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A novel fractional operator-based model for Parkinson’s disease: Analyzing abnormal beta-oscillation and the influence of synaptic parameters
Wang G., Huang N., Ahmad B.
Q2
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With the aggravation of the global aging trend, Parkinson’s disease has become a hot spot of scientific research all over the world. Abnormal β-oscillation in the basal ganglia region is considered to be a major inducement of Parkinson’s disease. In this paper, a new and more complete Parkinson’s model based on fractional operators is proposed to study the oscillation behavior of the basal ganglia region. The correctness of this new fractional model is validated by the simulation of Nambu and Tachibana’s experiment [A. Nambu, Y. Tachibana, Mechanism of parkinsonian neuronal oscillations in the primate basal ganglia: some considerations based on our recent work, Front. Syst. Neurosci., 8:74, 2014]. Then we carry out the Hopf bifurcation analysis of the fractional model and derive the critical conditions for periodic oscillation. The influence of important parameters on the oscillation behavior of the system is analyzed by numerical simulations. It is found that proper control of synaptic transmission delay and synaptic connection strength can improve the abnormal β-oscillation behavior in the basal ganglia region effectively. In addition, the fractional Parkinson’s model in this paper provides more flexibility for model fitting and parameter estimation. The choice of the fractional order α plays a crucial role in the analysis of system oscillation.
Control of the servo motor using feedback linearization and artificial gorilla troops optimizer
Jovanović R., Vesović M.
Q2
This paper establishes a nonlinear optimization strategy for position control of a direct current motor. When experimental evidence showed that the linear model does not sufficiently represent the system, the model is modified from linear to nonlinear, using friction-induced nonlinearity. In the course of the research, an analysis of the nonlinear feedback linearizing controller and the up to date gorilla troops optimization algorithm are carried out. The proposed algorithm is juxtapose with four others metaheuristic optimizations. Furthermore, performances with and without different types of disturbances are compared for individual desired output signals. The experimental results corroborate the nonlinear control’s robustness.
The finite-time ruin probabilities of a dependent bidimensional risk model with subexponential claims and Brownian perturbations
Xu C., Shen X., Wang K.
Q2
The paper considers a dependent bidimensional risk model with stochastic return and Brownian perturbations in which the price processes of the investment portfolio of the two lines of business are two geometric Lévy processes, and the claim-number processes of the two lines of business follows two different stochastic processes, which can be dependent. When the two components of each pair of claims from the two lines of business are strongly asymptotically independent and have subexponential distributions, the asymptotics of the finite-time ruin probability are obtained. Numerical studies are carried out to check the accuracy of the asymptotics of the finite-time ruin probability for the claims having regularly varying tail distributions.
On a novel type of generalized simulation functions with fixed point results for wide Ws-contractions
Roldán López de Hierro A.F., Sintunavarat W.
Q2
One of the most significant hypotheses in fixed point theory is the nonexpansivity condition of contractive mappings. This property is crucial as operators that do not satisfy this criterion may lack fixed points. In this paper, we propose a novel condition that, within the appropriate framework, can obviate the necessity of imposing the nonexpansivity requirement in the initial hypotheses. By employing this new condition, we illustrate how innovative results can be derived in this area. Finally, we examine the existence and uniqueness of a solution for an elastic beam equation with nonlinear boundary conditions grounded in the introduced fixed point results.
A discontinuous nonlinear singular elliptic problem with the fractional rho-Laplacian
Zineddaine G., Sabiry A., Kassidi A., Chadli L.S.
Q2
In this paper, we use the topological degree method, based on the abstract Hammerstein equation, to investigate the existence of weak solutions for a certain class of elliptic Dirichlet boundary value problems. These problems involve the fractional ρ-Laplacian operator and involve discontinuous nonlinearities in the framework of fractional Sobolev spaces.
Mitigating atmospheric carbon dioxide through deployment of renewable energy: A mathematical model
Jha A., Misra A.K.
Q2
In recent decades, the widespread reliance on fossil fuels has grown substantially, leading to a rise in atmospheric carbon dioxide (CO2), which poses a major global concern. In this study, we develop and analyze a novel mathematical model to examine the interactions between atmospheric CO2, human population, and energy demand. The model assumes that human activities and energy production from traditional sources (oil, coal, and gas) contribute to increasing CO2 level, while a shift in energy dependence from traditional to renewable sources (hydro, solar, etc.) occurs as a result of environmental awareness. We derive sufficient conditions for both local and global stability of the system’s interior equilibrium. Numerical simulations demonstrate that when reliance on renewable energy sources is low, the system can exhibit oscillatory dynamics and various bifurcations. However, beyond a critical threshold of renewable energy dependency, the system stabilizes around the interior equilibrium, leading to a reduction in atmospheric CO2. Additionally, an optimal control problem is formulated to reduce atmospheric CO2 level while minimizing the associated implementation costs.
Bifurcation in a Leslie–Gower system with fear in predators and strong Allee effect in prey
Wu R., Xiong W.
Q2
In this paper, we consider a modified Leslie–Gower predator–prey model with Allee effect on prey and fear effect on predators. Results show complex dynamical behaviors in the model with these factors. Existence of equilibrium points and their stability of the model are first given. Then it is found that, with the Allee and fear effects, the model exhibits various and different bifurcations, such as saddle-node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation. Theoretical analysis is verified through some numerical simulations.
Codimension-two bifurcation analysis of a discrete predator–prey system with fear effect and Allee effect
Lei C., Han X.
Q2
In this paper, we study the dynamic behavior of a discrete predator–prey model with fear effect and Allee effect by theoretical analysis and numerical simulation. Firstly, the existence and stability of the equilibrium points of the model are proved. Secondly, the existence of codimension-2 bifurcations (1 : 2, 1 : 3, and 1 : 4 strong resonances) in the case of two parameters is verified by bifurcation theory. In order to illustrate the complexity of the dynamic behavior of the model in the two-parameter space, we simulate the bifurcation diagrams, phase diagrams, maximum Lyapunov exponent diagrams, and isoperiodic diagram, and we verify the influence of model parameters on the population size.
Bifurcation analysis of a Leslie–Gower predator–prey system with fear effect and constant-type harvesting
Huangfu C., Li Z.
Q2
This paper investigates the effect of fear effect and constant-type harvesting on the dynamic of a Leslie–Gower predator–prey model. Initially, an analysis is carried out to identify all potential equilibria and evaluate their stability. Furthermore, the dynamic behavior at these points is examined, revealing various bifurcations such as saddle-node bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation. In particular, the model undergoes a degenerate Hopf bifurcation, which leads to the existence of two limit cycles. Additionally, we demonstrate that the Bogdanov–Takens bifurcation of codimension 2 occurs in this model. Ultimately, these findings are validated through numerical simulations, demonstrating that continuous harvesting or the significant fear effect is not conducive to either predator or prey surviving.
Notes on the losses compensation of a three-phase power system with DC offset based on geometric algebras
Brdečková J., Hrdina J.
Q2
In a three-phase symmetric power system, we propose a transformation that converts the original current to a current with minimal losses while preserving the standard constraints. The selected transformation is realized in suitable geometric algebra and is not time-dependent. The proposed transformation uses the group symmetry of conformal geometric algebra, mainly rotations and tations.
Estimations for the convex modular of the aliasing error of nonlinear sampling Kantorovich operators
Costarelli D., Natale M., Vinti G.
Q2
In this paper, we establish quantitative estimates for the nonlinear sampling Kantorovich operators in the general setting of modular spaces Lρ. To achieve this, we consider a notion of modulus of smoothness based on the convex modular functional ρ, which defines the space. The approach proposed is new in the sense that, in the literature, theorems for the order of approximation in Lρ are mainly qualitative, i.e., are proved considering functions belonging to Lipschitz classes; here the estimates are achieved for every function belonging to the whole Lρ. To show the effectiveness of the achieved results, several particular cases of modular spaces are presented in detail.
Effect of gravity on the pattern formation in aqueous suspensions of luminous Escherichia coli
Dapkūnas B., Baronas R., Šimkus R.
Q2
This paper presents a nonlinear two-dimensional-in-space mathematical model of self-organization of aqueous bacterial suspensions. The reaction–diffusion–chemotaxis model is coupled with the incompressible Navier–Stokes equations, which are subject to a gravitational force proportional to the relative bacteria density and include a cut-off mechanism. The bacterial pattern formation of luminous Escherichia coli is modelled near the inner lateral surface of a circular microcontainer, as detected by bioluminescence imaging. The simulated plume-like patterns are analysed to determine the values of the dimensionless model parameters, the Schmidt number, Rayleigh number and oxygen cut-off threshold, that closely match the patterns observed experimentally in a luminous E. coli colony. The numerical simulation at the transient conditions was carried out using the finite difference technique.
Turing pattern dynamics in a fractional-diffusion oregonator model under PD control
Li H., Yao Y., Xiao M., Wang Z., Rutkowski L.
Q2
In this paper, fractional-order diffusion and proportional-derivative (PD) control are introduced in oregonator model, and the Turing pattern dynamics is investigated for the first time. We take the cross-diffusion coefficient as the bifurcation parameter and give some necessary conditions for Turing instability of the fractional-diffusion oregonator model under PD control. At the same time, we construct the amplitude equations near the bifurcation threshold and determine the pattern formation of the fractional-diffusion oregonator model under PD controller. It is observed by numerical simulations that in the absence of control, the pattern formation changes with the variation of the cross-diffusion coefficient in two-dimensional space. Meanwhile, it is verified that the PD control has a significant impact on Turing instability, and the pattern structure can be changed by manipulating the control gain parameters for the fractional-diffusion oregonator model.
Fixed point results for single and multivalued three-points contractions
Jleli M., Petrov E., Samet B.
Q2
In this paper, we are concerned with the study of the existence of fixed points for single and multivalued three-points contractions. Namely, we first introduce a new class of singlevalued mappings defined on a metric space equipped with three metrics. A fixed point theorem is established for such mappings. The obtained result recovers that established recently by the second author [E. Petrov, Fixed point theorem for mappings contracting perimeters of triangles, J. Fixed Point Theory Appl., 25(3):74, 2023] for the class of single-valued mappings contracting perimeters of triangles. We next extend our study by introducing the class of multivalued three points contractions. A fixed point theorem, which is a multivalued version of that obtained in the above reference, is established. Some examples showing the validity of our obtained results are provided.
Existence of a positive solution with concave and convex components for a system of boundary value problems
Antoņuks A.
Q2
We prove the existence of at least one positive solution for a system of two nonlinear second-order differential equations with nonlocal boundary conditions. One component of the solution is a concave function, and the other one is a convex function. A recent hybrid Krasnosel’skiĭ–Schauder fixed point theorem is used to prove the existence of a positive solution. To illustrate the applicability of the obtained result, an example is considered.
A few generalizations of Kendall’s tau. Part II: Intrinsic meaning, properties, and computational aspects
Manstavičius M.
Q2
Continuing our investigation on generalizations of Kendall’s τ, started in Part I of the paper, here we elaborate on the intrinsic meaning and degree of such polynomial-type concordance measures, as well as present many examples of their computation. In particular, we interpret generalized Kendall’s τφ as the difference between the average capacities of concordance and discordance, and, for power-type distortion functions φ, we obtain polynomial-type concordance measures of various degree, which could stimulate further research of their characterization as achieved for degree-one polynomial-type concordance measures by Taylor, Edwards, and Mikusiński.
Convergence analysis of positive solution for Caputo–Hadamard fractional differential equation
Guo L., Li C., Qiao N., Zhao J.
Q2
By deriving the expression of Green function and some of its special properties and establishing appropriate substitution and appropriate cone, the existence of unique iterative positive, error estimation, and convergence rate of approximate solution are obtained for singular p-Laplacian Caputo–Hadamard fractional differential equation with infinite-point boundary conditions. Nonlinearities involve derivative terms that make our analysis difficult in the course of this research, and we deal with the difficulty of derivative terms by making appropriate substitutions. An example is given to demonstrate the validity of our main results.
On stability and convergence of difference schemes for one class of parabolic equations with nonlocal condition
Sapagovas M., Novickij J., Pupalaigė K.
Q2
In this paper, we construct and analyze the finite-difference method for a two-dimensional nonlinear parabolic equation with nonlocal boundary condition. The main objective of this paper is to investigate the stability and convergence of the difference scheme in the maximum norm. We provide some approaches for estimating the error of the solution. In our approach, the assumption of the validity of the maximum principle is not required. The assumption is changed to a weaker one: the difference problem’s matrix is the M-matrix. We present numerical experiments to illustrate and supplement theoretical results.
Positive solutions for a Hadamard-type fractional-order three-point boundary value problem on the half-line
Xu J., Cui Y., O’Regan D.
Q2
In this paper, we study a Hadamard-type fractional-order three-point boundary value problem on the half-line. Under some growth conditions concerning the spectral radius of the relevant linear operator, the existence and multiplicity of positive solutions is obtained using a fixed-point method. Our results improve and generalize some results in the literature.
Fixed point theory in RWC–Banach algebras
Banaś J., Krichen B., Mefteh B., O’Regan D.
Q2
In this paper, we prove some fixed point results for the sum and the product of nonlinear continuous operators acting on an RWC–Banach algebra. Our result is formulated in terms of topological conditions on the operators. An illustrative example on an RWC–Banach algebra, which is not a WC–Banach algebra, is provided.
New formulation of Lyapunov direct method for nonautonomous real-order systems
Lenka B.K., Bora S.N.
Q2
Lyapunov stability analysis of nonautonomous real-order systems is put forward here in the sense of Caputo in a new and different way. We introduce new theorems and inequalities that give stability of constant solutions in the domain of attraction to such systems when attached with random initial time placed on the real axis. We give some examples including an advanced nonlinear Lorenz system to illustrate the results.
Distributed constraint optimization for discrete-time multiagent systems with event-triggered communication
Gu M., Yu Z., Jiang H.
Q2
This paper investigates the distributed optimization problem (DOP) with equality constraint in discrete-time multiagent systems (MASs) in which the global optimization objective is constituted by the summation of local objective functions. Firstly, by employing the Lagrange multiplier method, we convert the convex optimization problem with equality constraint into a consensus problem of MASs. Secondly, to reduce the communication burden, a type of event-triggered control protocol is proposed to enable all agents achieving consensus. Thirdly, by employing the Lyapunov function method and a set of inequality techniques, we establish some sufficient conditions to ensure that all agents converge to consensus and successfully solve the original DOP. Finally, a numerical simulation example is presented to validate the effectiveness of the theoretical analysis.
Absolute exponential stability of switching time-delay Lurie systems with the application to switching Hopfield neural networks
Wenxiu Zhao W., Sun Y.
Q2
This paper investigates the problem of absolute exponential stability analysis for switching time-delay Lurie system (STDLS) with all modes unstable. By proposing a novel switching time-varying Lyapunov–Razumikhin function, a computable sufficient condition is formulated to guarantee absolute exponential stability of STDLS under mode-dependent range dwell-time (MDRDT) switching. Especially, theoretical results are applied to switching delay Hopfield neural network. Simulations are served to illustrate the developed theory.
On a sublinear nonlocal fractional problem
Molica Bisci G., Servadei R.
Q2
This paper deals with existence results of nonnegative solutions for a one-parameter sublinear elliptic boundary-value problem driven by the classical fractional Laplacian operator. The existence of a weak solution for any parameter λ beyond the first resonance has been proved by using a slight variation of the classical Mountain Pass result due to Ambrosetti and Rabinowitz.
Eigenvalue problems for a k-Hessian-type equation
Yang Z., Bai Z.
Q2
In this work, we focus on the eigenvalue problem for a class of k-Hessian-type equations. Under some suitable assumptions, we first determine the intervals of the parameter for the existence of nontrivial radial solutions. To this aim, we apply the eigenvalue theory and Jensen inequality. Finally, the behavior of the solutions with respect to the parameter is analyzed via Guo’s fixed point theorem.
