The dynamics of a nonlinear wave equation

We consider a wave equation in a bounded domain with linear dissipation and with a nonlinear source term. We give characterizations of all the solutions with respect to their qualitative properties: globality, boundedness, nonglobality, blow-up, and convergence to equilibria.     

Vortex dynamics for the nonlinear wave equation

Vortex dynamics for the nonlinear wave equation is a typical model of the “particle and field” theories of classical physics. The formal derivation of the dynamical law was done by J.Neu. He also made an interesting connection between vortex dynamics and the Dirac theory of electrons. Here we give a rigorous mathematical proof of this natural dynamical law. © 1999 John Wiley & Sons, Inc.     

Stability of nonlinear multiresolution analysis

Stability of nonlinear subdivision and multiresolution has recently been addressed in [1]. Here we give applications to convexity/monotonicity preserving schemes introduced in [2], [3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of the set of trajectories of nonlinear dynamics: Stability under interval initial conditions

For the set of equations of perturbed motion whose solutions satisfy interval initial conditions, we obtain sufficient conditions for the Lyapunov stability and the practical stability of these solutions. The analysis is performed on the basis of locally large scalar Lyapunov functions. As examples, we consider quasilinear and linear nonautonomous systems.     

Stability of planar diffusion wave for nonlinear evolution equation

It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f’(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.     

Stability of planar diffusion wave for the quasilinear wave equation with nonlinear damping

In this paper, we will show that under some smallness conditions, the planar diffusion wave v(√x11+) is stable for a quasilinear wave equation with nonlinear damping: vttt-f(v) + vt + g(vt) = 0, x =(xn1, x2,, xn)∈ R, where v(√x11+) is the unique similar solution to the one dimensional nonlinear heat equattion: vt-f(v)x1x1= 0, f ′(v) > 0, v(±∞, t) = v±, v+ = v-. We also obtain the L∞ time decay rate which rreads v-v L∞= O(1)(1 + t)-4, where r = min{3, n}. To get the main result, the energy method and a new inequality have been used.     

Stability of generalized equations under nonlinear perturbations

This paper studies solution stability of generalized equations over polyhedral convex sets. An exact formula for computing the Mordukhovich coderivative of normal cone operators to nonlinearly perturbed polyhedral convex sets is established based on a chain rule for the partial second-order subdifferential. This formula leads to a sufficient condition for the local Lipschitz-like property of the solution maps of the generalized equations under nonlinear perturbations.     

Stability of nonlinear viscoelastic systems under stochastic excitation

The dynamic behaviour of viscoelastic system with due account of finite deflections but under condition of small strains is described by the system of nonlinear integro-differential equations. On an example of a thin plate subjected to loads, which are assumed as random wide-band stationary noises and applied in the plate plane, the stability of nonlinear systems is considered. The stability in a case of finite deflections of the plate is considered as stability with respect to statistical moments of perturbations and almost sure stability. For the solution of the problem, a numerical method is offered, which is based on the statistical simulation of input stochastic stationary processes, which are assumed in the form of Gaussian ”colored” noises, and on the numerical solution of integro-differential or differential equations. The conclusion about the stability of the considered system is made on the basis of Lyapunov exponents. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability analysis of conformable fractional-order nonlinear systems

In this paper, we study the stability and asymptotic stability of conformable fractional-order nonlinear systems by using Lyapunov function.     

Stability and sensitivity analysis in multiobjective nonlinear programming

In this paper some results on stability and sensitivity analysis in multiobjective nonlinear programming are surveyed. Given a family of parametrized multiobjective programming problems, the perturbation map is defined as the set-valued map which associates to each parameter value the set of minimal points of the perturbed feasible set with respect to an ordering convex cone. The behavior of the perturbation map is analyzed both qualitatively and quantitatively. First, some sufficient conditions which guarantee the upper and lower semicontinuity of the perturbation map are provided. The contingent derivatives of the perturbation map are also studied. Moreover, it is shown that the results can be refined in the convex case.     

Stability analysis of nonlinear functional differential and functional equations

Sufficient conditions for the stability and asymptotic stability of the theoretical solutions to nonlinear systems of functional differential and functional equations are derived.     

Stability analysis based on nonlinear inhomogeneous approximation

The asymptotic stability of zero solutions for essentially nonlinear systems of differential equations in triangular inhomogeneous approximation is studied. Conditions under which perturbations do not affect the asymptotic stability of the zero solution are determined by using the direct Lyapunov method. Stability criteria are stated in the form of inequalities between perturbation orders and the orders of homogeneity of functions involved in the nonlinear approximation system under consideration.     

Stability conditions of nonlinear distributed processes under parametric excitation

We derive a stability criterion relative to a given measure on a finite time interval for distributed processes under parametric excitation. The corresponding theorem is proved by the comparison method combined with Lyapunov second method. Treating time as a parameter, we use the extremal properties of the Rayleigh quotient for self-adjoint operators in a Hilbert space, which in turn involves solving the eigenvalue problem generated by the linear operators corresponding to the original problem. The results are applied to establish sufficient conditions of technical stability relative to a given measure in the nonlinear problem of a hinged pole under the action of a continuous longitudinal force.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 107–116, 1986.     

Stability analysis in a nonlinear ecological model

A high dimension predator-prey model is considered in this paper. Some novel criteria are established for the existence and global asymptotic stability of a unique equilibrium of such model. The approaches are based on fixed point theory, matrix spectral theory and Lyapunov functional. The existence and stability conditions given in terms of spectral radius of explicit matrices are better than conditions obtained by using classic norms. Finally, an example and its simulations show the feasibility of our results.     

Stability analysis of age-structured disabled population dynamics

In this note, we present a characteristic of an ordered Banach space with base and then give a necessary and sufficient condition for the disabled population system to decay exponentially.