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Papers by Anastasiia Panchuk

Research paper thumbnail of Speculative behavior and chaotic asset price dynamics: On the emergence of a bandcount accretion bifurcation structure

Discrete & Continuous Dynamical Systems - B

We study a simple financial market model with interacting chartists and fundamentalists that may ... more We study a simple financial market model with interacting chartists and fundamentalists that may give rise to multiband chaotic attractors. In particular, asset prices fluctuate erratically around their fundamental values, displaying a significant bull and bear market behavior. An in-depth analytical and numerical study of our model furthermore reveals the emergence of a new bifurcation structure, a phenomenon that we call a bandcount accretion bifurcation structure. The latter consists of regions associated with chaotic dynamics only, the boundaries of which are not defined by homoclinic bifurcations, but mainly by contact bifurcations of particular type where two distinct critical points of certain ranks coincide.

Research paper thumbnail of Border collision bifurcations of chaotic attractors in one-dimensional maps with multiple discontinuities

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

In one-dimensional piecewise smooth maps with multiple borders, chaotic attractors may undergo bo... more 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 number of their bands (connected components).

Research paper thumbnail of Global dynamic scenarios in a discrete-time model of renewable resource exploitation: a mathematical study

Research paper thumbnail of Endogenous desired debt in a Minskyan business model

Chaos, Solitons & Fractals

Research paper thumbnail of Modeling learning and teaching interaction by a map with vanishing denominators: Fixed points stability and bifurcations

Chaos, Solitons & Fractals

Research paper thumbnail of A financial market model with two discontinuities: Bifurcation structures in the chaotic domain

Chaos (Woodbury, N.Y.), 2018

We continue the investigation of a one-dimensional piecewise linear map with two discontinuity po... more We continue the investigation of a one-dimensional piecewise linear map with two discontinuity points. Such a map may arise from a simple asset-pricing model with heterogeneous speculators, which can help us to explain the intricate bull and bear behavior of financial markets. Our focus is on bifurcation structures observed in the chaotic domain of the map's parameter space, which is associated with robust multiband chaotic attractors. Such structures, related to the map with two discontinuities, have been not studied before. We show that besides the standard bandcount adding and bandcount incrementing bifurcation structures, associated with two partitions, there exist peculiar bandcount adding and bandcount incrementing structures involving all three partitions. Moreover, the map's three partitions may generate intriguing bistability phenomena.

Research paper thumbnail of Bifurcation Structures in a Bimodal Piecewise Linear Map

Frontiers in Applied Mathematics and Statistics

In this paper we present an overview of the results concerning dynamics of a piecewise linear bim... more 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-called U-sequence is described. The boundaries of distinct regions belonging to these structures are obtained analytically using the skew tent map and the map replacement technique.

Research paper thumbnail of Some Aspects on Global Analysis of Discrete Time Dynamical Systems

Springer Proceedings in Complexity, 2016

Research paper thumbnail of Regular spiking in asymmetrically delay-coupled FitzHugh-Nagumo systems

Eprint Arxiv 0911 2071, Nov 11, 2009

We study two delay-coupled FitzHugh-Nagumo systems, introducing a mismatch between the delay time... more We study two delay-coupled FitzHugh-Nagumo systems, introducing a mismatch between the delay times, as the simplest representation of interacting neurons. We demonstrate that the presence of delays can cause periodic oscillations which coexist with a stable fixed point. Periodic solutions observed are of two types, which we refer to as a "long" and a "short" cycle, respectively.

Research paper thumbnail of CompDTIMe: Computing one-dimensional invariant manifolds for saddle points of discrete time dynamical systems

Research paper thumbnail of Role of reinvestment in a competitive market

Research paper thumbnail of Asymptotic dynamics of a piecewise smooth map modelling a competitive market

Mathematics and Computers in Simulation, 2015

Research paper thumbnail of Stability of cournot oligopoly when the competitors are numerous

Research paper thumbnail of Cournot equilibrium stability in a non-autonomous system modelling the oligopoly market

Research paper thumbnail of Role of reinvestment in a competitive market

Research paper thumbnail of Sudden transition from equilibrium stability to chaotic dynamics in a cautious tâtonnement model

Chaos, Solitons & Fractals, 2015

ABSTRACT

Research paper thumbnail of Stability of Periodic Clusters in Globally Coupled Maps

Research paper thumbnail of Dynamics of Industrial Oligopoly Market Involving Capacity Limits and Recurrent Investment

Dynamic Modeling and Econometrics in Economics and Finance, 2015

Research paper thumbnail of Partial synchronization in a system of globally coupled maps

Nonlinear Oscillations, 2004

We study the appearance of a chaotic partial synchronization in a system of globally coupled maps... more We study the appearance of a chaotic partial synchronization in a system of globally coupled maps. We analyze the structure of cluster zones for small values of the coupling parameter ε and conditions for the formation of chaotic attractors on cluster manifolds. We find a formula that describes the relationship between the transversal and longitudinal Lyapunov numbers for trajectories on the manifold and necessary conditions for the transversal stability of these trajectories.

Research paper thumbnail of Oligopoly model with recurrent renewal of capital revisited

Mathematics and Computers in Simulation, 2013

ABSTRACT The aim of the present paper is to investigate an oligopoly market, modelled by using CE... more ABSTRACT The aim of the present paper is to investigate an oligopoly market, modelled by using CES production function in combination with the isoelastic demand function. It is supposed that the competitors act not under constant, but eventually decaying returns, and thus, from time to time they need to renew their capital equipment, choosing its optimal amount according to the current market situation. It is shown that the asymptotic trajectories depend essentially on the value of the global capital durability, and are also sensitive to the initial choice of individual inactivity times. In particular, the firms may merge into different groups renewing their capitals simultaneously, which lead to distinct dynamical patterns. It is also studied how the capital wearing out rate influences the system behaviour.

Research paper thumbnail of Speculative behavior and chaotic asset price dynamics: On the emergence of a bandcount accretion bifurcation structure

Discrete & Continuous Dynamical Systems - B

We study a simple financial market model with interacting chartists and fundamentalists that may ... more We study a simple financial market model with interacting chartists and fundamentalists that may give rise to multiband chaotic attractors. In particular, asset prices fluctuate erratically around their fundamental values, displaying a significant bull and bear market behavior. An in-depth analytical and numerical study of our model furthermore reveals the emergence of a new bifurcation structure, a phenomenon that we call a bandcount accretion bifurcation structure. The latter consists of regions associated with chaotic dynamics only, the boundaries of which are not defined by homoclinic bifurcations, but mainly by contact bifurcations of particular type where two distinct critical points of certain ranks coincide.

Research paper thumbnail of Border collision bifurcations of chaotic attractors in one-dimensional maps with multiple discontinuities

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

In one-dimensional piecewise smooth maps with multiple borders, chaotic attractors may undergo bo... more 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 number of their bands (connected components).

Research paper thumbnail of Global dynamic scenarios in a discrete-time model of renewable resource exploitation: a mathematical study

Research paper thumbnail of Endogenous desired debt in a Minskyan business model

Chaos, Solitons & Fractals

Research paper thumbnail of Modeling learning and teaching interaction by a map with vanishing denominators: Fixed points stability and bifurcations

Chaos, Solitons & Fractals

Research paper thumbnail of A financial market model with two discontinuities: Bifurcation structures in the chaotic domain

Chaos (Woodbury, N.Y.), 2018

We continue the investigation of a one-dimensional piecewise linear map with two discontinuity po... more We continue the investigation of a one-dimensional piecewise linear map with two discontinuity points. Such a map may arise from a simple asset-pricing model with heterogeneous speculators, which can help us to explain the intricate bull and bear behavior of financial markets. Our focus is on bifurcation structures observed in the chaotic domain of the map's parameter space, which is associated with robust multiband chaotic attractors. Such structures, related to the map with two discontinuities, have been not studied before. We show that besides the standard bandcount adding and bandcount incrementing bifurcation structures, associated with two partitions, there exist peculiar bandcount adding and bandcount incrementing structures involving all three partitions. Moreover, the map's three partitions may generate intriguing bistability phenomena.

Research paper thumbnail of Bifurcation Structures in a Bimodal Piecewise Linear Map

Frontiers in Applied Mathematics and Statistics

In this paper we present an overview of the results concerning dynamics of a piecewise linear bim... more 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-called U-sequence is described. The boundaries of distinct regions belonging to these structures are obtained analytically using the skew tent map and the map replacement technique.

Research paper thumbnail of Some Aspects on Global Analysis of Discrete Time Dynamical Systems

Springer Proceedings in Complexity, 2016

Research paper thumbnail of Regular spiking in asymmetrically delay-coupled FitzHugh-Nagumo systems

Eprint Arxiv 0911 2071, Nov 11, 2009

We study two delay-coupled FitzHugh-Nagumo systems, introducing a mismatch between the delay time... more We study two delay-coupled FitzHugh-Nagumo systems, introducing a mismatch between the delay times, as the simplest representation of interacting neurons. We demonstrate that the presence of delays can cause periodic oscillations which coexist with a stable fixed point. Periodic solutions observed are of two types, which we refer to as a "long" and a "short" cycle, respectively.

Research paper thumbnail of CompDTIMe: Computing one-dimensional invariant manifolds for saddle points of discrete time dynamical systems

Research paper thumbnail of Role of reinvestment in a competitive market

Research paper thumbnail of Asymptotic dynamics of a piecewise smooth map modelling a competitive market

Mathematics and Computers in Simulation, 2015

Research paper thumbnail of Stability of cournot oligopoly when the competitors are numerous

Research paper thumbnail of Cournot equilibrium stability in a non-autonomous system modelling the oligopoly market

Research paper thumbnail of Role of reinvestment in a competitive market

Research paper thumbnail of Sudden transition from equilibrium stability to chaotic dynamics in a cautious tâtonnement model

Chaos, Solitons & Fractals, 2015

ABSTRACT

Research paper thumbnail of Stability of Periodic Clusters in Globally Coupled Maps

Research paper thumbnail of Dynamics of Industrial Oligopoly Market Involving Capacity Limits and Recurrent Investment

Dynamic Modeling and Econometrics in Economics and Finance, 2015

Research paper thumbnail of Partial synchronization in a system of globally coupled maps

Nonlinear Oscillations, 2004

We study the appearance of a chaotic partial synchronization in a system of globally coupled maps... more We study the appearance of a chaotic partial synchronization in a system of globally coupled maps. We analyze the structure of cluster zones for small values of the coupling parameter ε and conditions for the formation of chaotic attractors on cluster manifolds. We find a formula that describes the relationship between the transversal and longitudinal Lyapunov numbers for trajectories on the manifold and necessary conditions for the transversal stability of these trajectories.

Research paper thumbnail of Oligopoly model with recurrent renewal of capital revisited

Mathematics and Computers in Simulation, 2013

ABSTRACT The aim of the present paper is to investigate an oligopoly market, modelled by using CE... more ABSTRACT The aim of the present paper is to investigate an oligopoly market, modelled by using CES production function in combination with the isoelastic demand function. It is supposed that the competitors act not under constant, but eventually decaying returns, and thus, from time to time they need to renew their capital equipment, choosing its optimal amount according to the current market situation. It is shown that the asymptotic trajectories depend essentially on the value of the global capital durability, and are also sensitive to the initial choice of individual inactivity times. In particular, the firms may merge into different groups renewing their capitals simultaneously, which lead to distinct dynamical patterns. It is also studied how the capital wearing out rate influences the system behaviour.