Research

9 pages, 1121 KiB

Open AccessArticle

Towards Optimal Supercomputer Energy Consumption Forecasting Method

by Jiří Tomčala

Mathematics 2021, 9(21), 2695; https://doi.org/10.3390/math9212695 - 23 Oct 2021

Cited by 1 | Viewed by 1386

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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11 pages, 286 KiB

Open AccessArticle

Chaos on Fuzzy Dynamical Systems

by Félix Martínez-Giménez, Alfred Peris and Francisco Rodenas

Mathematics 2021, 9(20), 2629; https://doi.org/10.3390/math9202629 - 18 Oct 2021

Cited by 3 | Viewed by 1372

Abstract

Given a continuous map

f : X \to X

on a metric space, it induces the maps

\bar{f} : K (X) \to K (X)

, on the hyperspace of nonempty compact subspaces of X, and [...] Read more.

Given a continuous map

f : X \to X

on a metric space, it induces the maps

\bar{f} : K (X) \to K (X)

, on the hyperspace of nonempty compact subspaces of X, and

\hat{f} : F (X) \to F (X)

, on the space of normal fuzzy sets, consisting of the upper semicontinuous functions

u : X \to [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), \bar{f})

, and

(F (X), \hat{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)

13 pages, 505 KiB

Open AccessArticle

Nonlinear Compartment Models with Time-Dependent Parameters

by Jochen Merker, Benjamin Kunsch and Gregor Schuldt

Mathematics 2021, 9(14), 1657; https://doi.org/10.3390/math9141657 - 14 Jul 2021

Cited by 3 | Viewed by 1634

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

C^{0}

-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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12 pages, 7392 KiB

Open AccessArticle

Applications of the Network Simulation Method to Differential Equations with Singularities and Chaotic Behaviour

by Joaquín Solano, Francisco Balibrea and José Andrés Moreno

Mathematics 2021, 9(12), 1442; https://doi.org/10.3390/math9121442 - 21 Jun 2021

Cited by 8 | Viewed by 1575

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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11 pages, 2081 KiB

Open AccessCommunication

A Combined Energy Method for Flutter Instability Analysis of Weakly Damped Panels in Supersonic Airflow

by Xiaochen Wang, Zhichun Yang, Guiwei Zhang and Xinwei Xu

Mathematics 2021, 9(10), 1090; https://doi.org/10.3390/math9101090 - 12 May 2021

Cited by 2 | Viewed by 1557

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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19 pages, 10957 KiB

Open AccessArticle

Hidden Strange Nonchaotic Attractors

by Marius-F. Danca and Nikolay Kuznetsov

Mathematics 2021, 9(6), 652; https://doi.org/10.3390/math9060652 - 18 Mar 2021

Cited by 16 | Viewed by 3413

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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22 pages, 3456 KiB

Open AccessArticle

Performance Analysis on the Use of Oscillating Water Column in Barge-Based Floating Offshore Wind Turbines

by Payam Aboutalebi, Fares M’zoughi, Izaskun Garrido and Aitor J. Garrido

Mathematics 2021, 9(5), 475; https://doi.org/10.3390/math9050475 - 25 Feb 2021

Cited by 22 | Viewed by 2714

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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20 pages, 1362 KiB

Open AccessArticle

Stochastic Computing Implementation of Chaotic Systems

by Oscar Camps, Stavros G. Stavrinides and Rodrigo Picos

Mathematics 2021, 9(4), 375; https://doi.org/10.3390/math9040375 - 13 Feb 2021

Cited by 8 | Viewed by 2146

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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11 pages, 549 KiB

Open AccessArticle

Zero-Hopf Bifurcation in a Generalized Genesio Differential Equation

by Zouhair Diab, Juan L. G. Guirao and Juan A. Vera

Mathematics 2021, 9(4), 354; https://doi.org/10.3390/math9040354 - 10 Feb 2021

Cited by 1 | Viewed by 1721

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

R^{3}

, 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

R^{3}

, 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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17 pages, 321 KiB

Open AccessArticle

The Topological Entropy Conjecture

by Lvlin Luo

Mathematics 2021, 9(4), 296; https://doi.org/10.3390/math9040296 - 03 Feb 2021

Cited by 1 | Viewed by 1389

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

n_{J}

, where

n_{J}

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

n_{J}

, where

n_{J}

is the dimension of X associated with ∂. Therefore, we have

{\overset{ˇ}{H}}_{p} (X; Z)

, where

0 \leq p \leq n = n_{J}

. For a continuous self-map f on X, let

α \in J

be an open cover of X and

L_{f} (α) = {L_{f} (U) | U \in α}

. Then, there exists an open fiber cover

{\dot{L}}_{f} (α)

of

X^{f}

induced by

L_{f} (α)

. In this paper, we define a topological fiber entropy

e n t_{L} (f)

as the supremum of

e n t (f, {\dot{L}}_{f} (α))

through all finite open covers of

X^{f} = {L_{f} (U); U \subset X}

, where

L_{f} (U)

is the f-fiber of U, that is the set of images

f^{n} (U)

and preimages

f^{- n} (U)

for

n \in N

. Then, we prove the conjecture

log ρ \leq e n t_{L} (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

{\overset{ˇ}{H}}_{*} (X; Z) = ⨁_{i = 0}^{n} {\overset{ˇ}{H}}_{i} (X; Z) .

Full article

(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)

15 pages, 2183 KiB

Open AccessArticle

Estimation of Synchronization Errors between Master and Slave Chaotic Systems with Matched/Mismatched Disturbances and Input Uncertainty

by Chih-Hsueh Lin, Guo-Hsin Hu and Jun-Juh Yan

Mathematics 2021, 9(2), 176; https://doi.org/10.3390/math9020176 - 17 Jan 2021

Cited by 6 | Viewed by 1951

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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13 pages, 73602 KiB

Open AccessArticle

A Dynamic Duopoly Model: When a Firm Shares the Market with Certain Profit

by Sameh S. Askar

Mathematics 2020, 8(10), 1826; https://doi.org/10.3390/math8101826 - 17 Oct 2020

Cited by 7 | Viewed by 2508

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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21 pages, 1981 KiB

Open AccessArticle

Multiple Periodic Solutions and Fractal Attractors of Differential Equations with n-Valued Impulses

by Jan Andres

Mathematics 2020, 8(10), 1701; https://doi.org/10.3390/math8101701 - 03 Oct 2020

Viewed by 2171

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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