, C. Budd, A. R. Champneys, P. Kowalczyk, A. B. Nordmark, G. Olivar and P.T. Piiroinen, Bifurcations in Nonsmooth Dynamical Systems, SIAM Review, 50(4), pp.629-701, 2008

    [8] A. Colombo, M. di Bernardo, J. Hogan and P. Kowalczyk, Complex dynamics in a relay feedback system with hysteresis and delay, Journal of Nonlinear Science, 17(2), pp. 85-108, 2007

    [7] P. Kowalczyk, P.T. Piiroinen, Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator, Physica D, 237(8), pp.1053-1073, 2007

    [6] A. B. Nordmark and P. Kowalczyk, A codimension-two scenario of sliding solutions  in  grazing-sliding bifurcations,, Nonlinearity  19(1) pp. 1-26, 2006)

    [5] P. Kowalczyk, M. di Bernardo, A. R. Champneys, S.  J. Hogan, M. Homer, Yu. A. Kuznetsov, A. B. Nordmark and P. T. Piiroinen, Two-parameter nonsmooth bifurcations of limit cycles: classification and open problems, International Journal of bifurcation and chaos, Vol. 16 No. 3, pp.601-629, 2006

    [4] P. Kowalczyk and M. di Bernardo, Two-parameter degenerate sliding bifurcations in Filippov systems, Physica D, Vol. 204 pp. 204 - 229, 2005

    [3] P. Kowalczyk, Robust chaos and border-collision bifurcations in non-invertible piecewise-linear maps, Nonlinearity, Vol. 18 pp. 485-5042005 

    [2] M. di Bernardo, P. Kowalczyk,  and A. B. Nordmark,   Sliding bifurcations: A novel mechanism for the sudden onset of chaos in dry-friction oscillators, International Journal of Bifurcation and Chaos, Vol. 13, No. 10 pp. 2935-2948, 2003

    [1] M. di Bernardo, P. Kowalczyk,  and A. B. Nordmark,   Bifurcations of dynamical systems with sliding: derivation of normal-form mappings, Physica D,  volume 170, pp. 175-205, 2002

  • Conference proceedings
  • P. Kowalczyk, P. Glendinning, Micro-chaos in Relay Feedback Systems with Bang-Bang Control and Digital Sampling, To appear in the  proceedings of 18th Word Congress of the International Federation of Automatic Control, Milan 2011

    P. Kowalczyk, J. Sieber, Robustness of grazing-sliding bifurcations in Filippov type systems, In the proceedings of Second IFAC meeting related to analysis and control of chaotic systems, London 2009

    P. Kowalczyk, A. B. Nordmark, Bifurcations in non-smooth models of
    mechanical systems, In the proceeding of EUROMECH 500 conference on Non-smooth Problems in Vehicle Systems Dynamics - Analysis and Solutions, Lingby, Denmark 2008

    P. Kowalczyk, Grazing bifurcations: A mechanism for the sudden onset of robust chaos, In the proceedings of 10th Experimental Chaos Conference, Catania, Italy 2008

    Samia K. Genena, Daniel J. Pagano and P. Kowalczyk: HOSM Control of Stick-Slip Oscillations in Oil Well Drillstrings,in Proceeding to European Control Conference Kos 2007

    J. Sieber, P. Kowalczyk: Symmetric event collisions in dynamical systems with delayed switches . To appear in a special issue of "Discrete and Continuous Dynamical Systems Series B" (Proceedings of the sixth AIMS Conference on Dynamical Systems, Differential Equations and Applications, Poitiers 2007

    M. di Bernardo, A.R. Champneys, P. Kowalczyk, Corner-Collision and Grazing-Sliding: practical examples of border-collision bifurcations, Proc. IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Mechanics, Kluwer Academic, 2003

    M. di Bernardo and P. Kowalczyk, On the existence of stable asymmetric limit cycles and chaos in unforced symmetric relay feedback system, in Proceeding to European Control Conference Porto 2001

    P. Kowalczyk and M. di Bernardo, On a novel class of bifurcations in hybrid dynamical systemsthe case of relay feedback system, in  Proceedings of 4th International Workshop on Hybrid Systems Computation and Control, published by Springer-Verlag, pp. 361-374, 2001

    A. Sowa and P. Kowalczyk, Test chamber characteristics – an important factor determining required RF power of amplifier in radiated immunity tests, Accepted for the Electromagnetic Compatibility Conference Wroclaw 2000, Poland

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