Viktor Avrutin

• Dr. rer. nat. habil.
• Research Assistant at University of Stuttgart

Properties of one-dimensional (1D) piecewise monotone maps with a single discontinuity point, also known as Lorenz maps, have been the focus of many researchers from various theoretical and applied fields. Several bifurcation structures in the parameter space of these maps have been well described, particularly the period adding bifurcation structu...

We consider a discrete-time model of a population of agents participating in a minority game using a quantum cognition in an approach with binary choices. As the agents make decisions based on both their present and past states, the model is inherently two-dimensional, but can be reduced to a one-dimensional system governed by a bi-valued function....

Recently, we reported that a chaotic attractor in a discontinuous one-dimensional map may undergo a so-called exterior border collision bifurcation, which causes additional bands of the attractor to appear. In the present paper, we suppose that the chaotic attractor’s basin boundary contains a chaotic repeller, and discuss a bifurcation pattern con...

A 2D piecewise linear continuous two-parameter map known as the Lozi map is a special case of the 2D border collision normal form depending on four parameters. In the present paper, we investigate how the bifurcation structure of the Lozi map is incorporated into the bifurcation structure of the 2D border collision normal form using an analytical r...

We consider a 2D piecewise smooth map originating from an application (acting as a discrete-time model of a DC/DC converter with pulse-width modulated multilevel control). We focus on several non-trivial transformations occurring in the phase space of this map under parameter variation. In particular, we describe the effect of a fold border collisi...

In the present work, we investigate a period adding structure observed in a discontinuous 1D map modeling the behavior of a DC-AC H-bridge inverter with hysteresis control. Supported by experimental measurements from a laboratory prototype, we demonstrate that the structure is partially affected by bistability, so that for some parameter values the...

Maps with discontinuities can be shown to have many of the same properties of continuous maps if we include hidden orbits —solutions that include points lying on a discontinuity. We show here how the well known property that ‘period 3 implies chaos’ also applies to maps with discontinuities, but with a twist, namely that if a map has a hidden fixed...

This paper contributes to studying the bifurcations of closed invariant curves in piecewise-smooth maps. Specifically, we discuss a border collision bifurcation of a repelling resonant closed invariant curve (a repelling saddle-node connection) colliding with the border by a point of the repelling cycle. As a result, this cycle becomes attracting a...

Suddenly appearing high-frequency low-amplitude phase-restricted oscillations disturbing the waveform of signals in many kinds of power converters, also known as bubbling, have been recently shown to appear not at the points of smooth bifurcations, as previously assumed, but soon after. In the present work, we investigate further reasons for the on...

We consider the well-known Lozi map, which is a 2D piecewise linear map depending on two parameters. This map can be considered as a piecewise linear analog of the Hénon map, allowing to simplify the rigorous proof of the existence of a chaotic attractor. The related parameter values belong to a part of the parameter plane where the map has two sad...

In the present paper, we focus on the doubling of closed invariant curves associated with quasiperiodic dynamics. We consider a 5D map derived from a hybrid model originating from systems biology and containing a continuous part with time delay and pulse-modulated feedback. Using numerical bifurcation analysis, we show that doubling bifurcation tak...

In one-dimensional piecewise smooth maps with multiple borders, chaotic attractors may undergo border collision bifurcations, leading to a sudden change in their structure. We describe two types of such border collision bifurcations and explain the mechanisms causing the changes in the geometrical structure of the attractors, in particular, in the...

In this paper, we study the classical two-predators-one-prey model. The classical model described by a system of 3 ordinary differential equations can be reduced to a one-dimensional bimodal map. We prove that this map has at most two stable periodic orbits. Besides, we describe the structure of bifurcations of the map. Taking this mechanism into a...

One-dimensional maps with discontinuities are known to exhibit bifurcations somewhat different to those of continuous maps. Freed from the constraints of continuity, and hence from the balance of stability that is maintained through fold, flip, and other standard bifurcations, the attractors of discontinuous maps can appear as if from nowhere, and...

Vertical transmission of viral hepatitis B (HBV) is one of the main transmission ways in endemic regions. There is a relationship between structural changes in the placenta and the clinical course of HBV infection. The risk of vertical transmission of hepatitis B depends on the condition of the placenta. The goal of this work was to study the relat...

Chronic hepatitis B (CHB) is a significant public health problem worldwide. The aim of the present review is to summarize the actual trends in the management of CHB in pregnant women. The prevalence of hepatitis B virus (HBV) infection in pregnant women is usually comparable to that in the general population in the corresponding geographic area. Al...

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and study its properties. Further, we study how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where...

We study topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduce the maximal invariant measure and demonstrate that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allows us to extend some properties of IEMs (e.g. an estimate of the numbe...

The paper describes some aspects of sudden transformations of closed invariant curves in a 2D piecewise smooth map. In particular, using detailed numerically calculated phase portraits, we discuss transitions from smooth to piecewise smooth closed invariant curves. We show that such transitions may occur not only when a closed invariant curve colli...

In this paper we describe some nonlinear phenomena observed in a vibration machine with an unbalanced rotor driven by a DC motor controlled by a relay system. We demonstrate that this type of control leads to the appearance of several complex kinds of behavior caused by interactions between slow and fast dynamics. Our numerical results are verified...

Recently, it has been shown that DC–AC and AC–DC power converters whose dynamics is governed by two vastly different frequencies lead to a special class of piecewise smooth models characterized by a practically unpredictable number of switching manifolds between partitions in the state space associated with different dynamics. In the previous publi...

Onset of bubbling, i.e., a sudden appearance of high frequency oscillations disrupting the waveform of a slowly oscillating signal in a restricted phase interval is a serious problem for many applications in the field of power electronics. It has been shown in many publications that the appearance of bubbling is typically associated with a classica...

Physical experiments have long revealed that impact oscillators commonly exhibit large-amplitude chaos over a narrow band of parameter values close to grazing bifurcations. This phenomenon is not explained by the square-root singularity of the Nordmark map, which captures the local dynamics to leading order, because this map does not exhibit such d...

Recently, while studying the dynamics of power electronic DC/AC converters we have demonstrated that the behavior of these systems can be modeled by piecewise-smooth maps which belong to a specific class of models not investigated before. The characteristic feature of these maps is the presence of a very high number of switching manifolds (border p...

Objective: to examine the state of the immunity to measles in different age groups. Materials and methods: In 2018, 4444 people were examined at the Diagnostic Center (virological). Among them, 3783 people were examined using the passive haemagglutination test for measles (manufactured by Pasteur Research Institute of Epidemiology and Microbiology,...

In the present paper, we discuss bifurcations of chaotic attractors in piecewise smooth one-dimensional maps with a high number of switching manifolds. As an example, we consider models of DC/AC power electronic converters (inverters). We demonstrate that chaotic attractors in the considered class of models may contain parts of a very low density,...

The paper describes how several coexisting stable closed invariant curves embedded into each other can arise in a two-dimensional piecewise-linear normal form map. Phenomena of this type have been recently reported for a piecewise smooth map, modeling the behavior of a power electronic DC–DC converter. In the present work, we demonstrate that this...

In this work we investigated to which extent the evaluation results of the degree of hepatic fibrosis obtained by realtime elastography (RTE) method are compatible with the results of the transient elastography (TE) and with the APRI indexes. We also analyzed the factors which can influence the reliability of the fibrosis degree evaluation obtained...

Dynamical behaviors arising in a previously developed pulse-modulated mathematical model of non-basal testosterone regulation in the human male due to continuous exogenous signals are studied. In the context of endocrine regulation, exogenous signals represent, e.g., the influx of a hormone replacement therapy drug, the influence of the circadian r...

The paper describes a new scenario for the transition to complex dynamics in a vibrating system with an unbalanced rotor and a relay feedback control. We show that the transition from a regular dynamics without switching events in the relay element to an irregular dynamics which takes place completely in the hysteresis region occurs via a cascade o...

Despite the widespread prevalence of chronic hepatitis all over the world, the impact of these diseases on the pregnancy course and on the childbirth is still insufficiently investigated. Recently, some studies have been published, discussing the relationship between the state of the placenta and the risk of the mother to child transmission of hepa...

Power electronic DC/AC converters (inverters) play an important role in modern power engineering for a broad variety of applications including solar and wind energy systems as well as electric and hybrid cars drives. It is well known that the waveform of the output voltage (or current) of an inverter may be significantly distorted by phase restrict...

In this paper we present an overview of the results concerning dynamics of a piecewise linear bimodal map. The organizing principles of the bifurcation structures in both regular and chaotic domains of the parameter space of the map are discussed. In addition to the previously reported structures, a family of regions closely related to the so-calle...

A dangerous border collision bifurcation has been defined as the dynamical instability that occurs when the basins of attraction of stable fixed points shrink to a set of zero measure as the parameter approaches the bifurcation value from either side. This results in almost all trajectories diverging off to infinity at the bifurcation point, despit...

Power electronic DC/AC converters (inverters) play an important role in modern power engineering. These systems are also of considerable theoretical interest because their dynamics is influenced by the presence of two vastly different forcing frequencies. As a consequence, inverter systems may be modeled in terms of piecewise smooth maps with an ex...

The main purpose of the present survey is to contribute to the theory of dynamical systems defined by one-dimensional piecewise monotone maps. We recall some definitions known from the theory of smooth maps, which are applicable to piecewise smooth ones, and discuss the notions specific for the considered class of maps. To keep the presentation cle...

Power factor correction converters are used in many applications as AC-DC power supplies aiming at maintaining a near unity power factor. Systems of this type are known to exhibit nonlinear phenomena such as sub-harmonic oscillations and chaotic regimes that cannot be described by traditional averaged models. In this paper, we derive a time varying...

The goal of the present paper is to collect the results related to dynamics of a one-dimensional piecewise linear map widely known as the skew tent map. These results may be useful for the researchers working on theoretical and applied problems in the field of nonsmooth dynamical systems. In particular, we propose the complete description of the bi...

In this work, we investigate the dynamics of a piecewise linear 2D discontinuous map modeling a simple network showing the Braess paradox. This paradox represents an example in which adding a new route to a specific congested transportation network makes all the travelers worse off in terms of their individual travel time. In the particular case in...

Background Viral load measurement is necessary to estimate mother-to-child transmission risk for women with chronic hepatitis B (CHB), however, it is expensive. The present study aimed to investigate the relationship between HBsAg and hepatitis B virus (HBV) DNA levels, and to determine potential applications of HBsAg level monitoring for estimati...

In this work we investigate bifurcation structures in the chaotic domain of a piecewise linear bimodal map. The map represents a model of a circuit proposed to generate chaotic signals. For practical purposes it is necessary that the map generates robust broad-band chaos. However, experiments show that this requirement is fulfilled not everywhere....

Background and aims Hepatitis delta, caused by infection with the hepatitis D virus (HDV), is one of the most severe form of chronic viral hepatitis. It is estimated that worldwide about 15–20 million individuals are co-infected with hepatitis B virus (HBV) and HDV. Chronic hepatitis delta is associated with frequent development of liver cirrhosis...

Supported by experiments on a power electronic DC/AC converter, this paper considers an unusual transition from the domain of stable periodic dynamics (corresponding to the desired mode of operation) to chaotic dynamics. The behavior of the converter is studied by means of a 1D stroboscopic map derived from a non-autonomous ordinary differential eq...

In this work, we investigate the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map focusing on the regions associated with chaotic attractors (cyclic chaotic intervals). The boundaries of these regions corresponding to chaotic attractors with different number of intervals are identified. The result...

In Braess paradox the addiction of an extra resource creates a social dilemma in which the individual rationality leads to collective irrationality. In the literature, the dynamics has been analyzed when considering impulsive commuters, i.e., those who switch choice regardless of the actual difference between costs. We analyze a dynamical version o...

We study the dynamics of a growth model formulated in the tradition of Kaldor and Pasinetti where the accumulation of the ratio capital/workers is regulated by a two-dimensional discontinuous map with triangular structure. We determine analytically the border collision bifurcation boundaries of periodicity regions related to attracting cycles, show...

We study the bifurcation structure of the parameter space of a 1D continuous piecewise linear bimodal map which describes dynamics of a business cycle model introduced by Day-Shafer. In particular, we obtain the analytical expression of the boundaries of several periodicity regions associated with attracting cycles of the map (principal cycles and...

Objectives Viral load and HBsAg level are important markers used to diagnose hepatitis B and to determine its phase. The viral load's measurement is necessary to estimate mother-to-child transmission risk for women with chronic hepatitis B (CHB). Recently, the relationship between hepatitis B virus (HBV) viral load and other markers has been invest...

Power electronic DC/AC converters play an important role in modern power supply technology. As pa-rameters are varied, such converters may display a variety of unusual phenomena caused by the interaction of two in-ternal oscillatory modes (the ramp cycle and the external si-nusoidal reference signal). In this paper we consider a non-autonomous piec...

In this paper we study the effects of constraints on the dynamics of an adaptive segregation model introduced by Bischi and Merlone (2011). The model is described by a two dimensional piecewise smooth dynamical system in discrete time. It models the dynamics of entry and exit of two populations into a system, whose members have a limited tolerance...

We extend the analysis on the effects of the entry constraints on the dynamics of an adaptive segregation model of Shelling’s type when the two populations involved differ in numerosity, level of tolerance toward members of the other population, and speed of reaction. The model is described by a two-dimensional piecewise smooth dynamical system in...

Homoclinic orbits and heteroclinic connections are important in several contexts, in particular for a proof of the existence of chaos and for the description of bifurcations of chaotic attractors. In this work we discuss an algorithm for their numerical detection in smooth or piecewise smooth, continuous or discontinuous maps. The algorithm is base...

In this work, we classify the bifurcations of chaotic attractors in 1D piecewise smooth maps from the point of view of underlying homoclinic bifurcations of repelling cycles which are located before the bifurcation at the boundary of the immediate basin of the chaotic attractor.

We consider a two-parametric family of one-dimensional piecewise smooth maps with one discontinuity point. The bifurcation structures in a parameter plane of the map are investigated, related to codimension-2 bifurcation points defined by the intersections of two border collision bifurcation curves. We describe the case of the collision of two stab...

A chaotic attractor may consist of some number of bands (disjoint connected subsets). In continuous maps multi-band chaotic attractors are cyclic, that means every generic trajectory visits the bands in the same order. We demonstrate that in discontinuous maps multi-band chaotic attractors may be acyclic. Additionally, a simple criterion is propose...

When dealing with piecewise-smooth systems, the chaotic domain often does not contain any periodic inclusions, which is called “robust chaos”. Recently, the bifurcation structures in the robust chaotic domain of 1D piecewise-linear maps were investigated. It was shown that several regions of multi-band chaotic attractors emerge at the boundary betw...

We consider a family of piecewise linear bimodal maps having the outermost slopes positive and less than one. Three types of bifurcation structures incorporated into the parameter space of the map are described. These structures are formed by periodicity regions related to attracting cycles, namely, the skew tent map structure is associated with pe...

We give a brief overview of several bifurcation scenarios occurring in 1D piecewise monotone maps defined on two partitions, continuous or discontinuous. A collection of some basic blocks is proposed, which may be observed in particular bifurcation sequences of a system of interest both in regular and chaotic parameter domains.

In this work we report the recently discovered nested period incrementing bifurcation scenario. The investigated piecewise linear map is defined on three partitions of the unit interval, constant in the middle partition and therefore displays a rich variety of superstable orbits. These orbits are arranged according to an infinite binary tree of the...

Knowledge about the behavior of discontinuous piecewise-linear maps is important for a wide range of applications. An efficient way to investigate the bifurcation structure in 2D parameter spaces of such maps is to detect specific codimension-2 bifurcation points, called organizing centers, and to describe the bifurcation structure in their neighbo...

This work contributes to classify the dynamic behaviors of piecewise smooth systems in which border collision bifurcations characterize the qualitative changes in the dynamics. A central point of our investigation is the intersection of two border collision bifurcation curves in a parameter plane. This problem is also associated with the continuity...

In this work we consider the border collision bifurcations occurring in a one-dimensional piecewise linear map with two discontinuity points. The map, motivated by an economic application, is written in a generic form and considered in the stable regime, with all slopes between zero and one. We prove that the period adding structures occur in maps...

We give a brief overview of several bifurcation scenarios occurring in 1D piecewise monotone maps defined on two partitions, continuous or discontinuous. A collection of some basic blocks is proposed, which may be observed in particular bifurcation sequences of a system of interest both in regular and chaotic parameter domains.

Multiple attractor bifurcations occurring in piecewise smooth dynamical systems may lead to potentially damaging situations. In order to avoid these in physical systems, it is necessary to know their conditions of occurrence. Using the piecewise-linear 2D normal form, we investigate which types of multiple attractor bifurcations may occur and where...

Symbolic images represent a unified framework to apply several methods for the investigation of dynamical systems both discrete and continuous in time. By transforming the system flow into a graph, they allow it to formulate investigation methods as graph algorithms. Several kinds of stable and unstable return trajectories can be localized on this...

Typically, big bang bifurcation occurs for one (or higher)-dimensional piecewise-defined discontinuous systems whenever two border collision bifurcation curves collide transversely in the parameter space. At that point, two (feasible) fixed points collide with one boundary in state space and become virtual, and, in the one-dimensional case, the map...

We investigate the structure of the chaotic domain of a specific one-dimensional piecewise linear map with one discontinuity. In this system, the region of ârobust" chaos is embedded between two periodic domains. One of them is organized by the period-adding scenario whereas the other one by the period-increment scenario with coexisting attractors....

In this work we investigate a piecewise-linear dis- continuous map defined on three partitions. This map denoted as truncated tent map was specifically constructed in such a way that it shows a similar dynamic behavior like a piecewise- linear continuous map with an additional square root term with positive sign derived from the time continuous mod...

Based on a recently obtained Lemma about periodic orbits in linear systems with a piecewise-linear non-autonomous periodic control, we describe analytically the bifurcation structures in a ZAD-controlled buck converter. This analytical description shows that the period doubling bifurcation in this system may be both subcritical or supercritical. Co...

This article describes a novel bifurcation phenomenon occurring in the 2D parameter space of piecewise-linear maps. In the region of chaotic behavior we detect an infinite number of interior crises bounding the regions of multi-band attractors. This phenomenon, denoted as bandcount adding scenario, leads to a self-similar structure of the chaotic r...

Recently it has been demonstrated that the domain of robust chaos close to the periodic domain, which is organized by the period-adding structure, contains an infinite number of interior crisis bifurcation curves. These curves form the so-called bandcount adding scenario, which determines the occurrence of multi-band chaotic attractors. The analyti...

The complex bifurcation structure in the parameter space of the general piecewise-linear scalar map with a single discontinuity — nowadays known as nested period adding structure — was completely studied analytically by N. N. Leonov already 50 years ago. He used an elegant and very efficient recursive technique, which allows the analytical calculat...

50 years ago (1959) in a series of publications by Leonov, a detailed analytical study of the nested period adding bifurcation structure occurring in piecewise-linear discontinuous 1D maps was presented. The results obtained by Leonov are barely known, although they allow the analytical calculation of border-collision bifurcation subspaces in an el...

In this work we consider the discontinuous flat top tent map which represents an example for discontinuous piecewise-smooth maps, whereby the system function is constant on some interval. Such maps show several characteristics caused by this constant value which are still insufficiently investigated. In this work we demonstrate that in the disconti...

In piecewise-smooth dynamical systems, the regions of existence of a periodic orbit are typically parameter sub-spaces confined by border-collision bifurcations of this orbit. We demonstrate that additionally to the usual border-collision bifurcations occurring at finite points in the state space there exist also border-collision bifurcations occur...

We consider a one-dimensional continuous piecewise smooth nonlinear map with square-root singularity. For particular choice of the parameters, this map represents the normal form of discrete-time representation of impact oscillator near grazing bifurcation. Owing to the nonlinearity, the map exhibits smooth and nonsmooth bifurcations very closely r...

The study of bifurcations in a piecewise-smooth system is quite different from those occurring in smooth systems. In piecewise-linear systems, which we are considering in this paper, mainly border collision bifurcations (BCBs) and contact bifurcations occur. It is classified as border-collision any contact between an invariant set of a map with the...

In this work we reconsider the approach suggested about 50 years ago (1959) in a series of publications by N.N.Leonov for the calculation of border-collision bifurcation curves in piecewise-linear maps. We extend this approach and explain its applicability conditions.

When dealing with piecewise-smooth systems, the chaotic domain is often robust, that means does not contain any periodic inclusions. Recently, the bifurcation structures in the robust chaotic domain of 1D piecewise-linear maps was investigated. It was shown that several regions of multi-band chaotic attractors emerge at the boundary between the per...

Recently it was demonstrated that the domain of robust chaos close to the periodic domain, which is organized by the period-adding structure, contains an infinite number of interior crisis bifurcation curves. These curves form the so-called bandcount adding scenario, which determines the occurrence of multi-band chaotic attractors. The analytical c...

The object of the present work is to describe some bifurcations occurring in the 2D piecewise linear continuous map in canonical form involving the Poincare Equator (i.e. periodic points at infinity) and its relation to the regions of so called dangerous bifurcations (following Hassouneh et al. and Ganguli and Banerjee). It will be shown that such...

Considering the concept of multi-parametric bifurcations, the piecewise smooth linear normal form map can exhibit many typical bifurcation phenomena, which can not be observed in smooth dynamical systems. In this paper we investigate especially the interaction between smooth and non-smooth bifurcations in the multi-parametric domain.

Bifurcation structures in two-dimensional parameter spaces formed by chaotic attractors alone are still far away from being understood completely. In a series of three papers, we investigate the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous...

In this paper we report a new bifurcation phenomenon induced by interior crises, which explains the structure of multi-band chaotic attractors in the case of a piecewise-linear discontinuous map. This phenomenon, denoted as a fully developed bandcount adding scenario, leads to a self-similar structure of the chaotic region in parameter space.

Bifurcation structures in the two-dimensional parameter spaces formed by chaotic attractors alone are still far away from being understood completely. In a series of three papers, we investigate the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinu...

Bifurcation structures in 2D parameter spaces formed by chaotic attractors only are still far away from being understood completely. In a series of three papers we investigate the chaotic domain without periodic inclusions for a map which is considered by many authors as some kind of 1D canonical form for discontinuous maps. In the first part we re...