Search

In this paper, we begin with a biological interpretation of what the subsequent mathematical and numerical analyses of the FitzHugh-Nagumo model entail. The interaction between action potential variable and recovery variable is then revisited through linear stability analysis around the equilibrium and local stability conditions are determined. Analytical results are compared with numerical simulations. The study aims to show an alternative approach regarding Taylor polynomials and constructed finite difference scheme which play a key role in the numerical approach for the problem. The robustness of the schemes is investigated in terms of convergency and stability of the techniques. Moreover, the numerical simulations are shown. Consequently, a comprehensive investigation of the related model is examined. [ABSTRACT FROM AUTHOR]

Idiopathic pulmonary fibrosis (IPF) is a chronic lung disease characterized by excessive scarring and fibrosis due to the abnormal accumulation of extracellular matrix components, primarily collagen. This study aims to design and solve an optimal control problem to regulate M2 macrophage activity in IPF, thereby preventing fibrosis formation by controlling the anti-M1 agent. The research models the diffusion of M2 macrophages in inflamed tissue using a novel dynamical system with partial differential equation (PDE) constraints. The control problem is formulated to minimize fibrosis by regulating an anti-M1 agent. The study employs a two-step process of discretization followed by optimization, utilizing the Galerkin spectral method to transform the M2 diffusion PDE into an algebraic system of ordinary differential equations (ODEs). The optimal control problem is then solved using Pontryagin/s minimum principle, canonical Hamiltonian equations, and extended Riccati differential equations. The numerical simulations indicate that without control, M2 macrophage levels increase and stabilize, contributing to fibrosis. In contrast, the optimal control strategy effectively reduces M2 macrophages, preventing fibrosis formation within 120 days. The results highlight the potential of the proposed optimal control approach in modulating tissue repair processes and mitigating the progression of IPF. This study underscores the significance of targeting M2 macrophages and employing mathematical methods to develop innovative therapies for lung fibrosis. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and consider a new second order dynamical system for solving variational inequalities, which is second order boundary value problem. Using the forward finite difference schemes, we suggest some multi-step iterative methods for solving the variational inequalities. Convergence analysis is investigated under certain mild conditions. Since the variational inequalities are equivalent to the complementarity problems, our results can be used to develop new techniques for them. It is an interesting problem to compare these methods with other technique for solving variational inequalities and related optimizations for further research activities. [ABSTRACT FROM AUTHOR]

Accurate and efficient real-time simulation of nonlinear dynamic systems remains an important challenge in fields such as robotics, control systems and industrial processes, requiring innovative solutions for predictive modeling. In this work, we introduce a novel recurrent neural networks (RNN) architecture designed to simulate complex nonlinear dynamical systems. Our approach aims to predict system behavior at any time step and over any prediction horizon, using only the system's initial state and external inputs. The proposed architecture combines RNN with multilayer perceptron and incorporates an attention mechanism to process both previous state estimates and external inputs. By training without teacher forcing, our model can iteratively estimate the system's state over long prediction horizons. Experimental results on three public benchmarks show that our method outperforms other state-of-the-art solutions. We highlight the potential of our proposal for modeling and simulating nonlinear systems in real-world applications. [ABSTRACT FROM AUTHOR]

Meshfree collocation in reproducing kernel Hilbert spaces is an established method to solve generalized interpolation problems such as PDEs. It can be formulated as an optimization problem with equality constraints. In this paper, we consider optimization problems with both inequality and equality constraints for general linear operators, and develop a general theory of discretizing such problems. The unique solution of these discretized problems is obtained using quadratic optimization, and we show that the solutions of the discretized problems strongly converge to the unique solution of the original problem. The general theory is applied to compute complete Lyapunov functions for autonomous ordinary differential equations. [ABSTRACT FROM AUTHOR]

We consider a generalization of the classical results obtained by Freidlin and Wentzell to consider the case of time-dependent dissipative drifts. We show the convergence of diffusions with multiplicative noise in the zero limit of a diffusivity parameter to the related dynamical systems. We also derive the solution to the associated transport equation as an application. [ABSTRACT FROM AUTHOR]

A Universe containing uniform magnetic fields, strings, or domain walls is shown to have an ellipsoidal expansion. This case has motivations from observational cosmology especially the anomaly concerning the low quadrupole amplitude compared to the best-fit ΛCDM prediction in Planck data. It is shown that a Universe with eccentricity at decoupling of order 10−2 can reduce the quadrupole amplitude without affecting higher multipoles of the angular power spectrum of the temperature anisotropy. We study the evolution of ellipsoidal Universes using dynamical system techniques for the first time. The determined critical points vary between saddle and past attractors depending on dark energy state equation parameter wΛ, with no future attractors. Important results are shown with numerical integrations of this dynamical system done using several initial conditions. For instance, a tendency for high expansion differences between planar and perpendicular axes is observed which contradicts previous assumption on the evolution behavior of ellipsoid Universes. Considering dark energy as a Chaplygin gas solves this contradiction by controlling cosmic shear evolution. [ABSTRACT FROM AUTHOR]

We use data on imprisonment in Spain to fit a system of three ordinary differential equations that describes the temporal evolution of three different groups in the country: offenders that are not in prison, offenders that are in prison, and the rest of people. These remaining people are considered as susceptible, who may become offenders by their relationships. That is, crime is regarded to behave as a social epidemic. We first investigate the dynamics of the model to find out when criminality becomes extinct or endemic in the long run, depending on the basic reproduction number. Then, we estimate the parameters of the model and conduct a sensitivity analysis. Finally, a random error is incorporated, and nonlinear regression is carried out to gather the unexplained variability of the data. Our results report a satisfactory model fitting to the crime data, closely delineating their dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we consider some free boundary problems that arise in the mathematical modeling of cancer and its treatment. In these models, a parameter η is introduced to represent either the killing rate of the cancer cells by the T cells or a drug called checkpoint inhibitor. We study the long-time behavior of the tumor as the parameter η varies. [ABSTRACT FROM AUTHOR]

We introduce the Minimal $ \ell^2 $ Norm Discrete Multiplier Method (MN-DMM) to extend the practical applicability of the Discrete Multiplier Method, where conservative finite difference schemes can now be constructed procedurally. This method alleviates the potential need for significant manual effort at deriving globally defined conservative schemes for large dynamical systems with multiple conserved quantities. MN-DMM utilizes the Moore-Penrose pseudoinverse of the discrete multiplier matrix to simultaneously approximate the unique consistent conservative scheme with minimal $ \ell^2 $ norm and its solution. We show the wide applicability of MN-DMM and its relative ease of implementation compared to the original Discrete Multiplier Method on problems such as the planar restricted three-body problem, Lorenz system, three-species Lotka-Volterra systems, spherical point vortex problem, and geodesic curves on Schwarzschild spacetime metric. [ABSTRACT FROM AUTHOR]

In this article, we reconsider the classical target cell limited dynamical within-host HIV model, solely taking into account the interaction between $ {\rm{CD}}4^{+} $ T cells and virus particles. First, we summarize some analytical results regarding the corresponding dynamical system. For that purpose, we proved some analytical results regarding the system of differential equations as our first main contribution. Specifically, we showed non-negativity and boundedness of solutions, global existence in time and global uniqueness in time and examined stability properties of two possible equilibria. In particular, we demonstrated that the virus-free equilibrium and the plateau-phase equilibrium are locally asymptotically stable using the Routh–Hurwitz criterion under appropriate conditions. As our second main contribution, we underline our theoretical findings through some numerical experiments with standard Runge–Kutta time stepping schemes. We conclude this work with a summary of our main results and a suggestion of an extension for more complex dynamical systems with regard to HIV-infection.

TThis paper presents a comprehensive study in a collisionless plasma composed of charged state of heavy positive ion and light positive as wel as negative ion. By deriving the Korteweg-de Vries (KdV) equation and by using its standard solution we analyze the characteristics of the solitary profile under varying parameters. We found that the solution gives both rarefactive and compressive soliton. The compressive structures are formed for the slow mode, while rarefactive solitary structures are formed for the fast mode. Furthermore, with the application of planar dynamical systems bifurcation theory, the phase portraits have been analyzed. This dynamical system analysis allowed us to extract important information on the stability of these structures as represented by the KdV equation.

The HIV/AIDS model was dynamically analyzed in this study. The model has seven compartments: the uneducated, the educated, the HIV-positive who take antiretroviral therapy (ART), the HIV-positive who do not take ART, people receiving ART treatment, people with AIDS who do not receive any treatment (full-blown AIDS), and the recovered. This model takes into account the analysis of the multiple interactions between the uneducated and the educated subpopulations, the HIV-positive who take and who do not take ART. The free-disease and endemic equilibrium points, as well as the basic reproduction number [Formula: see text] as a limit condition for infection-free and endemic occurrence, were produced by a mathematical analysis. The center-manifold hypothesis was used to prove that a backward bifurcation exists. The free-disease and endemic equilibrium points coexist when [Formula: see text] This means that HIV/AIDS is still spreading. A basic reproduction number below one is insufficient to constitute a free-disease condition. In order to determine essential parameters that significantly contribute to HIV/AIDS transmission, we computed sensitivity index values using a sensitivity analysis. The HIV/AIDS model and bifurcation parameter both identified the rate of HIV transmission from uneducated individuals to HIV-positive individuals who do not receive ART as the most crucial parameter. A numerical simulation supports the dynamical analysis.

The Riemann wave equation presents appealing nonlinear equations applicable in sea-water and tsunami wave propagation, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, and homogeneous stationary media. This study focuses on deriving soliton solutions in optics and exploring their physical properties. A wave transformation is used to convert a partial differential equation into an ordinary differential equation, from which soliton solutions are obtained using the generalized Riccati equation mapping approach. The solutions encompass various types of solitons, including bright, dark, periodic, and kink solitons. A comparison of solutions from this analytical method enhances the understanding of the nonlinear model’s behavior, with implications in plasma physics, fluid dynamics, optics, and communication technology. Additionally, 2D and 3D graphs illustrate the physical phenomena of the solutions using appropriate constant parameters. The qualitative analysis of the undisturbed planar system involves examining phase portraits in bifurcation theory, followed by introducing an outward force to induce disruption and reveal chaotic phenomena. Chaotic trajectories in the perturbed system are detected through various plots, including 3D, 2D, power spectrum, and chaotic attractor, alongside Lyapunov exponents. Stability analysis under different initial conditions is conducted, and sensitivity assessments are performed using the Runge–Kutta method. The findings are innovative and have not been previously explored for this system, highlighting the reliability, simplicity, and effectiveness of these techniques in analyzing nonlinear models in mathematical physics and engineering.

Abstract This study aims to demonstrate the behaviors of a two degree-of-freedom (DOF) dynamical system consisting of attached mass to a nonlinear damped harmonic spring pendulum with a piezoelectric device. Such a system is influenced by a parametric excitation force on the direction of the spring’s elongation and an operating moment at the supported point. A negative-velocity-feedback (NVF) controller is inserted into the main system to reduce the undesired vibrations that affect the system’s efficiency, especially at the resonance state. The equations of motion (EOM) are derived by using Lagrangian equations. Through the use of the multiple-scales-strategy (MSS), approximate solutions (AS) are investigated up to the third order. The accuracy of the AS is verified by comparing them to the obtained numerical solutions (NS) through the fourth-order Runge-Kutta Method (RK-4). The study delves into resonance cases and solvability conditions to provide the modulation equations (ME). Graphical representations showing the time histories of the obtained solutions and frequency responses are presented utilizing Wolfram Mathematica 13.2 in addition to MATLAB software. Additionally, discusses the bifurcation diagrams, Poincaré maps, and Lyapunov exponent spectrums to show the various behavior patterns of the system. To convert vibrating motion into electrical power, a piezoelectric sensor is connected to the dynamical model, which is just one of the energy harvesting (EH) technologies with extensive applications in the commercial, industrial, aerospace, automotive, and medical industries. Moreover, the time histories of the obtained solutions with and without control are analyzed graphically. Finally, resonance curves are used to discuss stability analysis and steady-state solutions.

In this paper we introduce some weak dynamical properties by using subbases for the phase space. Among them, the notion of light chaos is the most significant. Several examples, which clarify the relationships between this kind of chaos and some classical notions, are given. Particular attention is also devoted to the connections between the dynamical properties of a system and the dynamical properties of the associated functional envelope. We show, among other things, that a continuous map f : X → X , where X is a metric space, is chaotic (in the sense of Devaney) if and only if the associated functional dynamical system is lightly chaotic.

Abstract This study dynamically investigates the mathematical Ivancevic option pricing governing system in terms of conformable fractional derivative, which illustrates a confined Brownian motion identified with a non-linear Schrödinger type equation. This model describes the controlled Brownian motion that comes with a non-linear Schrödinger type equation. The solution to comprehend the market price fluctuations for the suggested model is developed through the application of a mathematical strategy. The modified Kudryashov analytical method is applied to find the fractional analytical exact soliton solution. The restrictions on the parameters required for these solutions to exist were also the result of this approach. The dynamical insights are examined and significant aspects of the phenomenon under study are discussed through the use of the bifurcation analysis. In the related dynamical system, the phase portraits of market price fluctuations are displayed at equilibrium points and for different parameter values. Additionally, the chaos analysis was carried out to show the quasi-periodic and periodic chaotic patterns. In order to track changes in market price, the sensitivity analysis of the studied model is also looked at and presented at different initial conditions. It was discovered that the model experienced price fluctuations as a result of minute changes in initial conditions.

In a recent paper [Belykh et al., 2024], we proved that the bifurcation structure of a quadratic noninvertible map persists when the parameter increases from zero and the map turns into an invertible multidimensional Henon map. In this paper, we consider a similar problem for a generalized map which combines the Henon-type maps, the Poincaré return map for Shilnikov bifurcation of saddle-focus homoclinic orbit, the Lurie discrete time system, etc. We have obtained the expected result about the persistence of periodic orbits and their bifurcations when passing from a One-Dimensional (1D) endomorphism to the generalized map as a small parameter becomes nonzero. We have revealed the precise mechanism of change of homoclinic orbits and splitting of unstable manifolds as a result of the transition of 1D endomorphism to multidimensional map. Thereby we have derived the reconstruction rules of nonwandering set of orbits and bifurcations of the generalized map from those of 1D endomorphism. [ABSTRACT FROM AUTHOR]

When a finite group freely acts on a topological space, we can define its index and coindex. They roughly measure the size of the given action. We explore the interaction between this index theory and topological dynamics. Given a fixed-point free dynamical system, the set of p -periodic points admits a natural free action of ℤ / p ℤ for each prime number p. We are interested in the growth of its index and coindex as p → ∞. Our main result shows that there exists a fixed-point free dynamical system having the divergent coindex sequence. This solves a problem posed by M. Tsukamoto, M. Tsutaya and M. Yoshinaga, G -index, topological dynamics and marker property, preprint (2020), arXiv:2012.15372. [ABSTRACT FROM AUTHOR]

In this article, we first transform the split feasibility problem into the fixed point problem and focus on a novel dynamical system to solve the fixed point problem. The existence and uniqueness of the solution of the dynamical system are revealed. Under mild conditions, we derive that the solution of the proposed method globally converges to the solution of the split feasibility problem. Furthermore, the convergence time of the dynamical system can be upper bounded which is independent of any initial state. Finally, numerical experiments on signal processing and image restoration are presented to demonstrate the efficiency and competitiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This article explores the examination of the widely employed zig-zag optical lattice model for cold bosonic atoms, which is commonly utilized to depict nonlinear wave in fluid mechanics and plasma physics. The focus is on obtaining soliton solutions in optics and investigating their physical properties. A wave transformation is initially applied to convert a partial differential equation (PDE) into an ordinary differential equation (ODE). Soliton solutions are subsequently obtained through the application of two distinct methods, namely the generalized logistic equation method and the Sardar sub-equation method. These solutions include bright, dark, combined dark-bright, chirped type solitons, bell-shaped, periodic, W-shape, and kink solitons. In this paper, the solutions derived from two analytical approaches were compared to enhance the understanding of the behavior of the discussed nonlinear model. The obtained solutions have significant implications across various fields such as plasma physics, fluid dynamics, optics, and communication technology. Furthermore, 3D and 2D graphs are generated to depict the physical phenomena of the derived solutions by assigning appropriate constant parameters. The qualitative evaluation of the undisturbed planar system involves the analysis of phase portraits within bifurcation theory. Subsequently, the introduction of an outward force is carried out to induce disruption, and chaotic phenomena are unveiled. The detection of chaotic trajectory in the perturbed system is achieved through 3D plots, 2D plots, time scale plots, and Lyapunov exponents. Furthermore, stability analysis of the examined model is addressed under distinct initial conditions. Finally, the sensitivity assessment of the model under consideration is carried out using the Runge–Kutta method. The results of this study are innovative and have not been previously investigated for the system under consideration. The results obtained underscore the reliability, simplicity, and effectiveness of these techniques in analyzing a variety of nonlinear models found in mathematical physics and engineering disciplines. [ABSTRACT FROM AUTHOR]