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In this paper, we investigate numerically the dynamic snap-through motion of a shallow arch micro-beam that is electrostatically excited through three separated electrodes. It is excited by a dc voltage and a time-varying signal with a frequency tuned near its first in-plane frequency in both initial curvature and initial-counter curvature configurations. A reduced-order model based on the Galerkin method is utilized to discretize the governing equation of motion and to analyze the static and dynamic responses of the micro-beam. The phase portraits obtained from simulations reveal the presence of a stable response that oscillates back and forth around single and double equilibria until it eventually settles in a steady state zone near the first equilibrium point or encloses both equilibria. This can be tracked using a Long-Time Integration (LTI) and recording the time–history signal for a long period. Remarkably, this behavior can be controlled by adjusting the initial operating conditions mainly the signal frequency. We also observe the emergence of a chaotic attractor in the vicinity of the resonance frequency when the shallow arch beam is excited while it is in its initial-counter curvature. The simulated FFTs indicate the presence of the chaotic attractor in the vicinity of primary resonance due to dynamic snap-through motion or large excitation between two equilibria. We observed the existence of periodic orbits with single loop (P-1) and an full chaotic behavior around the two equilibria. The existence of this phenomenon can be attributed to the fact that the excitation voltage is positioned close to the snap-through threshold. [ABSTRACT FROM AUTHOR]

This paper provides a novel approach to studying the chaos control and synchronization of fractional-order hybrid Darcy–Brinkman systems (FOHDBs). We use the truncated Galerkin method to transform the system of partial differential equations (PDEs) into ordinary differential equations (ODEs) based on Fourier modes in a nonlinear low-dimensional Brinkman model. In the nonlinear complex system, hybrid nanoparticles and fluid flow physical characteristics are considered to be external disturbances. As a result, the implications of model uncertainty and external disruptions are fully absorbed into the problem, further complicating it and resulting in highly complex and unpredictable behaviors. In addition, To assure the occurrence of the sliding motion in a finite amount of time, an appropriate robust fractional sliding mode technique is proposed. Sliding motion is shown to occur in finite time using the fractional Lyapunov stability. Following that, we discussed control techniques for achieving infinitesimally close to an equilibrium of chaotic states and tracking errors. It is important to note that FOHDBs and external stochastic disturbances can be controlled using the proposed fractional nonsingular terminal sliding mode control technique. Finally, the effectiveness and applicability of the suggested finite-time control technique are graphically illustrated by the analytical and numerical simulations. [ABSTRACT FROM AUTHOR]

Highlights • We present a numerical algorithm for solving fractional Sturm–Liouville problems with variable coefficients subject to mixed boundary conditions. • Tedious manipulation of the series appearing in the implementation of the method have been simplified to obtain a system of algebraic equations for the coefficients. • Several examples are discussed to demonstrate the efficiency and accuracy of the present scheme. Abstract In this paper, we present a numerical technique for solving fractional Sturm–Liouville problems with variable coefficients subject to mixed boundary conditions. The proposed algorithm is a spectral Galerkin method based on fractional-order Legendre functions. Tedious manipulation of the series appearing in the implementation of the method have been carried out to obtain a system of algebraic equations for the coefficients. Our findings demonstrate the possibility of having no eigenvalues, finite number of eigenvalues or infinite number of eigenvalues depending on the fractional order. The convergence and effectiveness of the present algorithm are demonstrated through several numerical examples. [ABSTRACT FROM AUTHOR]

This paper presents an approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection circular plate of thermo-mechanical coupling by utilizing the criterion of the maximum Lyapunov exponent. The governing partial differential equation of the simply supported large deflection circular plate of thermo-mechanical coupling is first derived and simplified to a set of three ordinary differential equations by the Galerkin method. Several different features including time history, Power spectra, phase plot, Poincare map and bifurcation diagram are then numerically computed. These features are used to characterize the dynamic behavior of the plate subjected various geometric and excitation conditions. Numerical examples are presented to verify the validity of the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. The modeling results of numerical simulation indicate that the chaotic motion may occurs in the lateral loads &fmacr;=2.5, η1=1.1, β=0.5, δ=0.01 and #x03B5;∞=0.0007. As the thermo-elastic damping is great than a critical value, the dynamic motion of the thermal-couple plate is periodic. As the thickness parameter β of the concave circular plate is great than a critical value, the motion of the plate is periodic. The modeling result thus obtained by using the method proposed in this paper can be employed to predict the instability induced by the dynamics of the thermo-mechanical coupling circular plate in large deflection. [Copyright &y& Elsevier]

This paper introduces a new method based on Deep Galerkin Methods (DGMs) for solving high-dimensional stochastic Mean Field Games (MFGs). We achieve this by using two neural networks to approximate the unknown solutions of the MFG system and forward–backward conditions. Our method is efficient, even with a small number of iterations, and is capable of handling up to 300 dimensions with a single layer, which makes it faster than other approaches. In contrast, methods based on Generative Adversarial Networks (GANs) cannot solve MFGs with non-separable Hamiltonians. We demonstrate the effectiveness of our approach by applying it to a traffic flow problem, which was previously solved using the Newton iteration method only in the deterministic case. We compare the results of our method to analytical solutions and previous approaches, showing its efficiency. We also prove the convergence of our neural network approximation with a single hidden layer using the universal approximation theorem. • New approach: Deep Galerkin for mean field games with non-separable Hamiltonians. • We prove the convergence of our NN approximation with a single hidden layer. • Our method handles up to 300+ dimensions with a single layer for faster computing. • Efficiency demonstrated against prior approaches and traffic flow problem. • Our solution also performs well with multiple layers. [ABSTRACT FROM AUTHOR]

In this paper to approximate the Heydari–Hosseininia non-singular fractional derivative, we construct the L1-2 discretization by providing the error estimate. The error estimation of the L1 formula is also presented. The scheme uses the local discontinuous Galerkin method combing with the L1/L1-2 formula as spatial and time discretizations, respectively. To investigate the efficiency and accuracy of our scheme, variable-order fractional ordinary differential and 2-dimensional Sobolev equations are proposed. The scheme is second/third-order accurate in time for the L1/L1-2 formula, respectively. Utilizing k , the approximation degree, the rates of convergence in space are reported k + 1 when time step chosen τ = h k + 1 2 and τ = h k + 1 3 . Our argument is that new approximation L1-2 has less computational cost than the L1 discretization and numerical results would be given to confirm this reduction. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove the global existence of solutions to the initial boundary value problem of a viscous p-Laplace equation with nonlinear sources. The asymptotic behavior of solutions as the viscous coefficient k tends to zero is also investigated. In particular, we discuss the H 1-Galerkin finite element method for our problem and establish the error estimates for two semi-discrete approximate schemes. [Copyright &y& Elsevier]

• To confirm that the Galerkin technique is accurate, and can produce excellent predictions for PDE models with delays. • To construct full plot diagrams of the stability regions of Hopf points, and to determine the stable and unstable zones. • To show the influences of the diffusion and delay parameters on the model. • To provide examples of 2D phase-plane maps for the analytical and numerical simulation results. • To provide the bifurcation maps that show the periodic and chaotic behaviours of the nonlinear dynamical model. This paper considers both analytical and numerical solutions for a diffusive Schnakenberg model with gene-expression time delays, presenting an analysis of the effects of delays and diffusion on stability regions and bifurcation maps. A one-domain system was considered. Systems of delay ODEs were obtained using the Galerkin method. Theoretical conditions for the existence of steady-state and Hopf bifurcation curves were determined. In addition, Hopf bifurcation points and bifurcation stability regions were plotted in detail. The gene-expression time delays and diffusion rates influenced the stability regions for both reactant concentration controls in the system. The results showed that, with increases in time delays, the rates of the Hopf bifurcation points for both chemical concentrations controls decreased, while both parameters of the diffusion coefficient grew as the chemical control values increased. Numerical simulations for bifurcation diagrams, limit cycles, and periodic routes to chaotic behaviour planes confirm the solutions achieved from theory. [ABSTRACT FROM AUTHOR]

In this paper, we formulate the intricate process of Human Immunodeficiency Virus (HIV) infection in a mathematical model. In the formulation of the model, the density of CD 4 + T-cells is categorized into healthy and infected classes while the density of free HIV viruses in the human host's blood is considered a separate class. It is reported that when HIV enters a human's body, it infects a large amount of CD 4 + T-cells and unbalance the supply of new cells from the thymus. Therefore, we incorporate variable source term relying on the viral load for the supply of new cells instead of using a fixed value for the supply of these cells. We implement a continuous Galerkin Petrov time discretization scheme, particularly cGP(2)-scheme to visualize the dynamical behavior of the mentioned model and a detailed description of the effects of different physical parameters of interest are depicts graphically and discuss that how the level of healthy, infected CD 4 + T-cells and free HIV particles varies concerning the emerging parameters in the model. In the proposed cGP(2)-method, two unknowns terms can be found by solving a 2 × 2 block system. This method is accurate of order 3 in the whole time interval and shows even superconvergence of order 4 in the discrete-time points. Furthermore, determine the solution of the model utilizing the fourth-order Runge-Kutta (RK4)-method and find out the absolute errors between the solutions obtained from both approaches. Finally, the validity and reliability of the proposed scheme are verified by comparing the numerical and graphical results with the RK-4 method. We examine the accuracy of the cGP(2)-sheme and observe that it produces more accurate results at relatively larger step sizes as in comparison to RK-4 scheme. Correspondence between the findings of cGP(2)-method and RK4-method reveal that the new technique is a promising tool for obtaining approximate solutions to the nonlinear systems of real-world problems. [ABSTRACT FROM AUTHOR]

Abstract: The attitude stability of flexible spacecraft is discussed in this paper. The Euler’s equations of spacecraft and the equations of forced vibration of two flexible solar arrays are derived. The governing equation is discretized via the Galerkin method, and the stability of relative equilibrium of spacecraft in the orbit frame is determined based on the Kelvin–Tait–Chetayev theorem. The sufficient condition of stability in the analytical form for the n-term truncated modal is established based on the governing equation. [Copyright &y& Elsevier]

Abstract: This paper applies the Sinc-Galerkin method and He’s homotopy perturbation method to search for approximate solutions of a certain class of singular two-point boundary value problems. The sinc method converges exponentially to the exact solution. A new iterative scheme based on He’s homotopy perturbation method is proposed for the discussed problem. A numerical example is given to demonstrate the computational efficiency of the two methods. [Copyright &y& Elsevier]

Abstract: This paper presents a novel analytical approach utilizing fractal dimension criteria and the maximum Lyapunov exponent to characterize the conditions which can potentially lead to the chaotic motion of a simply supported thermo-mechanically coupled orthotropic rectangular plate undergoing large deflections. The study commences by deriving the governing partial differential equations of the rectangular plate, and then applies the Galerkin method to simplify these equations to a set of three ordinary differential equations. The associated power spectra, phase plots, Poincaré map, maximum Lyapunov exponents, and fractal and bifurcation diagrams are computed numerically. These features are used to characterize the dynamic behavior of the orthotropic rectangular plate under various excitation conditions. The maximum Lyapunov exponents and the correlation dimensions method indicate that chaotic motion of the orthotropic plate occurs at η1=1.0, , and for an external force of . The application of an external in-plane force of magnitude causes the orthotropic plate to perform bifurcation motion. Furthermore, when , aperiodic motion of the plate is observed. Hence, the dynamic motion of a thermo-mechanically coupled orthotropic rectangular plate undergoing large deflections can be controlled and manipulated to achieve periodic motion through an appropriate specification of the system parameters and loads. [Copyright &y& Elsevier]

Non-linear vibrations of beam-like model of two-dimensional axially moving web with time-dependent tension have been investigated in this paper. The beam model material as the Kelvin–Voigt element is considered. The Galerkin method and the fourth-order Runge–Kutta method were used to solve the governing non-linear partial–differential equation. The effects of the transport speed, the tension perturbation amplitude and the internal damping on the dynamic behaviour of the system were numerically investigated. The Poincare maps have been constructed to classify the vibrations. For small values of the transport speed and the amplitude of periodic perturbation the system is asymptotically stable with its response tending to zero. With the increase of parameters one can observe local pitchwork type bifurcation and the coexistence of attractors. For small values of internal damping chaotic motion occurs. Regular and chaotic motions occur when internal damping increases. [Copyright &y& Elsevier]

This paper investigates chaotic behaviors of an axially traveling viscoelastic string with geometric nonlinearity. The stress and the strain of the viscoelastic string obey the Boltzmann superposition principle. The Galerkin method is applied to truncate a nonlinear partial-differential–integral equation governing transverse motion into a set of ordinary differential–integral equations. For the string modeled as a standard linear solid, new auxiliary variables are introduced to transform those equations into ordinary differential equations. By use of the Poincare maps, the chaotic behaviors are presented based on the numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented for varying one of the following parameter: the axial traveling speed, the amplitude of tension fluctuation, the viscoelastic exponent and coefficient of the string, while other parameters are fixed. [Copyright &y& Elsevier]

• The dynamic equation of this model in established. • Dynamics of the unperturbed system under different system parameters are obtained analytically. • Chaotic feature on the location of the lumped mass is investigated. • Controllable frequencies decreasing monotonously with the location of the lumped mass are presented. • It is proved that the system may undergo chaotic motions through finite subharmonic bifurcations. With analytical and numerical methods, dynamic response of a composite laminated buckled beam with a lumped mass and an axial excitation is investigated in this paper. The dynamic equation of the first mode for this model, which is a perturbation of a Hamilton system with both quadratic and cubic nonlinearities as well as external and parametric excitations, is obtained with the Galerkin method. The distribution of the equilibriums, homoclinic orbits and periodic orbits for the unperturbed system under different system parameters are obtained analytically. Chaotic vibrations arising from homoclinic intersections are studied with Melnikov method. Chaotic feature on the location of the lumped mass is investigated in detail. It is presented that there exist controllable frequencies decreasing monotonously with the location of the lumped mass for this system. Subharmonic bifurcation is also studied with the subharmonic Melnikov method. It is numerically proved that the system may undergo chaotic motions through finite subharmonic bifurcations. Numerical simulations including the phase portraits, time histories, Poincar e ´ sections, Lyapunov spectrums and bifurcation diagram are given, which confirm the analytical results. [ABSTRACT FROM AUTHOR]