The Modulational Instability for a Generalized Korteweg–de Vries Equation

We study the spectral stability of a family of periodic standing wave solutions to the generalized Korteweg–de Vries in a neighborhood of the origin in the spectral plane using what amounts to a rigorous Whitham modulation theory calculation. In particular we are interested in understanding the role played by the null directions of the linearized operator in the stability of the traveling wave to perturbations of long wavelength. A study of the normal form of the characteristic polynomial of the monodromy map (the periodic Evans function) in a neighborhood of the origin in the spectral plane leads to two different instability indices. The first, an orientation index, counts modulo 2 the total number of periodic eigenvalues on the real axis. This index is a generalization of the one which governs the stability of the solitary wave. The second, a modulational instability index, provides a necessary and sufficient condition for the existence of a long-wavelength instability. This index is essentially the quantity calculated by Hǎrǎguş and Kapitula in the small amplitude limit. Both of these quantities can be expressed in terms of the map between the constants of integration for the ordinary differential equation defining the traveling waves and the conserved quantities of the partial differential equation. These two indices together provide a good deal of information about the spectrum of the linearized operator. We sketch the connection of this calculation to a study of the linearized operator—in particular we perform a perturbation calculation in terms of the Floquet parameter. This suggests a geometric interpretation attached to the vanishing of the orientation index previously mentioned.     

Stability analysis for two kinds of equations in two-species population dynamics

In this paper we study the stability for equilibrium points of equations in two-population dynamics.We discuss two predator-prey-patch models.Model1 is described by a differential equation.Model2 is described by an integral differential equation.We obtain the conditions for the stability of their equilibrium points.The results show that the overall population stability despite local extinction is realizable.     

Stability for Monostable Wave Fronts of Delayed Lattice Differential Equations

This paper is concerned with the stability of traveling wave fronts for delayed monostable lattice differential equations. We first investigate the existence non-existence and uniqueness of traveling wave fronts by using the technique of monotone iteration method and Ikehara theorem. Then we apply the contraction principle to obtain the existence, uniqueness, and positivity of solutions for the Cauchy problem. Next, we study the stability of a traveling wave front by using comparison theorems for the Cauchy problem and initial-boundary value problem of the lattice differential equations, respectively. We show that any solution of the Cauchy problem converges exponentially to a traveling wave front provided that the initial function is a perturbation of the traveling wave front, whose asymptotic behaviour at \(-\infty \) satisfying some restrictions. Our results can apply to many lattice differential equations, for examples, the delayed cellular neural networks model and discrete diffusive Nicholson’s blowflies equation.     

Global Hopf Bifurcation Analysis of a Nicholson’s Blowflies Equation of Neutral Type

We investigate Hopf bifurcations in a delayed Nicholson’s blowflies equation of neutral type, derived from the Gurtin–MacCamy model. A key parameter that determines the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Global extension of local Hopf branches is established by combining a global Hopf bifurcation theorem with a Bendixson criterion for higher dimensional ordinary differential equations. We show that a branch of slowly varying periodic solutions and a branch of fast oscillating periodic solutions coexist for all large delays.     

A Hartman-Grobman theorem for the Cahn-Hilliard and phase-field equations

In this paper, we study the structural stability of the Cahn-Hilliard equation and the phase-field equations. We show that the Cahn-Hilliard equation and the phase-field equations are topologically conjugate to a decoupled system of a linear equation of infinite dimension and an ordinary differential equation which is the reduced equation on the inertial manifold; particularly, the flow nearby hyperbolic stationary solutions is structurally stable.     

Nonlinear vibration behavior and resonance of a Cartesian manipulator system carrying an intermediate end effector

This paper studied the nonlinear vibration and resonance of a Cartesian manipulator system carrying an intermediate end effector under mixed excitations. The multiple scales method is applied to get the approximate solutions of this system of the second-order differential equation. Furthermore, the analytical solution obtained the amplitudes and phases of the response from the first-order differential equation governing. We extracted all worst resonance cases and studied it numerically. The numerical solutions and response amplitude of this system are also studied and discussed. We analyzed the stability of the steady-state solution of a Cartesian manipulator system using frequency response equations and phase plane technique at the worst resonance cases. Comparison between analytical and numerical solutions is obtained. We determined both bifurcation diagrams and stability using Poincaré maps. Also, the numerical results are obtained using MAPLE and MATLAB algorithms.     

Stability analysis of a noise control system in a duct by using delay differential equation

The paper deals with the criteria for the closed- loop stability of a noise control system in a duct. To study the stability of the system, the model of delay differential equation is derived from the propagation of acoustic wave governed by a partial differential equation of hyperbolic type. Then, a simple feedback controller is designed, and its closed- loop stability is analyzed on the basis of the derived model of delay differential equation. The obtained criteria reveal the influence of the controller gain and the positions of a sensor and an actuator on the closed-loop stability. Finally, numerical simulations are presented to support the theoretical results.     

Stability and Boundedness Results for Solutions of Certain Third Order Nonlinear Vector Differential Equations

In this paper, we investigate the asymptotic stability of the zero solution and boundedness of all solutions of a certain third order nonlinear ordinary vector differential equation. The results are proved using Lyapunov’s second (or direct method). Our results include and improve some well known results existing in the literature.     

The Analysis of Some Characteristic Equations Arising in Population and Epidemic Models

Epidemic models with a general infective period distribution are formulated as functional differential equations, as are population models with a general life span distribution. The analysis of the local stability properties of equilibria of such models leads to a characteristic equation involving the Laplace transform of the infective period (or life span) distribution. We obtain conditions under which all roots of the characteristic equation are in the left half plane, implying asymptotic stability of equilibrium, for every infective period distribution. We also consider the converse problem of describing when instability can occur for specific infective period distributions.     

Investigation of the stability for inviscid compressible swirling flow between concentric cylinders

The temporal stability on inviscid compressible swirling flow between two concentric cylinders is investigated. First, a linearized differential equation is derived. Two stability criteria are derived for compressible swirling flow by an analytic method analogous to Ludwieg ’s method. A finite-difference numerical method is then used to solve the eigenvalue problem of this differential equation, to get temporal growth rate and to check these stabilitv criteria derived. Finally.The effect of compressibility for stability is disscused.     

Global Stability of a Competitive Model with State-Dependent Delay

This article deals with a stage-structured model with state-dependent delay which is assumed to be an increasing function of the population density with lower and upper bound. Firstly, according to the principle of linearized stability (Theorem 3.6, Hartung et al. in Handbook of differential equations: ordinary differential equations, 2006), we study the local stability of system in combination with the positivity and boundedness of solutions. By using the comparison principle obtained and an iterative method, the global stability of the equilibria is completely analyzed. Our results show how the interaction between interspecific and intraspecific competition affects the coexistence of both species.     

On the aeroelastic stability and bifurcation structure of subsonic nonlinear thin panels subjected to external excitation

Dynamic behavior of panels exposed to subsonic flow subjected to external excitation is investigated in this paper. The von Karman’s large deflection equations of motion for a flexible panel and Kelvin’s model of structural damping is considered to derive the governing equation. The panel under study is two-dimensional and simply supported. A Galerkin-type solution is introduced to derive the unsteady aerodynamic pressure from the linearized potential equation of uniform incompressible flow. The governing partial differential equation is transformed to a series of ordinary differential equations by using Galerkin method. The aeroelastic stability of the linear panel system is presented in a qualitative analysis and numerical study. The fourth-order Runge-Kutta numerical algorithm is used to conduct the numerical simulations to investigate the bifurcation structure of the nonlinear panel system and the distributions of chaotic regions are shown in the different parameter spaces. The results shows that the panel loses its stability by divergence not flutter in subsonic flow; the number of the fixed points and their stabilities change after the dynamic pressure exceeds the critical value; the chaotic regions and periodic regions appear alternately in parameter spaces; the single period motion trajectories change rhythmically in different periodic regions; the route from periodic motion to chaos is via doubling-period bifurcation.     

An Integro-Differential Equation Modelling a Newtonian Dynamics and Its Scaling Limit

Summary We consider an integro-differential equation describing a Newtonian dynamics with long-range interaction for a continuous distribution of mass in R. First, we deduce unique existence and regularity properties of its solution locally in time, and then we investigate a scaling limit. As limit dynamics a nonlinear wave equation is determined. Technically, we rely on the connection of the Newtonian dynamics to a system of an integro-differential equation and a partial differential equation. Basic for our considerations is the study of the regularity properties of the solution of that system. For that purpose we exploit its similarity to a certain strongly hyperbolic system of partial differential equations. (Accepted August 21, 1995)     

A uniqueness criterion for fractional differential equations with Caputo derivative

We investigate the uniqueness of solutions to an initial value problem associated with a nonlinear fractional differential equation of order α∈(0,1). The differential operator is of Caputo type whereas the nonlinearity cannot be expressed as a Lipschitz function. Instead, the Riemann–Liouville derivative of this nonlinearity verifies a special inequality.     

On the Absolute Exponential Stability of Solutions of Systems of Linear Parabolic Differential Equations with One Delay

We investigate necessary conditions for the absolute exponential stability of a system of linear parabolic differential equations with one delay.     

Stability of axially accelerating viscoelastic beams: asymptotic perturbation analysis and differential quadrature validation

An asymptotic perturbation method is proposed to investigate stability of an axially accelerating viscoelastic beam. The material time derivative is used in the viscoelastic constitutive relation. The axial speed is characterized as a simple harmonic variation about the constant mean speed. The stability condition can be determined via the asymptotic perturbation method. The differential quadrature scheme is developed to solve numerically the equation of axially accelerating viscoelastic beams with simple supports. The stability boundaries are numerically located in the summation parametric resonance and the principal parametric resonance. Numerical examples show the effects of the beam viscoelasticity and the mean axial speed. The numerical calculations validate the analytical results in the principal parametric resonance.     

Sharp-Interface Nematic–Isotropic Phase Transitions without Flow

We derive a supplemental evolution equation for an interface between the nematic and isotropic phases of a liquid crystal when flow is neglected. Our approach is based on the notion of configurational force. As an application, we study the behavior of a spherical isotropic drop surrounded by a radially oriented nematic phase: our supplemental evolution equation then reduces to a simple ordinary differential equation admitting a closed-form solution. In addition to describing many features of isotropic-to-nematic phase transitions, this simplified model yields insight concerning the occurrence and stability of isotropic cores for hedgehog defects in liquid crystals.     

Nonlinear waves in electromigration dispersion in a capillary

We construct exact solutions to an unusual nonlinear advection–diffusion equation arising in the study of Taylor–Aris (also known as shear) dispersion due to electroosmotic flow during electromigration in a capillary. An exact reduction to a Darboux equation is found under a traveling-wave ansatz. The equilibria of this ordinary differential equation are analyzed, showing that their stability is determined solely by the (dimensionless) wave speed without regard to any (dimensionless) physical parameters. Integral curves, connecting the appropriate equilibria of the Darboux equation that governs traveling waves, are constructed, which in turn are shown to be asymmetric kink solutions (i.e., non-Taylor shocks). Furthermore, it is shown that the governing Darboux equation exhibits bistability, which leads to two coexisting non-negative kink solutions for (dimensionless) wave speeds greater than unity. Finally, we give some remarks on other types of traveling-wave solutions and a discussion of some approximations of the governing partial differential equation of electromigration dispersion.