Funzioni regolari di più variabili quaternioniche

Summary In this paper the stability analysis of an incompressible toroidal Hall Current plasma with resistivity and viscosity on the basis of the linearization of the governing equations and boundary condition is rigorously justified. A nonlinear local existence theorem for an initial-boundary value problem is first proved, and local stable and unstable invariant manifolds of a nonlinear resolving operator are then constructed. It is shown that linear stability implies nonlinear stability and global existence, and linear instability implies nonlinear instability in some sense.     

Orbital stability of bound states of nonlinear Schrödinger equations with linear and nonlinear optical lattices

We study the orbital stability and instability of single-spike bound states of semi-classical nonlinear Schrödinger (NLS) equations with critical exponent, linear and nonlinear optical lattices (OLs). These equations may model two-dimensional Bose-Einstein condensates in linear and nonlinear OLs. When linear OLs are switched off, we derive the asymptotic expansion formulas and obtain necessary conditions for the orbital stability and instability of single-spike bound states, respectively. When linear OLs are turned on, we consider three different conditions of linear and nonlinear OLs to develop mathematical theorems which are most general on the orbital stability problem.     

The Onset of Linear Instabilities in a Solid Combustion Model

This article concerns the onset of linear instability in a simple model of solid combustion in a semi-infinite two-dimensional strip of width l . The free boundary problem that describes the model involves initial and boundary conditions, including a nonlinear kinetic condition at the interface. The linear problem governing perturbations to a basic solution is solved by the method of images with the reaction front perturbation satisfying an integro-differential equation. This equation is then solved using Laplace transforms. Finally, we perform a stability analysis for the model by studying the solution of the reaction front perturbation. The inclusion of initial conditions enables us to show the development of linear instability from arbitrary initial small disturbances.     

Nonlinear stability of an axial electric field: Effect of interfacial charge relaxation

We introduce a nonlinear perturbation technique to third order, to study the stability between two cylindrical inviscid fluids, subjected to an axial electric field. The study takes into account the relaxation of electrical charges at the interface between the two fluids. At first order, a linear dispersion relation is obtained. Analytical and numerical results for the overstability and incipient instability conditions are given. For perfect dielectric fluids, the electric field has a stabilizing influence, while for leaky dielectric fluids, the electric field can have either a stabilizing or a destabilizing influence depending on the conductivity and permittivity ratios of the two fluids. At higher order, a nonlinear dispersion relation (nonlinear Ginzburg–Landau equation) is derived, describing the evolution of wave packets of the problem. For leaky dielectric fluids near the marginal state, a nonlinear diffusion equation (nonlinear incipient instability) is obtained. For perfect dielectric fluids, two cubic nonlinear Schrödinger equations are obtained. One of these equations to determine a nonlinear cutoff electric field separating stable and unstable disturbance, whereas the other is used to analyze the stability of the system. It is found that the nonlinear stability criterion depending on the ratio of permittivity, Such effects can only be explained successfully in the nonlinear sense, as the linear analysis unsuccessful to inform about them.     

Modulation instability of soliton trains in fiber communication systems

The linear stability problem for a soliton train described by the nonlinear Schrödinger equation is exactly solved using a linearization of the Zakharov-Shabat dressing procedure. This problem is reduced to finding a compatible solution of two linear equations. This approach allows the growth rate of the soliton lattice instability and the corresponding eigenfunctions to be found explicitly in a purely algebraic way. The growth rate can be expressed in terms of elliptic functions. Analysis of the dispersion relations and eigerfunctions shows that the solution, which has the form of a soliton train, is stable for defocusing media and unstable for focusing media with arbitrary parameters. Possible applications of the stability results to fiber communication systems are discussed.     

The effect of density-dependent dispersal on the stability of populations

The nonlinear stability of two populations diffusing in a common bounded domain with nonlinear density-dependent dispersal of a general type is studied, under Robin boundary conditions. The stability and instability of the positive equilibrium are studied by the linear eigenvalues method. The nonlinear stability is studied with the Lyapunov direct method. In a particular case, global stability has been obtained.     

A nonlinear stability analysis of a rotating double-diffusive magnetized ferrofluid

A nonlinear stability result for a double-diffusive magnetized ferrofluid layer rotating about a vertical axis for stress-free boundaries is derived via generalized energy method. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. The result is compared with the result obtained by linear instability theory. The critical magnetic thermal Rayleigh number given by energy theory is slightly less than those given by linear theory and thus indicates the existence of subcritical instability for ferrofluids. For non-ferrofluids, it is observed that the nonlinear critical stability thermal Rayleigh number coincides with that of linear critical stability thermal Rayleigh number. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M₃, solute gradient, S₁, and Taylor number, T_A1, on subcritical instability region have been analyzed. We also demonstrate coupling between the buoyancy and magnetic forces in the nonlinear stability analysis.     

Nonlinear stability and instability of relativistic Vlasov‐Maxwell systems

We prove the nonlinear stability or instability of certain periodic equilibria of the 1½D relativistic Vlasov‐Maxwell system. In particular, for a purely magnetic equilibrium with vanishing electric field, we prove its nonlinear stability under a sharp criterion by extending the usual Casimir‐energy method in several new ways. For a general electromagnetic equilibrium we prove that nonlinear instability follows from linear instability. The nonlinear instability is macroscopic, involving only the L¹‐norms of the electromagnetic fields. © 2006 Wiley Periodicals, Inc.     

On the orbital stability of pendulum-like oscillations and rotations of a symmetric rigid body with a fixed point

We deal with the problem of orbital stability of planar periodic motions of a dynamically symmetric heavy rigid body with a fixed point. We suppose that the center of mass of the body lies in the equatorial plane of the ellipsoid of inertia. Unperturbed periodic motions are planar pendulum-like oscillations or rotations of the body around a principal axis keeping a fixed horizontal position. Local coordinates are introduced in a neighborhood of the unperturbed periodic motion and equations of the perturbed motion are obtained in Hamiltonian form. Regions of orbital instability are established by means of linear analysis. Outside the above-mentioned regions, nonlinear analysis is performed taking into account terms up to degree 4 in the expansion of the Hamiltonian in a neighborhood of unperturbed motion. The nonlinear problem of orbital stability is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients. The orbital stability is investigated analytically in two limiting cases: small amplitude oscillations and rotations with large angular velocities when a small parameter can be introduced.     

On the stability of resonant rotation of a symmetric satellite in an elliptical orbit

We deal with the stability problem of resonant rotation of a symmetric rigid body about its center of mass in an elliptical orbit. The resonant rotation is a planar motion such that the body completes one rotation in absolute space during two orbital revolutions of its center of mass. In [1–3] the stability analysis of the above resonant rotation with respect to planar perturbations has been performed in detail.In this paper we study the stability of the resonant rotation in an extended formulation taking into account both planar and spatial perturbations. By analyzing linearized equations of perturbed motion, we found eccentricity intervals, where the resonant rotation is unstable. Outside of these intervals a nonlinear stability study has been performed and subintervals of formal stability and stability for most initial data have been found. In addition, the instability of the resonant rotation was established at several eccentricity values corresponding to the third and fourth order resonances.Our study has also shown that in linear approximation the spatial perturbations have no effect on the stability of the resonant rotation, whereas in a nonlinear system they can lead to its instability at some resonant values of the eccentricity.     

EVANS FUNCTIONS AND INSTABILITY OF A STANDING PULSE SOLUTION OF A NONLINEAR SYSTEM OF REACTION DIFFUSION EQUATIONS

In this paper, we consider a nonlinear system of reaction diffusion equations arising from mathematical neuroscience and two nonlinear scalar reaction diffusion equations under some assumptions on their coefficients. The main purpose is to couple together linearized stability criterion (the equivalence of the nonlinear stability, the linear stability and the spectral stability of the standing pulse solutions) and Evans functions to accomplish the existence and instability of standing pulse solutions of the nonlinear system of reaction diffusion equations and the nonlinear scalar reaction diffusion equations. The Evans functions for the standing pulse solutions are constructed explicitly.     

Stability of equilibrium solutions of a double power reaction-diffusion equation with a Dirac interaction

In this paper, information about the instability of equilibrium solutions of a nonlinear family of localized reaction-diffusion equations in dimension one is provided. More precisely, explicit formulas to the equilibrium solutions are computed and, via analytic perturbation theory, the exact number of positive eigenvalues of the linear operator associated to the stability problem is analyzed. In addition, sufficient conditions for blow up of the solutions of the equation are also discussed.     

Nonlinear evolution of the travelling waves at the surface of a thin viscoelastic falling film

The problem of hydrodynamic instability of a thin condensate viscoelastic liquid film flowing down on the outer surface of an axially moving vertical cylinder is investigated. In order to improve the accuracy of numerical results, the viscoelastic and heat transfer parameters have been included into the governing equations. Also, the analytical solutions are obtained by utilizing the long-wave perturbation method. The influence of some physical parameters is discussed in both linear and nonlinear steps of the problem. It has been revealed that the stability of the film flow is weakened when the radius of cylinder and the temperature difference are reduced. Moreover, it is found that the increment of down-moving motion of the cylinder can enhance the flow stability. Further, the thin film flow can be destabilized by the viscoelastic property. The results show that both supercritical stability and subcritical instability can take place within the film flow system given appropriate conditions. Moreover, the absence of Reynolds number leads to an obvious difference in the behavior of some physical parameters.     

Nonlinear Kelvin-Helmholtz instability of two miscible ferrofluids in porous media

The nonlinear theory of the Kelvin-Helmholtz instability is employed to analyze the instability phenomenon of two ferrofluids through porous media. The effect of both magnetic field and mass and heat transfer is taken into account. The method of multiple scale expansion is employed in order to obtain a dispersion relation for the first-order problem and a Ginzburg–Landau equation, for the higher-order problem, describing the behavior of the system in a nonlinear approach. The stability criterion is expressed in terms of various competing parameters representing the mass and heat transfer, gravity, surface tension, fluid density, magnetic permeability, streaming, fluid thickness and Darcy coefficient. The stability of the system is discussed in both theoretically and computationally, and stability diagrams are drawn.     

A nonlinear stability analysis for rotating magnetized ferrofluid heated from below saturating a porous medium

A nonlinear (energy) stability analysis is performed for a rotating magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. By introducing a generalized energy functional, a rigorous nonlinear stability result for a thermoconvective rotating magnetized ferrofluid is derived. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M ₃, medium permeability, D _a , and rotation, , on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter, M ₃, and Darcy number, D _a , the subcritical instability region between the two theories decreases quickly while with the increase of Taylor number, , the subcritical region expands. We also demonstrate coupling between the buoyancy and magnetic forces in the presence of rotation in nonlinear energy stability analysis as well as in linear instability analysis.      

A nonlinear stability analysis for rotating magnetized ferrofluid heated from below saturating a porous medium

A nonlinear (energy) stability analysis is performed for a rotating magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. By introducing a generalized energy functional, a rigorous nonlinear stability result for a thermoconvective rotating magnetized ferrofluid is derived. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M ₃, medium permeability, D _a , and rotation, T_A1T_{A_1}, on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter, M ₃, and Darcy number, D _a , the subcritical instability region between the two theories decreases quickly while with the increase of Taylor number, T_A1T_{A_1} , the subcritical region expands. We also demonstrate coupling between the buoyancy and magnetic forces in the presence of rotation in nonlinear energy stability analysis as well as in linear instability analysis.     

Nonlinear Kelvin-Helmholtz instability of two miscible ferrofluids in porous media

The nonlinear theory of the Kelvin-Helmholtz instability is employed to analyze the instability phenomenon of two ferrofluids through porous media. The effect of both magnetic field and mass and heat transfer is taken into account. The method of multiple scale expansion is employed in order to obtain a dispersion relation for the first-order problem and a Ginzburg–Landau equation, for the higher-order problem, describing the behavior of the system in a nonlinear approach. The stability criterion is expressed in terms of various competing parameters representing the mass and heat transfer, gravity, surface tension, fluid density, magnetic permeability, streaming, fluid thickness and Darcy coefficient. The stability of the system is discussed in both theoretically and computationally, and stability diagrams are drawn. Received: July 25, 2002; revised: April 16, 2003     

Nonlinear Stability Problem of a Rotating Doubly Diffusive Porous Layer

Lyapunov direct method is applied to study the non-linear conditional stability problem of a rotating doubly diffusive convection in a sparsely packed porous layer. For a Darcy number greater than or equal to 1000, and for any Prandtl number, Taylor number, and solute Rayleigh number it is found that the non-linear stability bound coincides with linear instability bound. For a Darcy number less than 1000, for a Prandtl number greater than or equal to one, and for a certain range of Taylor number, a coincidence between the linear and nonlinear (energy) stability thermal Rayleigh number values is still maintained. However, it is noted that for a Darcy number less than 1000, as the value of the solute Rayleigh number or the Taylor number increases, the coincidence domain between the two theories decreases quickly.     

Stability of a layer of dipolar fluid heated from below

The equations of Bleustein and Green [2] are formulated in a way suitable to describe the convective instability which occurs when a layer of dipolar fluid is heated from below. The linear instability boundary is shown to coincide with the nonlinear stability curve and the critical Rayleigh numbers describing this boundary are found; in particular, the non-dimensional micro-length is found to always stabilize.     

Receding Horizon Robust Control for Nonlinear Systems Based on Linear Differential Inclusion of Neural Networks

In this paper, we present a new receding horizon neural robust control scheme for a class of nonlinear systems based on the linear differential inclusion (LDI) representation of neural networks. First, we propose a linear matrix inequality (LMI) condition on the terminal weighting matrix for a receding horizon neural robust control scheme. This condition guarantees the nonincreasing monotonicity of the saddle point value of the finite horizon dynamic game. We then propose a receding horizon neural robust control scheme for nonlinear systems, which ensures the infinite horizon robust performance and the internal stability of closed-loop systems. Since the proposed control scheme can effectively deal with input and state constraints in an optimization problem, it does not cause the instability problem or give the poor performance associated with the existing neural robust control schemes.