Necessary and sufficient conditions of extremum for polynomials and power series in the case of two variables

The present paper is a continuation of the author's previous works, in which necessary and sufficient local extrema at a stationary point of a polynomial or a power series (and thus of an analytic function) are given. It is known that for the case of one variable, the necessary and sufficient conditions of the extremum are closing, i.e., they can be formulated as a single condition. The next most complicated case is the case with two variables, which is the one considered in this paper. In this case, many procedures, to which the verification of necessary and sufficient conditions is reduced, are based on the computation of real roots of a polynomial from one variable, as well as on the solution of some other rather simple practically realizable problems. An algorithm based on these procedures is described. Nevertheless, there are still cases where this algorithm "doesn't work". For such cases we propose the method of "substitution of polynomials with uncertain coefficients", using which, in particular, we have described an algorithm that allows us to unambiguously answer the question about the presence of a local minimum at a stationary point for a polynomial that is the sum of two A-quasi-homogeneous forms, where A - is a two-dimensional vector, whose components are natural numbers.Comment: 30 pages, 2 figure

Delayed exchange of stabilities in a class of singularly perturbed parabolic problems

We consider a class of singularly perturbed parabolic problems in case of exchange of stabilities, that is, the corresponding degenerate equation has two intersecting roots. By means of the technique of asymptotic lower and upper solutions we prove that the considered initial-boundary value problem has a unique solution exhibiting the phenomenon of delayed exchange of stabilities. Thus, the problem under consideration has a canard solution

Immediate exchange of stabilities in singularly perturbed systems

We study the initial value problem for singularly perturbed systems of ordinary differential equations whose associated systems have two transversally intersecting families of equilibria (transcritical bifurcation) which exchange their stabilities. By means of the method of upper and lower solutions we derive a sufficient condition for the solution of the initial value problem to exhibit an immediate exchange of stabilities. Concerning its asymptotic behavior with respect to ε we prove that an immediate exchange of stabilities implies a change of the asymptotic behavior from 0(ε) to 0(√ε) near the point of exchange of stabilities

Delayed exchange of stabilities in singularly perturbed systems

We consider a scalar nonautonomous singularly perturbed differential equation whose degenerate equation has two solutions which intersect at some point. These solutions represent families of equilibria of the associated equation where at least one of these families loses its stability at the intersection point. We study the behavior of the solution of an initial value problem of the singularly perturbed equation in dependence on the small parameter. We assume that the solution stays at the beginning near a stable branch of equilibria of the associated system where this branch loses its stability at some critical time tc. By means of the method of upper and lower solutions we determine the asymptotic delay t* of the solution for leaving the unstable branch. The obtained result holds for the case of transcritical bifurcation as well as for the case of pitchfork bifurcation. We consider some examples where we prove that a well-known result due to N.R. Lebovitz and R.J. Schaar about an immediate exchange of stabilities cannot be applied to singularly perturbed systems whose right hand side depends on ε

On immediate-delayed exchange of stabilities and periodic forced canards

We study scalar singularly perturbed non-autonomous ordinary differential equations whose associated equations feature the property of exchange of stabilities, i.e., the set of their equilibria consists of at least two intersecting curves. By means of the method of asymptotic lower and upper solutions we derive conditions guaranteeing that the solution of initial value problems exhibit the phenomenon of immediate exchange of stabilities as well as the phenomenon of delayed exchange of stabilities. We use this result to prove the existence of forced canard solutions

Exponential asymptotic stability via Krein--Rutman theorem for singularly perturbed parabolic periodic Dirichlet problems

We consider singularly perturbed semilinear parabolic periodic problems and assume the existence of a family of solutions. We present an approach to establish the exponential asymptotic stability of these solutions by means of a special class of lower and upper solutions. The proof is based on a corollary of the Krein-Rutman theorem

Analytic-numerical investigation of delayed exchange of stabilities in singularly perturbed parabolic problems

We consider a class of singularly perturbed parabolic problem in case of exchange of stabilities, that is, the corresponding degenerate equation has two intersecting roots. We present an analytic result about the phenomenon of delayed exchange of stabilities and compare it with numerical investigations of some examples

On existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations

We consider a singularly perturbed parabolic periodic boundary value problem for a reaction-advection-diffusion equation. We construct the interior layer type formal asymptotics and propose a modified procedure to get asymptotic lower and upper solutions. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution with an interior layer and estimate the accuracy of its asymptotics. Moreover, we are able to establish the asymptotic stability of this solution with interior laye

Singularly perturbed boundary value problems in case of exchange of stablities

We consider a mixed boundary value problem for a system of two second order nonlinear differential equations where one equation is singularly perturbed. We assume that the associated equation has two intersecting families of equilibria. This property excludes the application of standard results. By means of the method of upper and lower solutions we prove the existence of a solution of the boundary value problem and determine its asymptotic behavior with respect to the small parameter. The results can be used to study differential systems modelling bimolecular reactions with fast reaction rates

Singularly perturbed reaction-diffusion systems in case of exchange of stabilities

We study singularly perturbed elliptic and parabolic differential equations under the assumption that the associated equation has intersecting families of equilibria (exchange of stabilities). We prove by means of the method of asymptotic lower and upper solutions that the asymptotic behavior with respect to the small parameter changes near the curve of exchange of stabilities. The application of that result to systems modelling fast bimolecular reactions in a heterogeneous environment implies a transition layer (jumping behavior) of the reaction rate. This behavior has to be taken into account for identification problems in reaction systems