Nonlinear Coupled System Research Articles

Free-will arbitrary time adaptive chaos control for permanent magnet synchronous motor

Permanent magnet synchronous motor (PMSM) is a multivariable and highly coupled nonlinear system, which is prone to chaotic behavior during actual operation. In order to suppress the chaotic behavior of the PMSM, this work designs a free-will arbitrary time adaptive controller based on the inverse step derivation, adaptive control theorem and Lyapunov stability theory. Firstly, the corresponding coordination transformations are designed according to the PMSM. Secondly, the bounded problem of the controlled system is analyzed by using the Lyapunov stability theory, and the free-will arbitrary time adaptive controller is designed by backstepping derivation. Finally, numerical simulation verifies that the designed controller quickly stabilizes the system state variables to the output signals and proves that the controller is free-will arbitrary time stable for all closed-loop signals.

Existence of solutions to a strongly nonlinear parabolic–elliptic coupled system of infinite order

In this paper, we are concerned with the problem of existence of a capacity solution to the strongly nonlinear degenerate problem, namely, [Formula: see text] in [Formula: see text], where the operator [Formula: see text] is of the form [Formula: see text] By using a Schauder’s fixed point theorem, we establish the existence of weak solutions to a certain truncated problem of finite order. Then, we demonstrate that the sequence of solutions to this truncated problem converges (up to a subsequence) to a capacity solution to our problem of infinite order.

Fuzzy Intermittent Control for Nonlinear PDE-ODE Coupled Systems

Fuzzy Intermittent Control for Nonlinear PDE-ODE Coupled Systems

Effects of uneven subgrade settlement on the dynamics of train-CRTS III slab track nonlinear coupling system

Effects of uneven subgrade settlement on the dynamics of train-CRTS III slab track nonlinear coupling system

Parametrically excited shape distortion of a submillimeter bubble

The existence of finite amplitude shape distortion caused by parametrically excited surface instabilities for a gas bubble in water driven by a temporally periodic, spatially uniform pressure field in an axisymmetric geometry is investigated. Employing a nonlinear coupled system of equations which includes shape mode interactions to third order, the resultant spherical oscillations, translation, and shape distortion of the bubble are modelled, placing no restriction on the size of the spherical oscillations. The model accounts for viscous and thermal damping with compressibility effects. The existence of synchronous and higher order parametrically induced sustained, finite amplitude, periodic shape deformation is demonstrated. The excitement of an odd shape mode via the synchronous mechanism is shown to give rise to linear bubble self-propulsion. For larger driving amplitudes, it is shown that more than one shape mode can be parametrically excited at the same driving frequency but by different resonance mechanisms, leading to more involved shape deformation and the increased possibility of bubble self-propulsion.

Brain Connectivity Dynamics and Mittag–Leffler Synchronization in Asymmetric Complex Networks for a Class of Coupled Nonlinear Fractional-Order Memristive Neural Network System with Coupling Boundary Conditions

This paper investigates the long-time behavior of fractional-order complex memristive neural networks in order to analyze the synchronization of both anatomical and functional brain networks, for predicting therapy response, and ensuring safe diagnostic and treatments of neurological disorder (such as epilepsy, Alzheimer’s disease, or Parkinson’s disease). A new mathematical brain connectivity model, taking into account the memory characteristics of neurons and their past history, the heterogeneity of brain tissue, and the local anisotropy of cell diffusion, is proposed. This developed model, which depends on topology, interactions, and local dynamics, is a set of coupled nonlinear Caputo fractional reaction–diffusion equations, in the shape of a fractional-order ODE coupled with a set of time fractional-order PDEs, interacting via an asymmetric complex network. In order to introduce into the model the connection structure between neurons (or brain regions), the graph theory, in which the discrete Laplacian matrix of the communication graph plays a fundamental role, is considered. The existence of an absorbing set in state spaces for system is discussed, and then the dissipative dynamics result, with absorbing sets, is proved. Finally, some Mittag–Leffler synchronization results are established for this complex memristive neural network under certain threshold values of coupling forces, memristive weight coefficients, and diffusion coefficients.

A Qualitative Analysis of a Non-Linear Coupled System under Two Types of Fractional Derivatives along with Mixed Boundary Conditions

This work addresses the qualitative analysis of a novel non-linear coupled system of fractional differential problems (FDPs) using Caputo and Liouville–Riemann fractional derivatives. Fractional calculus has demonstrated significant applicability across various fields, including financial systems, optimal control, epidemiological models, chaotic systems, and engineering. The proposed model builds on existing research by formulating a non-linear coupled fractional boundary value problem with mixed boundary conditions. The primary advantages of our method include its ability to capture the dynamics of complex systems more accurately and its flexibility in handling different types of fractional derivatives. The model’s solution was derived using advanced mathematical techniques, and the results confirmed the existence and uniqueness of the solutions. This approach not only generalizes classical differential equation methods but also offers a robust framework for modeling real-world phenomena governed by fractional dynamics. The study concludes with the validation of the theoretical findings through illustrative examples, highlighting the method’s efficacy and potential for further applications.

Maximum-Norm Error Estimates of Fourth-Order Compact and ADI Compact Finite Difference Methods for Nonlinear Coupled Bacterial Systems

Maximum-Norm Error Estimates of Fourth-Order Compact and ADI Compact Finite Difference Methods for Nonlinear Coupled Bacterial Systems

FRACTAL-FRACTIONAL SIRS MODEL FOR THE DISEASE DYNAMICS IN BOTH PREY AND PREDATOR WITH SINGULAR AND NONSINGULAR KERNELS

In this study, we consider a mathematical model for the disease dynamics in both prey and predator by considering the Susceptible–Infected–Recovered–Susceptible (SIRS) model with the prey–predator Lotka–Volterra differential equations. Carrying capacity, predation, migration, and immunity loss are also taken into account for both species. Using the law of mass action, the physical model is transformed into a nonlinear coupled system of ODEs. The classical/integer order ODE system is then generalized through the fractal-fractional differential operators of power law and Mittag-Leffler kernels. For the model under consideration, we additionally check for positivity, boundedness, the basic reproduction number, and equilibrium points. Graphical results are obtained by use of a numerical approach, and the existence and uniqueness of the solution are also established theoretically. The graphical solutions has been displayed via 2D and 3D phase plots. It has been shown that when the fractional order reduces, the amplitude of the chaotic attractor dynamics shrinks, along with the range of limit cycles and periodic trajectories while a drop in the fractal dimension parameter causes an increase in the time period of the chaotic attractor dynamics. This study not only improves the understanding of epidemic breakouts in predator–prey systems, but also highlights the efficiency of fractal-fractional calculus in ecological modeling.

A Parallel Coloring Newton-Krylov Method for Multiphysics Coupling System in Nuclear Reactors

The Newton-Krylov method with the explicit Jacobian matrix is an efficient numerical method for solving the nuclear reactor nonlinear multiphysics coupling system. Compared with the Jacobian-free Newton-Krylov (JFNK) method, it has a better preconditioner matrix (the Jacobian matrix itself) and can achieve a more stable and faster convergence. How to compute the Jacobian matrix efficiently is a key issue for this method. The graph coloring algorithm is an essential technique and has been used to reduce the Jacobian computational burden by exploiting its sparsity. The fewer the coloring numbers in the Jacobian, the less the Jacobian computational cost will be. Besides, when computing the Jacobian in a distributed memory parallel environment, the parallel graph coloring algorithms are required because the Jacobian is distributed among processors. Currently, a popular parallel graph coloring algorithm has been used to color the Jacobian. However, this parallel graph coloring algorithm shows poor scalability in parallel. The coloring numbers will increase with the processors, resulting in poor Jacobian computational efficiency. In this paper, a more efficient parallel graph coloring method is proposed that aims to reduce the coloring numbers and improve Jacobian computation efficiency in parallel. The main feature of the new method is that the coloring numbers decrease with the increasing number of processors. A neutronics/thermal-hydraulic coupling problem arising from the simplified high-temperature gas coolant model is utilized to assess the performance of the newly proposed method. The results show that (1) the parallel coloring number is reduced significantly, (2) the Jacobian computed by the new method is completely correct and excellent parallel scalability is achieved, and (3) the parallel coloring Newton-Krylov method with explicit Jacobian is more efficient and more stable than the parallel JFNK due to a better preconditioner.

Existence of solutions to a strongly nonlinear elliptic coupled system of finite order

Existence of solutions to a strongly nonlinear elliptic coupled system of finite order

The switching dynamics of self-defocusing nonlinear coupled system with PT-symmetric Scarf II barrier potential

The switching dynamics of self-defocusing nonlinear coupled system with PT-symmetric Scarf II barrier potential

Frequency lock-in control and mitigation of nonlinear vortex-induced vibrations of an airfoil structure using a conserved-mass linear vibration absorber

We investigate the effectiveness of a vibration absorber on the vortex-induced vibration response of turbine blades during the frequency lock-in phase. A reduced-order model of a turbine blade and van der Pol oscillator is used to represent the fluid–blade interaction caused by the vortex shedding. A spring-mass-damper system is considered to model the vibration absorber. The advantage of the vibration absorber is demonstrated by simulating the nonlinear coupled four-degree-of-freedom aeroelastic system for the different sets of system parameters. We observe the dominance of a nonlinear vibration absorber over the linear vibration absorber only for the higher coupling parameter values. The analytical solution of the nonlinear coupled system is obtained through the method of multiple scales for the case of 1:1 internal resonance to identify the critical design parameters of the vibration absorber. We observe the high sensitivity of the system's frequency response to the distance of the vibration absorber from the elastic axis, along with the absorber's damping, stiffness, and mass. Finally, we perform a parametric analysis on the lock-in of the stability region to better understand the effect of the vibration absorber on the instability region.

Increase of power leads to a bilateral solution to a strongly nonlinear elliptic coupled system

Abstract In this paper, we analyze the following nonlinear elliptic problem A ( u ) = ρ ( u ) | ∇ φ | 2 in Ω , div ( ρ ( u ) ∇ φ ) = 0 in Ω , u = 0 on ∂ Ω , φ = φ 0 on ∂ Ω . $\begin{cases}A\left(u\right)=\rho \left(u\right)\vert \nabla \varphi {\vert }^{2}\,\text{in}\,{\Omega},\quad \hfill \\ \text{div}\left(\rho \left(u\right)\nabla \varphi \right)=0\,\text{in}\,{\Omega},\quad \hfill \\ u=0\,\text{on}\,\partial {\Omega},\quad \hfill \\ \varphi ={\varphi }_{0}\,\text{on}\,\partial {\Omega}.\quad \hfill \end{cases}$ where A(u) = −div a(x, u, ∇u) is a Leray-Lions operator of order p. The second member of the first equation is only in L 1(Ω). We prove the existence of a bilateral solution by an approximation procedure, the keypoint being a penalization technique.

Numerical simulation of the ferrohydrodynamics flow using an unconditionally stable second‐order scheme

AbstractIn this paper, we design a decoupled, linear, unconditionally stable and fully discrete numerical scheme for a ferrohydrodynamics system with second‐order temporal accuracy. This scheme is based on a second‐order backward difference formula for time derivative terms and linearization extrapolation for nonlinear terms, which produces a series of decoupled linear equations and solves effectively this nonlinear and multiphysical coupled system. Meanwhile, we show that the scheme is unconditionally stable. Finally, some numerical experiments are provided to verify the theoretical finding and illustrate the accuracy and efficiency of the proposed scheme.

Nonlinear coupled system in thin domains with corrugated boundaries for metabolic processes

Nonlinear coupled system in thin domains with corrugated boundaries for metabolic processes

The Existence Results of Solutions to the Nonlinear Coupled System of Hilfer Fractional Differential Equations and Inclusions

This paper is dedicated to studying the existence results of solutions to the nonlinear coupled system of Hilfer fractional differential equations and inclusions, with multi-strip and multi-point mixed boundary conditions. Through tools such as the Leray-Schauder alternative and the nonlinear alternative of Leray-Schauder type, continuous and measurable selection theorems, along with Leray-Schauder degree theory, the main results can be obtained. The Hilfer fractional differential system has practical implications for specific physical phenomena. Examples are provided to clarify the application of our main results.

Prescribed‐time global stabilization for MIMO nonlinear systems with a linear time‐varying control mechanism

AbstractIn this article, the prescribed‐time global stabilization problem is taken into consideration when it comes to multi‐input‐multi‐output (MIMO) systems with unknown mismatched coupling nonlinear items. It is assumed that the unknown nonlinearities satisfy the linear growth conditions. First, a class of MIMO coupling nonlinear systems with controllable canonical form are considered. Then, by wielding a time‐varying Lyapunov‐like function and properties of parametric Lyapunov equations, a high‐gain linear time‐varying state feedback control law is developed to attenuate the impact of unknown nonlinear terms. Meanwhile, all state signals and control inputs are globally bounded and the target system becomes stable within the prescribed‐time. Finally, the simulation results on an example system with 3 inputs and 11 state variables verify that the developed control mechanism accomplishes the goal of global stabilization in the pre‐defined time.

Numerical approximation of fractional SEIR epidemic model of measles and smoking model by using Fibonacci wavelets operational matrix approach

In the present article, we have considered two essential models (The epidemic model of measles and the smoking model). Across the globe, the primary cause of health problems is smoking. Measles can be controlled in infectious populations using the mathematical model representing the direct transmission of infectious diseases. The Caputo fractional derivative operator of order α∈[0,1] is used to determine the solution of the system of fractional differential equations in both models. We employed the Fibonacci wavelets operational matrix of integration and developed the Fibonacci wavelet collocation method (FWCM) to solve a nonlinear coupled system of fractional differential equations (NCSFDEs). The proposed approach transforms the given model into a system of algebraic equations. The Newton-Raphson method is considered to extract the unknown coefficient values. Numerical outcomes are obtained to illustrate the simplicity and effectiveness of the proposed method. Graphs and tables show how consistently and effectively the developed strategy works. Mathematical software called Mathematica has been used to perform all calculations. Theorems explain the convergence of this approach.

Upper semicontinuity of the attractor for a nonlinear hyperbolic-parabolic coupled system with fractional Laplacian

In this paper we establish the existence and uniqueness of global solutions (in time), as well as the existence, regularity and stability (upper semicontinuity) of the attractor for the semigroup generated by the solutions of a two-dimensional nonlinear hyperbolic-parabolic coupled system with fractional Laplacian. In addition, we also obtain the existence of an exponential attractor and show that this attractor has a finite fractal dimension in a space containing the phase space of the dynamical system.