Two-dimensional Nonlinear System Research Articles

On solvability of a two-dimensional symmetric nonlinear system of difference equations

We show that the system of difference equations xn+k=xn+lyn−efxn+l+yn−e−f,yn+k=yn+lxn−efyn+l+xn−e−f,n∈N0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ x_{n+k}=\\frac{x_{n+l}y_{n}-ef}{x_{n+l}+y_{n}-e-f},\\quad y_{n+k}= \\frac{y_{n+l}x_{n}-ef}{y_{n+l}+x_{n}-e-f},\\quad n\\in {\\mathbb{N}}_{0}, $$\\end{document} where k∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k\\in {\\mathbb{N}}$\\end{document}, l∈N0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$l\\in {\\mathbb{N}}_{0}$\\end{document}, l<k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$l< k$\\end{document}, e,f∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$e, f\\in {\\mathbb{C}}$\\end{document}, and xj,yj∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$x_{j}, y_{j}\\in {\\mathbb{C}}$\\end{document}, j=0,k−1‾\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$j=\\overline{0,k-1}$\\end{document}, is theoretically solvable and present some cases of the system when the general solutions can be found in a closed form.

On a solvable difference equations system of second order its solutions are related to a generalized Mersenne sequence

Abstract In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when a = b and p = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper.

Two-Dimensional System of Moment Equations and Macroscopic Boundary Conditions Depending on the Velocity of Movement and the Surface Temperature of a Body Moving in Fluid

This article is dedicated to the derivation of a two-dimensional system of moment equations depending on the velocity of movement and the surface temperature of a body submerged in fluid, and macroscopic boundary conditions for the system of moment equations approximating the Maxwell microscopic boundary condition for the particle distribution function. The initial-boundary value problem for the Boltzmann equation with the Maxwell microscopic boundary condition is approximated by a corresponding problem for the system of moment equations with macroscopic boundary conditions. The number of moment equations and the number of macroscopic boundary conditions are interconnected and depend on the parity of the approximation of the system of moment equations. The setting of the initial-boundary value problem for a non-stationary, nonlinear two-dimensional system of moment equations in the first approximation with macroscopic boundary conditions is presented, and the solvability of the above-mentioned problem in the space of functions continuous in time and square-integrable in spatial variables is proven.

Influence of nonlinearity to box-counting dimensions of spiral orbits for two-dimensional differential systems

This study focuses on spiral orbits approaching the origin of two-dimensional nonautonomous nonlinear systems. The main result explains that the fractal dimensions (box-counting dimensions) of spiral orbits of the systems on the phase plane can be derived from the power of the nonlinear term. The examples given show that when the coefficients are power functions, the balance between their power and the power of the nonlinear term determines the fractal dimension. In addition, some numerical simulations are also included.

An evolutive model of a boundedly rational consumer with changing preferences and reference group consumption

In this paper, two dynamic models, recently proposed to describe the adaptive repeated choices of a boundedly rational consumer, are joined together. One considers a consumer adjusting the consumption level of a given good over time according to the observed discrepancy between expected and realized utility gain and modifies the utility function according to past consumption experience, also including saturation effects when past consumption is excessive. The other one considers the same adjustment mechanism with constant preferences but with a behavioral effect that introduces a tendency (or bias) to imitate a reference group of consumers. Merging these two models, a two-dimensional nonlinear dynamical system is obtained which describes consumers that decide their next period consumption of a given good by following two different (sometimes contrasting) criteria: their own utility maximization on the one side and imitation of a reference group of consumers on the other side. This leads to a greater uncertainty with respect to the model without the behavioral bias. Such uncertainty is studied through a numerical exploration of the long-run dynamics, guided by some global dynamical features of the nonlinear model, such as the folding action of the critical curves that characterize the behavior of the iterated noninvertible map and the singularities related to the presence of a vanishing denominator, namely focal points and prefocal curves. So, the aim of the paper is twofold: on the one side, it tries to contribute to the literature on the economic theory of boundedly rational consumers represented by evolutionary and behavioral approaches; On the other side, it tries to contribute to the recent literature about the global analysis of discrete dynamical systems characterized by contact bifurcations leading to the creation of complex topological structures of the attractors and their basins of attraction.

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.

Comparing the spectra of wandering exponents of a nonlinear two-dimensional system and a first approximation system

Comparing the spectra of wandering exponents of a nonlinear two-dimensional system and a first approximation system

Six classes of nonlinear two-dimensional systems of difference equations which are solvable

Six classes of nonlinear two-dimensional systems of difference equations which are solvable

Comparing the Spectra of Oscillation Exponents of a Nonlinear System and the First Approximation System

We study the oscillation exponents of differential systems. It is established that there is no dependence between the spectra of oscillation exponents of a nonlinear system and the system of its first approximation; namely, a two-dimensional nonlinear system is constructed such that the spectra of oscillation exponents of its restriction to any open neighborhood of zero on the phase plane consist of all rational numbers in the interval and the spectra of the linear system of its first approximation consist of only one element.

A Unique Continuation Result for a 2D System of Nonlinear Equations for Surface Waves

In this paper, we establish a result of unique continuation for a special two-dimensional nonlinear system that models the evolution of long water waves with small amplitude in the presence of surface tension. More precisely, we will show that if (eta ,Phi ) = (eta (x,y, t),Phi (x,y, t)) is a solution of the nonlinear system, in a suitable function space, and (eta ,Phi ) vanishes on an open subset Omega of mathbb {R}^2 times [-T,T], then (eta ,Phi )equiv 0 in the horizontal component of Omega . To state such property, we use a Carleman-type estimate for a differential operator mathcal {L} related to the system. We prove the Carleman estimate using a particular version of the well known Treves’ inequality.

Distributed event-based H∞ consensus filtering for 2-D T-S fuzzy systems over sensor networks subject to DoS attacks

This paper concentrates on the H∞ consensus filtering issue of a kind of nonlinear two-dimensional (2-D) systems over sensor networks, which are described by the Takagi-Sugeno fuzzy model. With consideration of the limited communication bandwidth and the malicious network attack behavior, a distributed event-triggered mechanism is considered for the sensor network under DoS attacks. Different from the implicit assumption of many previous results that detail information of the non-periodic DoS attacks is available, a switching-like distributed event-triggered mechanism is introduced under the 2-D framework, which could schedule the triggering strategy flexibly in the attack silent or active regions. By means of the switching system analysis method and the discontinuous Lyapunov functions, this paper aims to develop a proper distributed attack-resilient H∞ consensus filter based on the information from the local sensor and its neighbors. A set of sufficient conditions are presented so that the estimation error system can reach exponential stability and meet an expected H∞ performance. Finally, a numerical example is given to demonstrate feasibility of the filter designing method.

Fuzzy Observer-Based Output Feedback Control of Continuous-Time Nonlinear Two-Dimensional Systems

The fuzzy observer-based output feedback control of continuous-time nonlinear two-dimensional systems is studied by T-S fuzzy models in this work. The plant information of the two-dimensional systems evolves along two independent directions dynamically. The nonlinear two-dimensional systems are firstly expressed by Takagi-Sugeno fuzzy models with parameter uncertainties. By a Lyapunov method together with some convexification techniques, two methods are developed for fuzzy observer-based output feedback controller synthesis of the underlying fuzzy two-dimensional systems, and novel output feedback controller synthesis results are proposed within a convex optimization setup. Simulation studies are provided to illustrate the validity of the proposed methods.

The Stochastic Structural Modulations in Collapsing Maccari’s Model Solitons

The two-dimensional Maccari nonlinear system performs the energy and wave dynamical features in fiber communications and modern physical science as hydrodynamic and space plasma. Several new forms of solutions for the Maccari’s model are constructed by a unified solver method that mainly depends on He’s variations method. The obtained solutions identify new wave stochastic structures with important features in energy physics such as rational explosive, breather, dispersive, explosive dissipated, dark solitons and blow-up (shock structure). It was elucidated that the random effects amend the energy wave strength or the collapsing due to model medium turbulence. Finally, the produced stochastic structures may be vital in some of these relationships between dispersions, nonlinearity and dissipative effects. The predominant energy waves that are collapsing or being forced may be applied to electrostatic auroral Langmuir structures and energy-generating ocean waves.

Oscillation and Nonoscillation for Two-dimensional Nonlinear Systems of Ordinary Differential Equations

For the two-dimensional nonlinear system \[ u' = a(t) |v|^{1/\alpha} \operatorname{sgn}v, \quad v' = - b(t) |u|^{\alpha} \operatorname{sgn}u \] with $\alpha \gt 0$, $a,b \in C[t_{0},\infty)$, $a(t) \geq 0$ ($t \geq t_{0}$), new oscillation criteria and nonoscillation criteria are given in both cases $\int_{t_{0}}^{\infty} a(s) \, ds = \infty$ and $\int_{t_{0}}^{\infty} a(s) \, ds \lt \infty$. One of the main results is an analogue of the Hartman–Wintner oscillation theorem. Our results generalize Li and Yeh's results for second order half-linear scalar equations.

Degenerate Resonances and Synchronization of Quasiperiodic Oscillations

We consider quasiperiodic nonconservative perturbations of two-dimensional nonlinear Hamiltonian systems. We introduce the notion of a degenerate resonance and study the topology of a degenerate resonance zone with odd and even orders of degeneracy. A special attention is paid the oscillation synchronization while an invariant torus is passing through the resonance zone. We establish the existence of synchronization intervals with respect to a control parameter. The study is based on the analysis of a pendulum–type averaged system determining the dynamics in the resonance zone.

Stability Analysis of Equilibrium Point and Limit Cycle of Two-Dimensional Nonlinear Dynamical Systems—A Tutorial

The equilibrium state of a dynamical system can be divided into the equilibrium point and limit cycle. In this paper, the stability analysis of the equilibrium point and limit cycle of dynamical systems are presented through different and all possible approaches, and those approaches are compared as well. In particular, the author presented the stability analysis of the equilibrium point through phase plane approach, Lyapunov–LaSalle energy-based approach, and linearization approach, respectively, for two-dimensional nonlinear system, while the stability analysis of the limit cycle is analyzed by using the LaSalle local invariant set theorem and Poincaré–Bendixson theorem, which is only valid in two-dimensional systems. Different case studies are used to demonstrate the stability analysis of equilibrium point and limit cycle.

Approximation of limit cycle of differential systems with variable coefficients

The behavior of the approximate solutions of two-dimensional nonlinear differential systems with variable coefficients is considered. Using a property of the approximate solution, so called conditional Ulam stability of a generalized logistic equation, the behavior of the approximate solution of the system is investigated. The obtained result explicitly presents the error between the limit cycle and its approximation. Some examples are presented with numerical simulations.

Set-Membership Estimation for Nonlinear 2-D Systems With Missing Measurements

This brief addresses the set-membership estimation issue for two-dimensional nonlinear systems with missing measurements and unknown-but-bounded disturbances. The phenomenon of missing measurements is modeled by a deterministic approach with values of 0 and 1. The considered nonlinearities are treated by Taylor series expansion, in which the truncation errors are transformed subtly into norm-bounded parameter uncertainties. A set-membership estimator is established, which ensures that the real system state is included in a certain ellipsoidal region of the estimation state at each step. Then, to identify the local best estimation performance, an optimization algorithm is devised aiming at ellipsoidal minimization (in the sense of matrix trace). The numerical results further confirm the efficacy of the proposed algorithm.

On a two-dimensional nonlinear system of difference equations close to the bilinear system

&lt;abstract&gt;&lt;p&gt;We consider the two-dimensional nonlinear system of difference equations&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ x_n = x_{n-k}\frac{ay_{n-l}+by_{n-(k+l)}}{cy_{n-l}+dy_{n-(k+l)}},\quad y_n = y_{n-k}\frac{{\alpha} x_{n-l}+{\beta} x_{n-(k+l)}}{{\gamma} x_{n-l}+{\delta} x_{n-(k+l)}}, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;for $ n\in{\mathbb N}_0, $ where the delays $ k $ and $ l $ are two natural numbers, and the initial values $ x_{-j}, y_{-j} $, $ 1\le j\le k+l $, and the parameters $ a, b, c, d, {\alpha}, {\beta}, {\gamma}, {\delta} $ are real numbers. We show that the system of difference equations is solvable by presenting a method for finding its general solution in detail. Bearing in mind that the system of equations is a natural generalization of the corresponding one-dimensional difference equation, whose special cases appear in the literature from time to time, our main result presented here also generalizes many results therein.&lt;/p&gt;&lt;/abstract&gt;

Theoretical Aspects on Heat Transfer and Fluid Flow in Micro Channels

The past ten years have seen a lot of research on the issues of heat transfer and fluid flow in micro-channels. With an emphasis on upcoming research requirements, a critical analysis of the state of research as it stands is offered. Following a brief introduction, the study discusses six themes related to transport phenomena in micro-channels: condensation, cooling of electronics, single-phase gas flow, augmentation of single-phase liquid flow and flow boiling, and micro-scale heat exchangers. In this study, we investigate the three-dimensional heat transfer and water flow properties in a set of rectangular micro-channel heat sinks for advanced electronic systems. Over the past ten years, mini/micro channel type compact heat exchangers have received a lot of attention. The ability to make heat exchangers smaller, lighter, and cheaper than those in use now is the primary motivator. Additionally, emerging applications that call for the cooling of small things, such electronics and micro-electro-mechanical devices, need for heat exchangers with tiny channels. Theoretically, it is examined how a nanofluid moves and transfers heat through a horizontal micro channel while being affected by a magnetic field and an electric double layer (EDL). The flow problem for a micro channel with a large aspect ratio is handled as a two-dimensional nonlinear system. The magnetic field and EDL body force are taken into account while calculating momentum equation