Stability results for a class of non-linear parabolic equations

Summary We study the asymptotic behaviour of the solutions of the equation u_t=Au+λu−|u|^αu. Denoting by λ₀ the principal eigenvalue of the second-order differential operator A, we shall prove that if λ ⩽ λ₀ the only equilibrium solution, namely zero, is asymptotically stable, whereas, if λ>λ₀, the nontrivial equilibrium solutions without internal zeros are asymptotically stable. Attractivity and stability are proved both in the L²-norm and in the H ₀ ¹ -norm. Entrata in Redazione il 15 ottobre 1976.     

On the influence of parameters determining anisotropic spaces of smooth functions on characteristics of the embedding theorems

An anisotropic Sobolev and Nikol'skii-Besov space on a domain G is determined by its integro-differential (shortly, ID) parameters. On the other hand, the geometry of G is characterized by the set Λ(G) of all vectors λ=(λ₁,..., λ_n) such that G satisfies the λ-horn condition. We study the dependence of the totality of possible embeddings upon the set Λ(G) and theID-parameters of the space. We consider only embeddings with q≥pⁱ, where pⁱ are the integral parameters of the space and q is the integral embedding parameter. For a given space, we introduce its initial matrix A₀ determined by theID-parameters. A₀ turns out to be a Z-matrix. On the basis of a natural classification of Z-matrices, a classification of anisotropic spaces is introduced. This classification allows one to restate the existence of an embedding with q≥pⁱ in terms of certain specific properties of A₀. Let A₀ be a nondegenerate M-matrix. Any vector λ∈Λ(G) gives rise to a certain set of admissible values of the embedding parameters. We call λ optimal if this set is the largest possible. It turns out that the optimal vector λ _G ^* is determined by Λ(G) and A₀, and may be found by a linear optimization procedure. The following cases are possible: a) , b) , c) λ _G ^* does not exist. In case a) the set of admissible values of the embedding parameters is the biggest, while in case c) no embeddings with q≥pⁱ exist. In case b) the so-called saturation phenomenon occurs, i.e., certain variations of some differential parameters of the space do not change the set of admissible values of the embedding parameters. The latter fact has some applications to the problem of extension of all functions belonging to the given space from G to Eⁿ. Bibliography: 20 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 22–94. Translated by A. A. Mekler.     

Random walks,Kleinian groups,and bifurcation currents

Let (ρ _λ ) _λ∈Λ be a holomorphic family of representations of a finitely generated group G into PSL(2,ℂ), parameterized by a complex manifold Λ. We define a notion of bifurcation current in this context, that is, a positive closed current on Λ describing the bifurcations of this family of representations in a quantitative sense. It is the analogue of the bifurcation current introduced by DeMarco for holomorphic families of rational mappings on ℙ¹. Our definition relies on the theory of random products of matrices, so it depends on the choice of a probability measure μ on G.     

Bifurcations of rough heteroclinic loop with two saddle points

The bifurcation problems of rough 2-point-loop are studied for the case p11 > λ11, p21 < λ21, P11p21 <λ111λ21. where - pi1 < 0 and λi1 > 0 are the pair of principal eigenvalues of unperturbed system at saddle point pi, i = 1,2. Under the transversal and nontwisted conditions, the authors obtain some results of the existence of one 1-periodic orbit, one 1-periodic and one 1-homoclinic loop, two 1-periodic orbits and one 2-fold 1-periodic orbit. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graph is drawn.     

Decay properties and quasi-stationary distributions for stopped Markovian bulk-arrival and bulk-service queues

We consider decay properties including the decay parameter, invariant measures and quasi-stationary distributions for a Markovian bulk-arrival and bulk-service queue which stops when the waiting line is empty. Investigating such a model is crucial for understanding the busy period and other related properties of the Markovian bulk-arrival and bulk-service queuing processes. The exact value of the decay parameter λ _C is first obtained. We show that the decay parameter can be easily expressed explicitly. The invariant measures and quasi-distributions are then revealed. We show that there exists a family of invariant measures indexed by λ∈[0,λ _C ]. We then show that under some mild conditions there exists a family of quasi-stationary distributions also indexed by λ∈[0,λ _C ]. The generating functions of these invariant measures and quasi-stationary distributions are presented. We further show that this stopped Markovian bulk-arrival and bulk-service queueing model is always λ _C -transient. Some deep properties regarding λ _C -transience are examined and revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper.     

Families of periodic orbits in resonant reversible systems

We study the dynamics near an equilibrium point p ₀ of a Z ₂(ℝ)-reversible vector field in ℝ²ⁿ with reversing symmetry R satisfying R ² = I and dimFix(R) = n. We deal with one-parameter families of such systems X _λ such that X ₀ presents at p ₀ a degenerate resonance of type 0: p: q. We are assuming that the linearized system of X ₀ (at p ₀) has as eigenvalues: λ₁ = 0 and λ _j = ±iα _j , j = 2, … n. Our main concern is to find conditions for the existence of one-parameter families of periodic orbits near the equilibrium.     

On the period of the limit cycles appearing in one-parameter bifurcations

The generic isolated bifurcations for one-parameter families of smooth planar vector fields {X_μ} which give rise to periodic orbits are: the Andronov-Hopf bifurcation, the bifurcation from a semi-stable periodic orbit, the saddle-node loop bifurcation and the saddle loop bifurcation. In this paper we obtain the dominant term of the asymptotic behaviour of the period of the limit cycles appearing in each of these bifurcations in terms of μ when we are near the bifurcation. The method used to study the first two bifurcations is also used to solve the same problem in another two situations: a generalization of the Andronov-Hopf bifurcation to vector fields starting with a special monodromic jet; and the Hopf bifurcation at infinity for families of polynomial vector fields.     

Eigenvalue bifurcation in multiparameter families of non-self-adjoint operator matrices

We consider two-point non-self-adjoint boundary eigenvalue problems for linear matrix differential operators. The coefficient matrices in the differential expressions and the matrix boundary conditions are assumed to depend analytically on the complex spectral parameter λ and on the vector of real physical parameters p. We study perturbations of semi-simple multiple eigenvalues as well as perturbations of non-derogatory eigenvalues under small variations of p. Explicit formulae describing the bifurcation of the eigenvalues are derived. Application to the problem of excitation of unstable modes in rotating continua such as spherically symmetric MHD α ²-dynamo and circular string demonstrates the efficiency and applicability of the approach.     

Interior transition layers of solutions to the perturbed elliptic Sine-Gordon equation on an interval

We consider the perturbed elliptic Sine-Gordon equation on an interval-u″t+γsinu(t)=μf(u(t)),t ∈I := (-T, T),u(t) > 0,t ∈I,u(±T)=0 where λ, μ>0 are parameters andT>0 is a constant. By applying variational methods subject to the constraint depending on λ, we obtain eigenpairs (μ,u)=(μ(λ),u _λ) which solve this eigenvalue problem for a given λ>0. Then we study the asymptotic behavior ofu _λ and μ(λ) as λ→∞. Especially, we study the location of interior transition layers ofu _λ as λ→∞. This research has been supported by the Japan Society for the Promotion of Science.     

Decay parameter and related properties of 2-type branching processes

We consider the decay parameter, invariant measures/vectors and quasi-stationary dis- tributions for 2-type Markov branching processes. Investigating such properties is crucial in realizing life period of branching models. In this paper, some important properties of the generating functions for 2-type Markov branching q-matrix are firstly investigated in detail. The exact value of the decay parameter λC of such model is given for the communicating class C = Z+2 \ 0. It is shown that this λC can be directly ...     

Monotone maximum likelihood estimate for hazard function

A wide class of reliability theory models or lifetime data can be described as follows. Assume that the lifetime distribution function is F(t, θ)=F₀(λ(θ)t), where θ is the parameter characterizing some inner properties of a product and λ(θ) is an unknown increasing function. The paper deals with methods of estimation of λ(θ) from the sample (t _i ,θ _i ),i = 1, ...,n, for the case of exponentialF ₀. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 46–51, Perm, 1991.     

Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary

In this paper we perform an extensive study of the existence, uniqueness (or multiplicity) and stability of nonnegative solutions to the semilinear elliptic equation − Δu = λ u − u ^p in Ω, with the nonlinear boundary condition ∂u/∂ν = u ^r on ∂Ω. Here Ω is a smooth bounded domain of with outward unit normal ν, λ is a real parameter and p, r > 0. We also give the precise behavior of solutions for large |λ| in the cases where they exist. The proofs are mainly based on bifurcation techniques, sub-supersolutions and variational methods.      

A sturm-liouville problem with physical and spectral parameters in boundary conditions

We study the spectral probleml(u)=−u″+q(x)u(x)=λu(x),u′(0)=0, u′(π)=mλu(π), where λ andm are a spectral and a physical parameter. Form<0, we associate with the problem a self-adjoint operator in Pontryagin space II₁. Using this fact and developing analytic methods of the theory of Sturm-Liouville operators, we study the dynamics of eigenvalues and eigenfunctions of the problems asm→−0. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 163–172, August, 1999.     

Normal Vectors on Manifolds of Critical Points for Parametric Robustness of Equilibrium Solutions of ODE Systems

Summary. {Equilibrium solutions of systems of parameterized ordinary differential equations \dot x = f(x, α) , x ∈ R ⁿ , α∈ R ^m can be characterized by their parametric distance to manifolds of critical solutions at which the behavior of the system changes qualitatively. Critical points of interest are bifurcation points and points at which state variable constraints or output constraints are violated. We use normal vectors on manifolds of critical points to measure the distance between these manifolds and equilibrium solutions as suggested in I. Dobson [J. Nonlinear Sci., 3:307-327, 1993], where systems of equations to calculate normal vectors on codimension-1 bifurcations were presented. We present a scheme to derive systems of equations to calculate normal vectors on manifolds of critical points which (i) generalizes to bifurcations of arbitrary codimension, (ii) can be applied to state variable constraints and output constraints, (iii) implies that the normal vector defining system of equations is of size c ₁ n+ c ₂ m+ c ₃ , c _i ∈ R , i.e., no bilinear terms nm or higher-order terms occur, (iv) reduces the number of equations for normal vectors on Hopf bifurcation manifolds compared to previous work, and (v) simplifies the proof of regularity of the normal vector system. As an application of this scheme, we present systems of equations for normal vectors to manifolds of output/state variable constraints, to manifolds of saddle-node, Hopf, cusp, and isola bifurcations, and we give illustrative examples of their use in engineering applications.} Received September 27, 2000; accepted December 10, 2001 Online publication March 11, 2002 Communicated by Y. G. Kevrekidis Communicated by Y. G. Kevrekidis rid="     

New asymptotic formula for eigenvalues of nonlinear Sturm-Liouville problems

We study the nonlinear Sturm-Liouville problem

Control of the Planar Takens–Bogdanov Bifurcation with Applications

It is well-known that on a versal deformation of the Takens–Bogdanov bifurcation is possible to find dynamical systems that undergo saddle-node, Hopf, and homoclinic bifurcations. In this document a nonlinear control system in the plane is considered, whose nominal vector field has a double-zero eigenvalue, and then the idea is to find under which conditions there exists a scalar control law such that be possible establish a priori, that the closed-loop system undergoes any of the three bifurcations: saddle-node, Hopf or homoclinic. We will say then that such system undergoes the controllable Takens–Bogdanov bifurcation. Applications of this result to the averaged forced van der Pol oscillator, a population dynamics, and adaptive control systems are discussed.     

Statistical inference for a bivariate exponential distribution based on grouped data

Consider the bivarate exponential distribution due to Marshall and Olkin [2], whose survival function is F(x,y)=exp[-λ1x-λ2y-λ12 max(x,y)] (x≥0,y≥0)with unknown pexameters λ1＞0,λ2＞0 andλ12≥0. Based on grouped data, a new estimstor for ,λ1, ,λ2 and ,λ12 is derlved and its asymptotic perties are discussed.Bealdes, some test procedures of equalmarginals and independence are gven. A simulation result is given, too.     

Maxwell equations in complex form,squaring procedure and separating the variables

The Riemann–Silberstein–Majorana–Oppenheimer complex approach to the Maxwell electrodynamics is investigated within the matrix formalism. Within the squaring procedure we construct four types of formal solutions of the Maxwell equations on the base of scalar D’Alembert solutions. General problem of separating physical electromagnetic solutions in the linear space λ₀Ψ⁰ + λ₁Ψ¹ + λ₂Ψ² + λ₃Ψ³ is investigated, the Maxwell equations reduce to a new form including parameters λ _a . Several particular cases, plane waves and cylindrical waves, are considered in detail. Possible extension of the technique to a curved space–time models is discussed.     

The semiflow of a reaction diffusion equation with a singular potential

We study the semiflow defined by a semilinear parabolic equation with a singular square potential . It is known that the Hardy-Poincaré inequality and its improved versions, have a prominent role on the definition of the natural phase space. Our study concerns the case 0 < μ ≤ μ*, where μ* is the optimal constant for the Hardy-Poincaré inequality. On a bounded domain of , we justify the global bifurcation of nontrivial equilibrium solutions for a reaction term f(s) = λs − |s|^2γ s, with λ as a bifurcation parameter. We remark some qualitative differences of the branches in the subcritical case μ < μ* and the critical case μ = μ*. The global bifurcation result is used to show that any solution , initiating form initial data tends to the unique nonnegative equilibrium.