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

Deadline for manuscript submissions: closed (30 September 2021) | Viewed by 31303

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Prof. Dr. J. A. Tenreiro Machado <sup>†</sup>

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Institute of Engineering, Department of Electrical Engineering, Polytechnic Institute of Porto, R. Dr. Roberto Frias, 4249-015 Porto, Portugal
Interests: nonlinear dynamics; complexity; fractional calculus; modeling; control; entropy; genomics
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Dr. António Lopes

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Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200–465 Porto, Portugal
Interests: complex systems modelling; automation and robotics; fractional order systems modelling and control; data analysis and visualization
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Special Issue Information

Dear Colleagues,

The modeling and control of nonlinear dynamic systems is still a challenging problem in mathematics and engineering. Despite much investigation being carried out so far, nonlinear and complex phenomena are not yet fully understood, due to their considerable randomness and a diversity of reasons underlying the energy dissipation involving the dynamic effects. In fact, the present knowledge and scientific tools are still far from capturing the overall richness of the system dynamics.

This Special Issue focuses on the modeling and control of nonlinear dynamic systems. Manuscripts on nonlinear dynamics, advanced control systems, complex dynamics, fractional calculus and its applications, fractals and chaos, multibody systems, robotics, and modelling, among others, are welcome.

The Special Issue will bring together contributions from researchers in different topics of engineering, mathematics, physics, biology, geophysics, and other sciences. Papers describing original theoretical research as well as new experimental results are expected.

Prof. Dr. J. A. Tenreiro Machado
Prof. Dr. António M. Lopes
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

nonlinear dynamics
complexity
advanced control systems
fractional calculus
signal processing
time series
complex networks
electro-mechanical structures
finance end economy systems
biomathematics
evolutionary computing
chaos and fractals

Published Papers (13 papers)

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Research

25 pages, 1014 KiB

Open AccessArticle

Dynamics Analysis and Optimal Control for a Delayed Rumor-Spreading Model

by Chunru Li and Zujun Ma

Mathematics 2022, 10(19), 3455; https://0-doi-org.brum.beds.ac.uk/10.3390/math10193455 - 22 Sep 2022

Cited by 3 | Viewed by 1167

Abstract

In this work, we analyze a delayed rumor-propagation model. First, we analyze the existence and boundedness of the solution of the model. Then, we give the conditions for the existence of the rumor-endemic equilibrium. Regrading the delay as a bifurcating parameter, we explore the local asymptotic stability and Hopf bifurcation of the rumor-endemic equilibrium. By a Lyapunov functional technique, we examine the global asymptotically stability of the rumor-free and the rumor-endemic equilibria. We provide two control variables in the rumor-spreading model with time delay, and get the optimal solution via the optimal procedures. Finally, we present some numerical simulations to verify our theoretical predictions. They illustrate that the delay is a crucial issue for system, and it can lead to not just Hopf bifurcation but also chaos. Full article

(This article belongs to the Special Issue Nonlinear Dynamics)

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10 pages, 264 KiB

Open AccessFeature PaperArticle

New Irregular Solutions in the Spatially Distributed Fermi–Pasta–Ulam Problem

by Sergey Kashchenko and Anna Tolbey

Mathematics 2021, 9(22), 2872; https://0-doi-org.brum.beds.ac.uk/10.3390/math9222872 - 12 Nov 2021

Cited by 2 | Viewed by 1168

Abstract

For the spatially-distributed Fermi–Pasta–Ulam (FPU) equation, irregular solutions are studied that contain components rapidly oscillating in the spatial variable, with different asymptotically large modes. The main result of this paper is the construction of families of special nonlinear systems of the Schrödinger type—quasinormal [...] Read more.

t \to \infty

. On their basis, results are obtained on the asymptotics in the main solution of the FPU equation and on the interaction of waves moving in opposite directions. The problem of “perturbing” the number of N elements of a chain is considered. In this case, instead of the differential operator, with respect to one spatial variable, a special differential operator, with respect to two spatial variables appears. This leads to a complication of the structure of an irregular solution. Full article

(This article belongs to the Special Issue Nonlinear Dynamics)

17 pages, 917 KiB

Open AccessArticle

Stability Switches and Double Hopf Bifurcation Analysis on Two-Degree-of-Freedom Coupled Delay van der Pol Oscillator

by Yani Chen and Youhua Qian

Mathematics 2021, 9(19), 2444; https://0-doi-org.brum.beds.ac.uk/10.3390/math9192444 - 01 Oct 2021

Cited by 2 | Viewed by 1577

Abstract

In this paper, the normal form and central manifold theories are used to discuss the influence of two-degree-of-freedom coupled van der Pol oscillators with time delay feedback. Compared with the single-degree-of-freedom time delay van der Pol oscillator, the system studied in this paper has richer dynamical behavior. The results obtained include: the change of time delay causing the stability switching of the system, and the greater the time delay, the more complicated the stability switching. Near the double Hopf bifurcation point, the system is simplified by using the normal form and central manifold theories. The system is divided into six regions with different dynamical properties. With the above results, for practical engineering problems, we can perform time delay feedback adjustment to make the system show amplitude death, limit loop, and so on. It is worth noting that because of the existence of unstable limit cycles in the system, the limit cycle cannot be obtained by numerical solution. Therefore, we derive the approximate analytical solution of the system and simulate the time history of the interaction between two frequencies in Region IV. Full article

(This article belongs to the Special Issue Nonlinear Dynamics)

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18 pages, 999 KiB

Open AccessArticle

On the Poisson Stability to Study a Fourth-Order Dynamical System with Quadratic Nonlinearities

by Alexander N. Pchelintsev

Mathematics 2021, 9(17), 2057; https://0-doi-org.brum.beds.ac.uk/10.3390/math9172057 - 26 Aug 2021

Cited by 3 | Viewed by 1648

Abstract

This article discusses the search procedure for Poincaré recurrences to classify solutions on an attractor of a fourth-order nonlinear dynamical system, using a previously developed high-precision numerical method. For the resulting limiting solution, the Lyapunov exponents are calculated, using the modified Benettin’s algorithm [...] Read more.

(This article belongs to the Special Issue Nonlinear Dynamics)

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25 pages, 1887 KiB

Open AccessArticle

Schistosomiasis Model Incorporating Snail Predator as Biological Control Agent

by Wahyudin Nur, Trisilowati, Agus Suryanto and Wuryansari Muharini Kusumawinahyu

Mathematics 2021, 9(16), 1858; https://0-doi-org.brum.beds.ac.uk/10.3390/math9161858 - 05 Aug 2021

Cited by 4 | Viewed by 2110

Abstract

Schistosomiasis is a parasitic disease caused by the schistosoma worm. A snail can act as the intermediate host for the parasite. Snail-population control is considered to be an effective way to control schistosomiasis spread. In this paper, we discuss the schistosomiasis model incorporating a snail predator as a biological control agent. We prove that the solutions of the model are non-negative and bounded. The existence condition of equilibrium points is investigated. We determine the basic reproduction number when the predator goes to extinction and when the predator survives. The local stability condition of disease-free equilibrium point is proved using linearization, and the Lienard–Chipart and Routh–Hurwitz criteria. We use center-manifold theory to prove the local stability condition of the endemic equilibrium points. Furthermore, we constructed a Lyapunov function to investigate the global stability condition of the disease-free equilibrium points. To support the analytical results, we presented some numerical simulation results. Our findings suggest that a snail predator as a biological control agent can reduce schistosomiasis prevalence. Moreover, the snail-predator birth rate plays an essential role in controlling schistosomiasis spread. Full article

(This article belongs to the Special Issue Nonlinear Dynamics)

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15 pages, 3262 KiB

Open AccessArticle

LMI-Observer-Based Stabilizer for Chaotic Systems in the Existence of a Nonlinear Function and Perturbation

by Hamede Karami, Saleh Mobayen, Marzieh Lashkari, Farhad Bayat and Arthur Chang

Mathematics 2021, 9(10), 1128; https://0-doi-org.brum.beds.ac.uk/10.3390/math9101128 - 16 May 2021

Cited by 21 | Viewed by 2805

Abstract

In this study, the observer-based state feedback stabilizer design for a class of chaotic systems in the existence of external perturbations and Lipchitz nonlinearities is presented. This manuscript aims to design a state feedback controller based on a state observer by the linear matrix inequality method. The conditions of linear matrix inequality guarantee the asymptotical stability of the system based on the Lyapunov theorem. The stabilizer and observer parameters are obtained using linear matrix inequalities, which make the state errors converge to the origin. The effects of the nonlinear Lipschitz perturbation and external disturbances on the system stability are then reduced. Moreover, the stabilizer and observer design techniques are investigated for the nonlinear systems with an output nonlinear function. The main advantages of the suggested approach are the convergence of estimation errors to zero, the Lyapunov stability of the closed-loop system and the elimination of the effects of perturbation and nonlinearities. Furthermore, numerical examples are used to illustrate the accuracy and reliability of the proposed approaches. Full article

(This article belongs to the Special Issue Nonlinear Dynamics)

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16 pages, 11082 KiB

Open AccessArticle

Nonlinear Dynamics and Control of a Cube Robot

by Teh-Lu Liao, Sian-Jhe Chen, Cheng-Chang Chiu and Jun-Juh Yan

Mathematics 2020, 8(10), 1840; https://0-doi-org.brum.beds.ac.uk/10.3390/math8101840 - 19 Oct 2020

Cited by 3 | Viewed by 4726

Abstract

The paper aims to solve problems of the mathematical modeling and realization of a cube robot capable of self-bouncing and self-balancing. First, the dynamic model of the cube robot is derived by using the conservation of the angular momentum and the torque equilibrium theory. Furthermore, the controllability of the cube robot is analyzed and the angle of the cube robot is derived from the attitude and heading reference system (AHRS). Then the parallel proportional–integral–derivative (PID) controller is proposed for the balancing control of the self-designed cube robot. As for the bounce control of the cube robot, a braking system triggered by the servo motor is designed for converting the kinetic energy to the potential energy. Finally, the experimental results are included to demonstrate that the cube robot can complete the actions of self-bouncing and self-balancing with good robustness to external disturbances. Full article

(This article belongs to the Special Issue Nonlinear Dynamics)

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16 pages, 516 KiB

Open AccessArticle

Unpredictable Solutions of Linear Impulsive Systems

by Marat Akhmet, Madina Tleubergenova, Mehmet Onur Fen and Zakhira Nugayeva

Mathematics 2020, 8(10), 1798; https://0-doi-org.brum.beds.ac.uk/10.3390/math8101798 - 16 Oct 2020

Cited by 7 | Viewed by 2448

Abstract

We consider a new type of oscillations of discontinuous unpredictable solutions for linear impulsive nonhomogeneous systems. The models under investigation are with unpredictable perturbations. The definition of a piecewise continuous unpredictable function is provided. The moments of impulses constitute a newly determined unpredictable discrete set. Theoretical results on the existence, uniqueness, and stability of discontinuous unpredictable solutions for linear impulsive differential equations are provided. We benefit from the B-topology in the space of discontinuous functions on the purpose of proving the presence of unpredictable solutions. For constructive definitions of unpredictable components in examples, randomly determined unpredictable sequences are newly utilized. Namely, the construction of a discontinuous unpredictable function is based on an unpredictable sequence determined by a discrete random process, and the set of discontinuity moments is realized by the logistic map. Examples with numerical simulations are presented to illustrate the theoretical results. Full article

(This article belongs to the Special Issue Nonlinear Dynamics)

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14 pages, 298 KiB

Open AccessArticle

On Complete Monotonicity of Solution to the Fractional Relaxation Equation with the nth Level Fractional Derivative

by Yuri Luchko

Mathematics 2020, 8(9), 1561; https://0-doi-org.brum.beds.ac.uk/10.3390/math8091561 - 11 Sep 2020

Cited by 8 | Viewed by 1613

Abstract

In this paper, we first deduce the explicit formulas for the projector of the nth level fractional derivative and for its Laplace transform. Afterwards, the fractional relaxation equation with the nth level fractional derivative is discussed. It turns out that, under some conditions, the solutions to the initial-value problems for this equation are completely monotone functions that can be represented in form of the linear combinations of the Mittag–Leffler functions with some power law weights. Special attention is given to the case of the relaxation equation with the second level derivative. Full article

(This article belongs to the Special Issue Nonlinear Dynamics)

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24 pages, 386 KiB

Open AccessArticle

Revisiting the 1D and 2D Laplace Transforms

by Manuel Duarte Ortigueira and José Tenreiro Machado

Mathematics 2020, 8(8), 1330; https://0-doi-org.brum.beds.ac.uk/10.3390/math8081330 - 10 Aug 2020

Cited by 27 | Viewed by 2406

Abstract

The paper reviews the unilateral and bilateral, one- and two-dimensional Laplace transforms. The unilateral and bilateral Laplace transforms are compared in the one-dimensional case, leading to the formulation of the initial-condition theorem. This problem is solved with all generality in the one- and two-dimensional cases with the bilateral Laplace transform. The case of fractional-order systems is also included. General two-dimensional linear systems are introduced and the corresponding transfer function is defined. Full article

(This article belongs to the Special Issue Nonlinear Dynamics)

24 pages, 3011 KiB

Open AccessArticle

Computer Analysis of Human Belligerency

by José A. Tenreiro Machado, António M. Lopes and Maria Eugénia Mata

Mathematics 2020, 8(8), 1201; https://0-doi-org.brum.beds.ac.uk/10.3390/math8081201 - 22 Jul 2020

Cited by 3 | Viewed by 1805

Abstract

War is a cause of gains and losses. Economic historians have long stressed the extreme importance of considering the economic potential of society for belligerency, the role of management of chaos to bear the costs of battle and casualties, and ingenious and improvisation methodologies for emergency management. However, global and inter-temporal studies on warring are missing. The adoption of computational tools for data processing is a key modeling option with present day resources. In this paper, hierarchical clustering techniques and multidimensional scaling are used as efficient instruments for visualizing and describing military conflicts by electing different metrics to assess their characterizing features: time, time span, number of belligerents, and number of casualties. Moreover, entropy is adopted for measuring war complexity over time. Although wars have been an important topic of analysis in all ages, they have been ignored as a subject of nonlinear dynamics and complex system analysis. This paper seeks to fill these gaps in the literature by proposing a quantitative perspective based on algorithmic strategies. We verify the growing number of events and an explosion in their characteristics. The results have similarities to those exhibited by systems with increasing volatility, or evolving toward chaotic-like behavior. We can question also whether such dynamics follow the second law of thermodynamics since the adopted techniques reflect a system expanding the entropy. Full article

(This article belongs to the Special Issue Nonlinear Dynamics)

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18 pages, 2135 KiB

Open AccessArticle

A New Generalized Taylor-Like Explicit Method for Stiff Ordinary Differential Equations

by Essam R. El-Zahar, José Tenreiro Machado and Abdelhalim Ebaid

Mathematics 2019, 7(12), 1154; https://0-doi-org.brum.beds.ac.uk/10.3390/math7121154 - 01 Dec 2019

Cited by 8 | Viewed by 2394

Abstract

A new generalised Taylor-like explicit method for stiff ordinary differential equations (ODEs) is proposed. The algorithm is presented in its component and vector forms. The error and stability analysis of the method are developed showing that it has an arbitrary high order of convergence and the L-stability property. Moreover, it is verified that several integration schemes are special cases of the new general form. The method is applied on stiff problems and the numerical solutions are compared with those of the classical Taylor-like integration schemes. The results show that the proposed method is accurate and overcomes the shortcoming of the classical Taylor-like schemes in their component and vector forms. Full article

(This article belongs to the Special Issue Nonlinear Dynamics)

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25 pages, 1352 KiB

Open AccessArticle

Ethanol Prices and Agricultural Commodities: An Investigation of Their Relationship

by Sergio Adriani David, Claudio M. C. Inácio, Jr. and José A. Tenreiro Machado

Mathematics 2019, 7(9), 774; https://0-doi-org.brum.beds.ac.uk/10.3390/math7090774 - 22 Aug 2019

Cited by 6 | Viewed by 2704

Abstract

Brazil is an important player when it comes to biofuel and agricultural production. The knowledge of the price relationship between these markets has increasing importance. This paper adopts several tools, namely the Bai–Perron test of breakpoints, the Johansen cointegration test and the vector error correction model exploited by the orthogonal impulse response and the forecast error variance decomposition, for investigating the price transmission among the ethanol and the main Brazil’s agricultural commodities (sugar, cotton, arabica coffee, robusta coffee, live cattle, corn and soybean). The data series cover the period from January 2011 up to December 2018. The results suggest a stronger price transmission from the ethanol commodity to the agricultural commodities, rather than the opposite situation. Full article

(This article belongs to the Special Issue Nonlinear Dynamics)

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