Marek Lampart

VSB - Technical University of Ostrava · Department of Applied Mathematics and IT4 Innovations

Entropy serves as a measure of chaos in systems by representing the average rate of information loss about a phase point’s position on the attractor. When dealing with a multifractal system, a single exponent cannot fully describe its dynamics, necessitating a continuous spectrum of exponents, known as the singularity spectrum. From an investor’s p...

Entropy serves as a measure of chaos in systems by representing the average rate of information loss about a phase point's position on the attractor. When dealing with a multifractal system, a single exponent cannot fully describe its dynamics, necessitating a continuous spectrum of exponents, known as the singularity spectrum. From an investor's p...

The goal is to investigate the dynamics of banks’ share prices and related financials that lead to potential disruptions to credit and the economy. We adopt a classic macroeconomic equilibrium model with households, banks, and non-financial companies and explain both market valuations and endogenous debt constraints in terms of risk. Heterogeneous...

In the present paper, we demonstrate the possibilities of designing quantum computing circuits using a specific swarm intelligence algorithm — iSOMA in the form of three experiments. All simulations are based on a simple sample of a quantum computing circuit from the Qiskit environment, which was used as a comparison circuit with the results of the...

In this paper, we analyze the interpretable models from real gasification datasets of the project “Centre for Energy and Environmental Technologies” (CEET) discovered by symbolic regression. To evaluate CEET models based on input data, two different statistical metrics to quantify their accuracy are usually used: Mean Square Error (MSE) and the Pea...

The objective of this paper is the study of the dynamical properties analysis of an original specification of the classical Cournot heterogeneous model with optimal response; specifically, a new approach that considers ordinal utility instead of cardinal monetary amounts is proposed where the classical decision of quantity is disentangled from the...

In the field of mechanical engineering, conveyors and moving belts are frequently used machine parts. In many working regimes, they are subjected to sudden loading, which can be a source of irregular motion in the impacting bodies and undesirable behavior in the working machine. This paper deals with a mechanical model where colisions between an im...

In this work, the combination of the 0–1 test for chaos and approximate entropy is applied to a newly established mechanical model instead of the Lyapunov exponent exploration on huge simulations reached on the supercomputer Salomon (Czech Republic). This procedure is applied to the mechanical systems modeled by a system of non-autonomous ordinary...

Conveyors and moving belts are frequently used in the field of mechanical engineering. In many operating regimes they are subjected to impact loading, which can induce irregular motion and undesirable behaviour of the working machine. This paper focuses on the impacts between an impact body (the cylinder in our case) and a moving belt. Results of t...

The main aim of this paper is to detect embedded dynamics of the Györgyi-Field model of the Belousov–Zhabotinsky chemical reaction. The corresponding three-variable model given as a set of nonlinear ordinary differential equations depends on one parameter, the flow rate. As certain values of this parameter can give rise to chaos, an analysis was pe...

In this paper it is numerically shown that the dynamics of a heterogeneous Cournot oligopoly model depending on two bifurcation parameters can exhibit hidden and self-excited attractors. The system has a single equilibrium and a line of equilibria. The bifurcation diagrams show that the system admits several attractor coexistence windows, where the...

The main aim of this paper focuses on chaos suppression (control) and stimulation (anti-control) of a heterogeneous Cournot oligopoly model. This goal is reached by applying the theory of dynamical systems, namely impulsive control. The main aim was to demonstrate, through massive numerical simulations and estimation of the maximal Lyapunov exponen...

In this paper it is numerically proved that a heterogeneous Cournot oligopoly model presents hidden and self-excited attractors. The system has a single equilibrium and a line of equilibria. The bifurcation diagrams show that the system admits several attractors coexistence windows, where the hidden attractors can be found. Intensive numerical test...

The main aim of this paper is to detect dynamical properties of the Gy\"orgyi-Field model of the Belousov-Zhabotinsky chemical reaction. The corresponding three-variable model given as a set of nonlinear ordinary differential equations depends on one parameter, the flow rate. As certain values of this parameter can give rise to chaos, the analysis...

Pendulums and similar systems, such as links of chains, bodies hanging on ropes, kinematic chains forming working parts of manipulators, and robotic devices, are frequently used in industrial applications. They often cooperate in tubes or working spaces limited by walls or other rigid obstacles. This was the inspiration to carry out this study on t...

The main aim of this paper is the study of dynamical properties of the Laplacian-type coupled map lattice induced by the logistic family on a periodic lattice depending on two parameters: the state parameter of the logistic map and the coupling constant. For this purpose, tools like maximal Lyapunov exponent, approximate entropy, and the 0–1 test f...

Many components of machines and other technological devices (chains and bodies hanging on the ropes) can be modeled as multiple pendulums situated in tubes or holes of limited space. The investigation of motion of such systems represents the substance of many real technological problems; therefore, it stirred up motivation to perform research on vi...

The main aim of this paper is to study the evolution of the transmembrane potential on the cardiac cell under different rates and amplitudes of stimulation. For modeling this potential, the modification of the Fenton‐Karma model was applied. It is a phenomenological model with 3 degrees of freedom that corresponds to nondimensional transmembrane po...

Presence of partial discharge pattern implies the fault behavior on insulation system of medium voltage overhead lines, especially with covered conductors (CC). This paper covers the examination of Approximation and Sample entropy as a signal complexity measures on partial discharge patterns of several kinds of faults. These features are calculated...

This research was motivated by a real technological problem of vibrations of bodies hanging on chains or ropes in tubes or spaces limited by walls or other bodies. The studied system has two degrees of freedom. It is formed by two pendulums moving between two walls. Its movement is governed by a set of nonlinear ordinary differential equations. The...

One-dimensional dynamical systems attract researches for more than half a century and the topic was inspired by many real problems. Mainly piecewise linear and polynomial maps were considered and researched under several motivations from different scientific fields. As a main aim of the paper, the Logistic (polynomials of the second order) and the...

In this paper, Partial Discharge pattern as an indicator of the fault state of insulation systems of medium voltage overhead lines with covered conductors are described, analyzed, and their dynamical properties are researched. Application of data obtained in natural environment with huge variety of noise interferences, affected by various weather c...

The aim of this paper is a study on relations between ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-chaos and the structure of ω\documentclas...

This paper concentrates on the vibrations attenuation of a rotor driven by a DC motor and its frame flexibly coupled with a baseplate by linear cylindrical helical springs and damped by an element that can work either in inertia or impact regime. The system oscillation is governed by three mutually coupled second-order ordinary differential equatio...

The main aim of the paper is to research dynamic properties of a mechanical system consisting of a ball jumping between a movable baseplate and a fixed upper stop. The model is constructed with one degree of freedom in the mechanical oscillating part. The ball movement is generated by the gravity force and non-harmonic oscillation of the baseplate...

We study the dynamics of Laplacian-type coupling induced by logistic family fμ(x)=μx(1−x), where μ∈[0,4], on a periodic lattice, that is the dynamics of maps of the form F(x,y)=((1−ε)fμ(x)+εfμ(y),(1−ε)fμ(y)+εfμ(x)) where ε>0 determines strength of coupling. Our main objective is to analyze the structure of attractors in such systems and especially...

The goal of this paper is to concentrate on dynamics and vibration attenuation of a machine coupled with a baseplate by nonlinear disc springs and damped by a double-impact-element. The system motion is governed by a set of three mutually coupled second-order ordinary differential equations. Analysis of behavior of the linearized system shows that...

The goal of this paper is to concentrate on dynamics and vibration attenuation of the electromechanical system flexibly coupled with a baseplate and damped by a double impact element. The model is constructed with four degrees of freedom in the mechanical oscillating part, three translational and one rotational. The system movement is reported by f...

The main aim of this paper is focused on vibration attenuation of the electromechanical system flexibly coupled with a baseplate and damped by an impact element. The model is constructed with three degrees of freedom in the mechanical oscillating part, two translational and one rotational. The system movement is described by three mutually coupled...

It was proved by M. Babilonová-Štefánková (Int J Bifurc Chaos 13(7):1695–1700, 2003) that each bitransitive continuous map f of the interval is conjugated to a map g which is distributionally chaotic with a distributionally scrambled set D. The goal of this chapter is to improve this result, by showing that D is formed by points that are recurrent...

The notion of omega chaos was introduced by S. Li in 1993 for continuous maps of compact metric spaces by three conditions: 1. the set difference of omega limit sets is uncountable, 2. intersection of omega limit sets is nonempty and 3. each omega limit set of the point from the omega scrambled set is not contained in the set of all periodic points...

This paper focuses on vibrations attenuation of an electromechanical system flexibly coupled with a baseplate by cylindrical helical springs and damped by an element that can work either in inertia or impact regime. The model is constructed with three degrees of freedom in the mechanical oscillating part, two translational and one rotational. The s...

The main aim of this paper is to focus on analysis of the dynamic properties of the electromechanical system with an impact element. This model is constructed with three degrees of freedom in the mechanical oscillating part, two translational and one rotational, and is completed with an electric circuit. The mathematical model of the system is repr...

The main aim of this paper is to focus on dynamics of the electromechanical system flexibly coupled with a baseplate and damped by an impact element. The model is constructed with three degrees of freedom in the mechanical oscillating part, two translational and one rotational. The system movement is described by three mutually coupled second-order...

The main aim of the paper is to focus on analysis of dynamical properties of the electromechanical system with impact element. This model is constructed with three degrees of freedom in mechanical oscillating part, two translational and one rotational. The mathematical model of the system is represented by three coupled second-order ordinary differ...

Many phenomena coming from the biology, economy, engineering are modeled using discrete dynamical systems. The concept of backward orbit is an essential concept for understanding the dynamics of the system. In the literature various definitions of the concept of the α–limit point (respectively set) have been historically used. The aim of this paper...

Many phenomena coming from the biology, economy, engineering are modeled using discrete dynamical systems. The concept of backward orbit is an essential concept for understanding the dynamics of the system. In the literature various definitions of the concept of the alpha-limit point (respectively set) have been historically used. The aim of this p...

The aim of this paper is to analyze the dynamics of a four dimensional system described in [the first author and R. G. Rubio, Appl. Math. Lett. 23, No. 8, 836–838 (2010; Zbl 1189.91062)], which generalizes the classical Cournot competition in a local way. In particular, by considering an aggregation parameter of production costs, we are able to des...

The main aim of the paper is to focus on the analysis of the dynamical properties of the electromechanical system with impact element. This model is constructed with three degrees of freedom in the mechanical oscillating part, two translational and one rotational. The mathematical model of the system is represented by three coupled second-order ord...

In this paper a Cournot-like model is constructed with an iso-elastic demand function for n competitors. The Cournot equilibrium is constructed for general constant unit costs. Finally, it is proved that for identical unit costs the Cournot point is a sink for two or three competitors and a saddle for more than four players.

In this paper we study the structure of negative limit sets of maps on the unit interval. We prove that every alpha-limit set is an omega-limit set, while the converse is not true in general. Surprisingly, it may happen that the space of all alpha-limit sets of interval maps is not closed in the Hausdorff metric (and thus some omega-limit sets are...

Any continuous map T on a compact metric space X induces in a natural way a continuous map T¯ on the space K(X) of all non-empty compact subsets of X. Let T be a homeomorphism on the interval or on the circle. It is proved that the topological entropy of the induced set valued map T¯ is zero or infinity. Moreover, the topological entropy of T¯|C(X)...

In this paper we present a lattice dynamical system stated by Kaneko in (Phys Rev Lett, 65: 1391–1394, 1990) which is related to the Belusov-Zhabotinskii reaction. We prove that this CML (Coupled Map Lattice) system has positive topological entropy for zero coupling constant. KeywordsCoupled map lattice-Positive topological entropy

In this paper we present a lattice dynamical system stated by Kaneko in (Phys Rev Lett, 65: 1391–1394, 1990) which is related to the Belusov–Zhabotinskii chemical reaction. We prove that this CML (Coupled Map Lattice) system is chaotic in the sense of Li–Yorke and in the sense of Devaney for zero coupling constant. Some problems on the dynamics of...

The main aim of the present paper is to describe some relations between specification property and ω-chaos. In particular, we study how properties of factor maps can be used to transfer this kind of chaos by a semiconjugacy with a shift space.

Let f be a continuous self–map of a compact metric space X. The transformation f induces in a natural way a self–map f defined on the hyperspace K(X) of all nonempty closed subsets of X. We study which of the most usual notions of chaos for dynamical systems induced by f are inherited by f and vice versa. We consider distributional chaos, Li–Yorke...

The dynamics of the transformation F : (x,y) → (x(4 -x -y), xy) defined on the plane triangle Δ of vertices (0,0), (0,4) and (4,0) plays an important role in the behaviour of the Lotka-Volterra map. In 1993, A. N. Sharkovskiǐ (Proc. Oberwolfach 20/1993) stated some problems on it, in particular a question about the trasitivity of F was posed. The m...

The forcing relations between notions of distributional, Li–Yorke and ω chaos were studied by many authors. In this paper we summarize all known connections between these three different types of chaos and fulfill the results for general compact metric spaces by the construction of a selfmap on a compact perfect set which is ω chaotic, not distribu...

Given the plane triangle with vertices (0,0), (0,4) and (4,0) and the transformation F:(x,y)↦(x(4-x-y),xy) introduced by A. N. Sharkovskij [Low dimensional dynamics, Tagungsbericht 20/1993, Proceedings of Mathematisches Forschungsinstitut Oberwolfach, 17 (1993)], we prove the existence of the following objects: a unique invariant curve of spiral ty...

In L. Wang, Z. Chu and G. Liao [Topology Appl. 138, No. 1–3, 97–107 (2004; Zbl 1034.37010)] it was stated that there is an uncountable subset T of the shift space S such that T⊂R(σ)∖UR(σ) (where R(·) denotes the set of recurrent points and UR(·) the set of uniformly recurrent points), and that σ is uniquely ergodic on T. We prove that the second pa...

In the present paper we study Li and Yorke chaos on several spaces in connection with the cardinality of its scrambled sets. We prove that there is a map on a Cantor set and a map on a two-dimensional arcwise connected continuum (with empty interior) such that each scrambled set contains exactly two points.

In this paper we consider relations between chaos in the sense of Li and Yorke, and ω-chaos. The main aim is to show how important the size of scrambled sets is in definitions of chaos. We provide an example of an ω-chaotic map on a compact metric space which is chaotic in the sense of Li and Yorke, but any scrambled set contains only two points. C...

We deal with two types of chaos: the well known chaos in the sense of Li and Yorke and ω-chaos which was introduced in [S. Li, Trans. Amer. Math. Soc. 339 (1993)]. In this paper we prove that every bitransitive map f ϵ C(I, I) is conjugate to g ϵ C(I, I), which satisfies the following conditions, 1. there is a c-dense ω-scrambled set for g, 2. ther...

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