Special Issue "Dynamical Systems and Their Applications Methods"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Dynamical Systems".

Deadline for manuscript submissions: closed (31 August 2021).

Special Issue Editor

Prof. Dr. Marek Lampart
E-Mail Website
Guest Editor
IT4Innovations, Department of Applied Mathematics, VSB - Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava, Czech Republic
Interests: dynamical systems; chaos; 0-1 test for chaos; hidden attractor; multistability
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Special Issue Information

In the past decades, non-linear phenomena have attracted the attention of researchers across many scientific fields. The development of technological devices and theories have led to new scientific results in the area of dynamical systems. These novel outcomes open new and prospective possibilities in widespread fields of science and engineering.

This Special Issue invites papers that focus on recent and novel developments in the theory of dynamical systems and their applications; especially on analytical, numerical, and experimental results showing non-linear phenomena with regular and irregular patterns.

This Special Issue will accept high-quality papers containing original research results and survey articles of exceptional merit in the following fields:

  • Dynamical systems
  • Differential and difference equations
  • Chaos, chaos control and anticontrol
  • Stability, multi-stability, hidden and self-excited attractors
  • Entropy, 0–1 test for chaos, Lyapunov exponent
  • Time-series, time series forecasting, and predictability
  • Real-world dynamical systems application

Prof. Marek Lampart
Guest Editor

Manuscript Submission Information

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Keywords

  • Dynamical systems
  • Differential and difference equations
  • Chaos, chaos control and anticontrol
  • Stability, multi-stability, hidden and self-excited attractors
  • Entropy, 0–1 test for chaos, Lyapunov exponent
  • Time-series, time series forecasting, and predictability
  • Real-world dynamical systems application

Published Papers (13 papers)

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Research

Article
Towards Optimal Supercomputer Energy Consumption Forecasting Method
Mathematics 2021, 9(21), 2695; https://doi.org/10.3390/math9212695 - 23 Oct 2021
Viewed by 362
Abstract
Accurate prediction methods are generally very computationally intensive, so they take a long time. Quick prediction methods, on the other hand, are not very accurate. Is it possible to design a prediction method that is both accurate and fast? In this paper, a [...] Read more.
Accurate prediction methods are generally very computationally intensive, so they take a long time. Quick prediction methods, on the other hand, are not very accurate. Is it possible to design a prediction method that is both accurate and fast? In this paper, a new prediction method is proposed, based on the so-called random time-delay patterns, named the RTDP method. Using these random time-delay patterns, this method looks for the most important parts of the time series’ previous evolution, and uses them to predict its future development. When comparing the supercomputer infrastructure power consumption prediction with other commonly used prediction methods, this newly proposed RTDP method proved to be the most accurate and the second fastest. Full article
(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)
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Article
Chaos on Fuzzy Dynamical Systems
Mathematics 2021, 9(20), 2629; https://doi.org/10.3390/math9202629 - 18 Oct 2021
Cited by 1 | Viewed by 343
Abstract
Given a continuous map f:XX on a metric space, it induces the maps f¯:K(X)K(X), on the hyperspace of nonempty compact subspaces of X, and [...] Read more.
Given a continuous map f:XX on a metric space, it induces the maps f¯:K(X)K(X), on the hyperspace of nonempty compact subspaces of X, and f^:F(X)F(X), on the space of normal fuzzy sets, consisting of the upper semicontinuous functions u:X[0,1] with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems (X,f), (K(X),f¯), and (F(X),f^). In particular, we considered several dynamical properties related to chaos: Devaney chaos, A-transitivity, Li–Yorke chaos, and distributional chaos, extending some results in work by Jardón, Sánchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451–463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics). Full article
(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)
Article
Nonlinear Compartment Models with Time-Dependent Parameters
Mathematics 2021, 9(14), 1657; https://doi.org/10.3390/math9141657 - 14 Jul 2021
Viewed by 399
Abstract
A nonlinear compartment model generates a semi-process on a simplex and may have an arbitrarily complex dynamical behaviour in the interior of the simplex. Nonetheless, in applications nonlinear compartment models often have a unique asymptotically stable equilibrium attracting all interior points. Further, the [...] Read more.
A nonlinear compartment model generates a semi-process on a simplex and may have an arbitrarily complex dynamical behaviour in the interior of the simplex. Nonetheless, in applications nonlinear compartment models often have a unique asymptotically stable equilibrium attracting all interior points. Further, the convergence to this equilibrium is often wave-like and related to slow dynamics near a second hyperbolic equilibrium on the boundary. We discuss a generic two-parameter bifurcation of this equilibrium at a corner of the simplex, which leads to such dynamics, and explain the wave-like convergence as an artifact of a non-smooth nearby system in C0-topology, where the second equilibrium on the boundary attracts an open interior set of the simplex. As such nearby idealized systems have two disjoint basins of attraction, they are able to show rate-induced tipping in the non-autonomous case of time-dependent parameters, and induce phenomena in the original systems like, e.g., avoiding a wave by quickly varying parameters. Thus, this article reports a quite unexpected path, how rate-induced tipping can occur in nonlinear compartment models. Full article
(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)
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Article
Applications of the Network Simulation Method to Differential Equations with Singularities and Chaotic Behaviour
Mathematics 2021, 9(12), 1442; https://doi.org/10.3390/math9121442 - 21 Jun 2021
Cited by 1 | Viewed by 393
Abstract
In this paper, we deal with some applications of the network simulation method (NMS) to the non-linear differential equations derived of a parametric family associated to stated problems by Newton in and others like the parabolic mirror and van der Pol non-linear equation. [...] Read more.
In this paper, we deal with some applications of the network simulation method (NMS) to the non-linear differential equations derived of a parametric family associated to stated problems by Newton in and others like the parabolic mirror and van der Pol non-linear equation. We underly the efficientcy of the (NMS) method, compare it with Matlab procedures and present figures of solutions of the equations obtained by it on the mentioned problems. Additionally, we introduce also the electric-electronic circuits we have designed to be able of obtaining the solutions of the referred equations. Full article
(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)
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Communication
A Combined Energy Method for Flutter Instability Analysis of Weakly Damped Panels in Supersonic Airflow
Mathematics 2021, 9(10), 1090; https://doi.org/10.3390/math9101090 - 12 May 2021
Viewed by 433
Abstract
A combined energy method is proposed to investigate the flutter instability characteristics of weakly damped panels in the supersonic airflow. Based on the small damping assumption, the motion governing partial differential equation (PDE) of the panel aeroelastic system, is built by adopting the [...] Read more.
A combined energy method is proposed to investigate the flutter instability characteristics of weakly damped panels in the supersonic airflow. Based on the small damping assumption, the motion governing partial differential equation (PDE) of the panel aeroelastic system, is built by adopting the first-order piston theory and von Karman large deflection plate theory. Then by applying the Galerkin procedure, the PDE is discretized into a set of coupled ordinary differential equations, and the system reduced order model (ROM) with two degrees of freedom is obtained. Considering that the panel aeroelastic system is non-conservative in the physical nature, and assuming that the panel exhibits a single period oscillation on the flutter occurrence, the non-conservative energy balance principle is applied to the linearized ROM within one single oscillation period. The obtained result shows that the ROM modal coordinate amplitudes ratio is regulated by the modal damping coefficients ratio, though each modal damping coefficient is small. Furthermore, as the total damping dissipation energy can be eliminated due to its smallness, the He’s energy balance method is applied to the undamped ROM, therefore the critical non-dimensional dynamic pressure on the flutter instability occurrence, and the oscillation circular frequency amplitude relationship (linear and nonlinear form) are derived. In addition, the damping destabilization paradoxical influence on the system flutter instability is investigated. The accuracy and efficiency of the proposed method are validated by comparing the results with that obtained by using Routh Hurwitz criteria. Full article
(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)
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Article
Hidden Strange Nonchaotic Attractors
Mathematics 2021, 9(6), 652; https://doi.org/10.3390/math9060652 - 18 Mar 2021
Cited by 3 | Viewed by 897
Abstract
In this paper, it is found numerically that the previously found hidden chaotic attractors of the Rabinovich–Fabrikant system actually present the characteristics of strange nonchaotic attractors. For a range of the bifurcation parameter, the hidden attractor is manifestly fractal with aperiodic dynamics, and [...] Read more.
In this paper, it is found numerically that the previously found hidden chaotic attractors of the Rabinovich–Fabrikant system actually present the characteristics of strange nonchaotic attractors. For a range of the bifurcation parameter, the hidden attractor is manifestly fractal with aperiodic dynamics, and even the finite-time largest Lyapunov exponent, a measure of trajectory separation with nearby initial conditions, is negative. To verify these characteristics numerically, the finite-time Lyapunov exponents, ‘0-1’ test, power spectra density, and recurrence plot are used. Beside the considered hidden strange nonchaotic attractor, a self-excited chaotic attractor and a quasiperiodic attractor of the Rabinovich–Fabrikant system are comparatively analyzed. Full article
(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)
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Article
Performance Analysis on the Use of Oscillating Water Column in Barge-Based Floating Offshore Wind Turbines
Mathematics 2021, 9(5), 475; https://doi.org/10.3390/math9050475 - 25 Feb 2021
Cited by 4 | Viewed by 727
Abstract
Undesired motions in Floating Offshore Wind Turbines (FOWT) lead to reduction of system efficiency, the system’s lifespan, wind and wave energy mitigation and increment of stress on the system and maintenance costs. In this article, a new barge platform structure for a FOWT [...] Read more.
Undesired motions in Floating Offshore Wind Turbines (FOWT) lead to reduction of system efficiency, the system’s lifespan, wind and wave energy mitigation and increment of stress on the system and maintenance costs. In this article, a new barge platform structure for a FOWT has been proposed with the objective of reducing these undesired platform motions. The newly proposed barge structure aims to reduce the tower displacements and platform’s oscillations, particularly in rotational movements. This is achieved by installing Oscillating Water Columns (OWC) within the barge to oppose the oscillatory motion of the waves. Response Amplitude Operator (RAO) is used to predict the motions of the system exposed to different wave frequencies. From the RAOs analysis, the system’s performance has been evaluated for representative regular wave periods. Simulations using numerical tools show the positive impact of the added OWCs on the system’s stability. The results prove that the proposed platform presents better performance by decreasing the oscillations for the given range of wave frequencies, compared to the traditional barge platform. Full article
(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)
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Article
Stochastic Computing Implementation of Chaotic Systems
Mathematics 2021, 9(4), 375; https://doi.org/10.3390/math9040375 - 13 Feb 2021
Cited by 1 | Viewed by 926
Abstract
An exploding demand for processing capabilities related to the emergence of the Internet of Things (IoT), Artificial Intelligence (AI), and big data, has led to the quest for increasingly efficient ways to expeditiously process the rapidly increasing amount of data. These ways include [...] Read more.
An exploding demand for processing capabilities related to the emergence of the Internet of Things (IoT), Artificial Intelligence (AI), and big data, has led to the quest for increasingly efficient ways to expeditiously process the rapidly increasing amount of data. These ways include different approaches like improved devices capable of going further in the more Moore path but also new devices and architectures capable of going beyond Moore and getting more than Moore. Among the solutions being proposed, Stochastic Computing has positioned itself as a very reasonable alternative for low-power, low-area, low-speed, and adjustable precision calculations—four key-points beneficial to edge computing. On the other hand, chaotic circuits and systems appear to be an attractive solution for (low-power, green) secure data transmission in the frame of edge computing and IoT in general. Classical implementations of this class of circuits require intensive and precise calculations. This paper discusses the use of the Stochastic Computing (SC) framework for the implementation of nonlinear systems, showing that it can provide results comparable to those of classical integration, with much simpler hardware, paving the way for relevant applications. Full article
(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)
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Article
Zero-Hopf Bifurcation in a Generalized Genesio Differential Equation
Mathematics 2021, 9(4), 354; https://doi.org/10.3390/math9040354 - 10 Feb 2021
Viewed by 522
Abstract
The purpose of the present paper is to study the presence of bifurcations of zero-Hopf type at a generalized Genesio differential equation. More precisely, by transforming such differential equation in a first-order differential system in the three-dimensional space R3, we are [...] Read more.
The purpose of the present paper is to study the presence of bifurcations of zero-Hopf type at a generalized Genesio differential equation. More precisely, by transforming such differential equation in a first-order differential system in the three-dimensional space R3, we are able to prove the existence of a zero-Hopf bifurcation from which periodic trajectories appear close to the equilibrium point located at the origin when the parameters a and c are zero and b is positive. Full article
(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)
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Article
The Topological Entropy Conjecture
Mathematics 2021, 9(4), 296; https://doi.org/10.3390/math9040296 - 03 Feb 2021
Cited by 1 | Viewed by 545
Abstract
For a compact Hausdorff space X, let J be the ordered set associated with the set of all finite open covers of X such that there exists nJ, where nJ is the dimension of X associated with . [...] Read more.
For a compact Hausdorff space X, let J be the ordered set associated with the set of all finite open covers of X such that there exists nJ, where nJ is the dimension of X associated with . Therefore, we have Hˇp(X;Z), where 0pn=nJ. For a continuous self-map f on X, let αJ be an open cover of X and Lf(α)={Lf(U)|Uα}. Then, there exists an open fiber cover L˙f(α) of Xf induced by Lf(α). In this paper, we define a topological fiber entropy entL(f) as the supremum of ent(f,L˙f(α)) through all finite open covers of Xf={Lf(U);UX}, where Lf(U) is the f-fiber of U, that is the set of images fn(U) and preimages fn(U) for nN. Then, we prove the conjecture logρentL(f) for f being a continuous self-map on a given compact Hausdorff space X, where ρ is the maximum absolute eigenvalue of f*, which is the linear transformation associated with f on the Čech homology group Hˇ*(X;Z)=i=0nHˇi(X;Z). Full article
(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)
Article
Estimation of Synchronization Errors between Master and Slave Chaotic Systems with Matched/Mismatched Disturbances and Input Uncertainty
Mathematics 2021, 9(2), 176; https://doi.org/10.3390/math9020176 - 17 Jan 2021
Cited by 2 | Viewed by 607
Abstract
This study is concerned with robust synchronization for master–slave chaotic systems with matched/mismatched disturbances and uncertainty in the control input. A robust sliding mode control (SMC) is presented to achieve chaos synchronization even under the influence of matched/mismatched disturbances and uncertainty of inputs. [...] Read more.
This study is concerned with robust synchronization for master–slave chaotic systems with matched/mismatched disturbances and uncertainty in the control input. A robust sliding mode control (SMC) is presented to achieve chaos synchronization even under the influence of matched/mismatched disturbances and uncertainty of inputs. A proportional-integral (PI) switching surface is introduced to make the controlled error dynamics in the sliding manifold easy to analyze. Furthermore, by using the proposed SMC scheme even subjected to input uncertainty, we can force the trajectories of the error dynamics to enter the sliding manifold and fully synchronize the master–slave systems in spite of matched uncertainties and input nonlinearity. As for the mismatched disturbances, the bounds of synchronization errors can be well estimated by introducing the limit of the Riemann sum, which is not well addressed in previous works. Simulation experiments including matched and mismatched cases are presented to illustrate the robustness and synchronization performance with the proposed SMC synchronization controller. Full article
(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)
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Article
A Dynamic Duopoly Model: When a Firm Shares the Market with Certain Profit
Mathematics 2020, 8(10), 1826; https://doi.org/10.3390/math8101826 - 17 Oct 2020
Cited by 2 | Viewed by 789
Abstract
The current paper analyzes a competition of the Cournot duopoly game whose players (firms) are heterogeneous in a market with isoelastic demand functions and linear costs. The first firm adopts a rationally-based gradient mechanism while the second one chooses to share the market [...] Read more.
The current paper analyzes a competition of the Cournot duopoly game whose players (firms) are heterogeneous in a market with isoelastic demand functions and linear costs. The first firm adopts a rationally-based gradient mechanism while the second one chooses to share the market with certain profit in order to update its production. It trades off between profit and market share maximization. The equilibrium point of the proposed game is calculated and its stability conditions are investigated. Our studies show that the equilibrium point becomes unstable through period doubling and Neimark–Sacker bifurcation. Furthermore, the map describing the proposed game is nonlinear and noninvertible which lead to several stable attractors. As in literature, we have provided an analytical investigation of the map’s basins of attraction that includes lobes regions. Full article
(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)
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Article
Multiple Periodic Solutions and Fractal Attractors of Differential Equations with n-Valued Impulses
Mathematics 2020, 8(10), 1701; https://doi.org/10.3390/math8101701 - 03 Oct 2020
Viewed by 1118
Abstract
Ordinary differential equations with n-valued impulses are examined via the associated Poincaré translation operators from three perspectives: (i) the lower estimate of the number of periodic solutions on the compact subsets of Euclidean spaces and, in particular, on tori; (ii) weakly locally [...] Read more.
Ordinary differential equations with n-valued impulses are examined via the associated Poincaré translation operators from three perspectives: (i) the lower estimate of the number of periodic solutions on the compact subsets of Euclidean spaces and, in particular, on tori; (ii) weakly locally stable (i.e., non-ejective in the sense of Browder) invariant sets; (iii) fractal attractors determined implicitly by the generating vector fields, jointly with Devaney’s chaos on these attractors of the related shift dynamical systems. For (i), the multiplicity criteria can be effectively expressed in terms of the Nielsen numbers of the impulsive maps. For (ii) and (iii), the invariant sets and attractors can be obtained as the fixed points of topologically conjugated operators to induced impulsive maps in the hyperspaces of the compact subsets of the original basic spaces, endowed with the Hausdorff metric. Five illustrative examples of the main theorems are supplied about multiple periodic solutions (Examples 1–3) and fractal attractors (Examples 4 and 5). Full article
(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)
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