The stability of travelling fronts for general scalar viscous balance law

This paper is concerned with the existence and stability of travelling front solutions for some general scalar viscous balance law. By shooting methods we prove the existence of some class of travelling fronts for any positive viscosity. Further by analytic semigroup theory and detailed spectral analysis, we show that the travelling fronts obtained are asymptotically stable in some appropriate exponentially weighted space. Especially for all sufficiently small viscosity, the travelling waves are proved to be uniformly exponentially stable in the same weighted space.     

The ratio of q-like orthogonal polynomials

It is shown that the ratio of orthogonal polynomials with exponentially increasing recurrence coefficients is closely related to some new orthogonal q-polynomials of which the recurrence coefficients converge exponentially fast to zero. These q-polynomials are investigated in detail.     

On the Asymptotic and Numerical Analyses of Exponentially III-Conditioned Singularly Perturbed Boundary Value Problems

Asymptotic and numerical methods are used to study several classes of singularly perturbed boundary value problems for which the underlying homogeneous operators have exponentially small eigenvalues. Examples considered include the familiar boundary layer resonance problems and some extensions and certain linearized equations associated with metastable internal layer motion. For the boundary layer resonance problems, a systematic projection method, motivated by the work of De Groen [1], is used to analytically calculate high-order asymptotic solutions. This method justifies and extends some previous results obtained from the variational method of Grasman and Matkowsky [2]. A numerical approach, based on an integral equation formulation, is used to accurately compute boundary layer resonance solutions and their associated exponentially small eigenvalues. For various examples, the numerical results are shown to compare very favorably with two-term asymptotic results. Finally, some Sturm-Liouville operators with exponentially small spectral gap widths are studied. One such problem is applied to analyzing metastable internal layer motion for a certain forced Burgers equation.     

BOUNDARY FEEDBACK STABILIZATION OF NONUNIFORM TIMOSHENKO BEAM WITH A TIPLOAD

g1. IntroductionThe purpose of this paPer is to study the stabilization problem of TimoshenkO beamattached with a load of mass M at one end and forced by linear boundary feedback conirols.The system to be investigated in this paper is described as follows (see [1] for example):lthe::1::1'{;::f'5'>', (11)Here u1(t) and u2(t) are the boundary feedback coatrols of force and momellt respectivelythe meanings of all the other variables, functions and coefficients aJre the same as thosedescribed in…     

On exponential ill-conditioning and internal layer behavior

In the limit of small difTusivity the internal layer behavior associated with the initial-boundary value problems for a viscous shock equation and a reaction diffusion equation is analyzed.As a result of the occurrence of exponentially small eigenvalues for the linearized problems the steady state internal layer solutions are shown to very sensitive to small perturbations.For the time dependent problems the small eigenvalues give rise to exponentially slow internal layer motion.Accurate numerical methods are used to compute the steady state internal layer solutions and the slow internal layer motion.The relationship between the viscous shock problem and some exponentially ill-conditioned linear singular perturbation problems is discussed.     

EXPONENTIAL STABILITY FOR A CLASS OF SWITCHED STOCHASTIC SYSTEMS

In this paper, the stability properties for a class of switched stochastic systems with commutative componentwise subsystem matrices are studied. Under some switching law, the trivial solutions of the above systems are proved to be exponentially stable in mean square and almost sure exponentially stable if the random perturbations are sufficiently "small".     

具有动态边界的非均匀Euler-Bernoulli梁的边界反馈镇定

本讨论一端带有重物的Euler-Bernoulli梁的边界反馈镇定问题。在生物的质量忽略不计而只考虑重物的转动惯量的情况下，证明了同时在梁的自由端施加力和力矩反馈，闭环系统的能量可被指数镇定。进而对于系统只有力反馈或只有力矩反馈的情况，得到了闭环系统（指数）稳定的充分必要条件。     

Nonlinear stability of traveling wave fronts for delayed reaction diffusion systems

This paper is concerned with nonlinear stability of traveling wave fronts for a delayed reaction diffusion system. We prove that the traveling wave front is exponentially stable to perturbation in some exponentially weighted L^∞ spaces, when the difference between initial data and traveling wave front decays exponentially as x→−∞, but the initial data can be suitable large in other locations. Moreover, the time decay rates are obtained by weighted energy estimates.     

Exponentially Small Splitting for the Pendulum: A Classical Problem Revisited

In this paper, we study the classical problem of the exponentially small splitting of separatrices of the rapidly forced pendulum. Firstly, we give an asymptotic formula for the distance between the perturbed invariant manifolds in the so-called singular case and we compare it with the prediction of Melnikov theory. Secondly, we give exponentially small upper bounds in some cases in which the perturbation is bigger than in the singular case and we give some heuristic ideas how to obtain an asymptotic formula for these cases. Finally, we study how the splitting of separatrices behaves when the parameters are close to a codimension-2 bifurcation point.     

Partial differential equations with delayed random perturbations: existence uniqueness and stability of solutions

We consider a stochastic non–linear Partial Differential Equation with delay which may be regarded as a perturbed equation. First, we prove the existence and the uniqueness of solutions. Next, we obtain some stability results in order to prove the following: if the unperturbed equation is exponentially stable and the stochastic perturbation is small enough then, the perturbed equations remains exponentially stable. We impose standard assumptions on the differential operators and we use strong and mild solutions     

Fixed points and exponential stability for stochastic Volterra-Levin equations

In this paper we study a stochastic Volterra-Levin equation. By using fixed point theory, we give some conditions for ensuring that this equation is exponentially stable in mean square and is also almost surely exponentially stable. Our result generalizes and improves on the results in [14], [1] and [30].     

The exponential behavior of the stochastic three-dimensional primitive equations with multiplicative noise

In this article, we study the stability of weak solutions to the stochastic three-dimensional (3D) primitive equations (PEs) with multiplicative noise. In particular, we prove that under some conditions on the forcing terms, the weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions. We also prove a result related to the stabilization of these equations.     

Exponential convergence for a class of Nicholson’s blowflies model with multiple time-varying delays

This paper is concerned with the generalized Nicholson’s blowflies model with multiple time-varying delays which is defined on the nonnegative function space. By using some differential inequalities, some new results are established to ensure that all solutions of the model converge exponentially to zero equilibrium point. Moreover, an example is given to illustrate our main results.     

Exponential mixing for smooth hyperbolic suspension flows

We present some simple examples of exponentially mixing hyperbolic suspension flows. We include some speculations indicating possible applications to suspension flows of algebraic Anosov systems. We conclude with some remarks about generalizations of our methods.     

Exponential Decay of Equal-Time Four-Point Correlation Functions in the Hubbard Model on the Copper-Oxide Lattice

For the Hubbard model on the two-dimensional copper-oxide lattice, equal-time four-point correlation functions at positive temperature are proved to decay exponentially in the thermodynamic limit if the magnitude of the on-site interactions is smaller than some power of temperature. This result especially implies that the equal-time correlation functions for singlet Cooper pairs of various symmetries decay exponentially in the distance between the Cooper pairs in high temperatures or in low-temperature weak-coupling regimes. The proof is based on a multi-scale integration over the Matsubara frequency.     

Stability of periodically switched discrete-time linear singular systems

In this paper we present some necessary and sufficient conditions for the stability of periodically switched discrete-time linear index-1 singular system, (PSSS). In particular, it is proved that, if at least one subsystem of a PSSS is asymptotically stable, then there is a switching rule, so that the whole system is also uniformly exponentially stable. Furthermore, for a periodically switched control system with no stable subsystems, there exist a switching rule and feedback matrices, such that the obtained PSSS is uniformly exponentially stable.     

Stability to weak dissipative Bresse system

In this paper we study the Bresse system with frictional dissipation working only on the angle displacement. Our main result is to prove that this dissipative mechanism is enough to stabilize exponentially the whole system provided the velocities of waves propagations are the same. This result is significative only from the mathematical point of view since in practice the velocities of waves propagations are always different. In that direction we show that when the velocities are not the same, the system is not exponentially stable and we prove that the solution in this case goes to zero polynomially, with rates that can be improved by taking more regular initial data. Finally, we give some numerical result to verify our analytical results.     

New results on the parametric stability of nonlinear systems

In this paper, we derive some new results on the parametric stability of nonlinear systems. Explicitly, we derive a necessary and sufficient condition for a nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. Next, we derive some new results on the parametric stability of discrete-time nonlinear systems. As in the continuous case, we derive a necessary and sufficient condition for a discrete-time nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the discrete-time nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. We illustrate our results with some classical examples from the bifurcation theory.     

Exponential stability of Cohen–Grossberg neural networks with delays

The exponential stability characteristics of the Cohen–Grossberg neural networks with discrete delays are studied in this paper, without assuming the symmetry of connection matrix as well as the monotonicity and differentiability of the activation functions and the self-signal functions. By constructing suitable Lyapunov functionals, the delay-independent sufficient conditions for the networks converge exponentially towards the equilibrium associated with the constant input are obtained. By employing Halanay-type inequalities, some sufficient conditions for the networks to be globally exponentially stable are also derived. It is not doubt that our results are significant and useful for the design and applications of the Cohen–Grossberg neural networks.     

On One Class of Systems of Differential Equations with Periodic Coefficients in Linear Terms

Siberian Mathematical Journal - Under consideration is some class of systems of nonlinear differential equations. The exponentially dichotomous linear part of the systems is assumed to have...