Parametric Poincaré-Perron theorem with applications

We prove a parametric generalization of the classical Poincaré-Perron theorem on stabilizing recurrence relations, where we assume that the varying coefficients of a recurrence depend on auxiliary parameters and converge uniformly in these parameters to their limiting values. As an application, we study convergence of the ratios of families of functions satisfying finite recurrence relations with varying functional coefficients. For example, we explicitly describe the asymptotic ratio for two classes of biorthogonal polynomials introduced by Ismail and Masson.     

Poincaré–Sobolev equations in the hyperbolic space

We study the a priori estimates, existence/nonexistence of radial sign changing solution, and the Palais–Smale characterisation of the problem ${-\Delta_{{\mathbb B}^{N}}u - \lambda u = |u|^{p-1}u, u\in H^1({\mathbb B}^{N})}$ in the hyperbolic space ${{\mathbb B}^{N}}$ where ${1 < p\leq\frac{N+2}{N-2}}$ . We will also prove the existence of sign changing solution to the Hardy–Sobolev–Mazya equation and the critical Grushin problem.     

Well-posedness of the Poincar�� problem for a multidimensional hyperbolic equation with the Chaplygin operator

In this paper, the unique solvability of the Poincaré problem for multidimensional hyperbolic equation with the Chaplygin operator in the domain with a deviation from the characteristic is proved. In the theory of partial differential equations of the hyperbolic type, boundary-value problems with the data on the whole boundary of the domain serve as an example of problems that are not well posed [3].     

Generalized doubling meets Poincaré

We establish a modified segment inequality on metric spaces that satisfy a generalized volume doubling property. This leads to Sobolev and Poincaré inequalities for such spaces. We also give several examples of spaces that satisfy the generalized doubling condition.     

Orbites hétérocliniques au voisinage d'une bifurcation de Poincaré dégénérée pour les champs de vecteurs de ℝ2

We consider a ²-ordinary differential equation, where the fixed point (0, 0) presents a degenerate Poincaré-bifurcation of resonancek(=2k) and dominanced(=k–1). We prove the existence of a 2-dimensional linear manifoldV in the parameter space. , on which the perturbed dominant differential system (SD) possesses heteroclinic orbits between fixed points. The numerical continuation of the local stable or unstable manifolds of the saddle fixed points shows that for any neighborhood, in , of a point ofV corresponding to a saddle heteroclinic orbit, there exists only one stable (resp. unstable) periodic orbit close to the stable — in the Andronov sense [1]-(resp. unstable) heteroclinic orbit. Applications are given fork=4 andk=6.     

Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles

In the present paper, we advance considerably the current knowledge on the topic of bifurcations of heteroclinic cycles for smooth, meaning C ^∞, parametrized families {g _t ∣t∈ℝ} of surface diffeomorphisms. We assume that a quadratic tangency q is formed at t=0 between the stable and unstable lines of two periodic points, not belonging to the same orbit, of a (uniformly hyperbolic) horseshoe K (see an example at the Introduction) and that such lines cross each other with positive relative speed as the parameter evolves, starting at t=0 and the point q. We also assume that, in some neighborhood W of K and of the orbit of tangency o(q), the maximal invariant set for g ₀=g _t=0 is K∪o(q), where o(q) denotes the orbit of q for g ₀. We then prove that, when the Hausdorff dimension HD(K) is bigger than one, but not much bigger (see (H.4) in Section 1.2 for a precise statement), then for most t, |t| small, g _t is a non-uniformly hyperbolic horseshoe in W, and so g _t has no attractors in W. Most t, and thus most g _t , here means that t is taken in a set of parameter values with Lebesgue density one at t=0.     

A Poincaré-Sobolev type inequality on compact Riemannian manifolds with boundary

In this paper we study a Poincaré-Sobolev type inequality on compact Riemannian n-manifolds with boundary where the exponent growth is critical. Two constants have to be determined. We show that, contrary to the classical Sobolev inequality, the first best constant in this inequality does not depend on the dimension only, but depends on the geometry. It can be represented as the minimum of a given energy functional. We study the nonlinear PDE associated to this functional which involves the geometry of the boundary. For a star-shaped domain D in whose boundary has positive Ricci curvature, we give explicitly two Sobolev constants corresponding to the embedding in . This result is used to obtain an explicit geometrical lower bound for . Received November 15, 1999 / Published online April 12, 2001     

Time symmetry preserving perturbations of systems,and Poincaré mappings

In the present paper, we obtain necessary and sufficient conditions under which two differential systems have the same symmetries described by a reflecting function. Under these conditions, the systems in question have a common shift operator along solutions of these systems on a symmetric time interval [?ω, ω]. Therefore, the mappings over the period [?ω, ω] coincide for such systems provided that these systems are 2ω-periodic.     

Transverse intersections between invariant manifolds of doubly hyperbolic invariant tori, via the Poincaré-Mel’nikov method

We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptic-hyperbolic type, having a homoclinic orbit. More precisely, we consider an (n + 2)-degree-of-freedom near integrable Hamiltonian with n centers and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the center manifold near the equilibrium, there is a Cantorian family of hyperbolic KAM tori, and we study the homoclinic intersections between the stable and unstable manifolds associated to such tori. We establish that, in general, the manifolds intersect along transverse homoclinic orbits. In a more concrete model, such homoclinic orbits can be detected, in a first approximation, from nondegenerate critical points of a Mel’nikov potential. We provide bounds for the number of transverse homoclinic orbits using that, in general, the potential will be a Morse function (which gives a lower bound) and can be approximated by a trigonometric polynomial (which gives an upper bound).     

Melnikov函数和Poincaré映射

本文中我们给出了Melnikov函数和Poincaré映射的关系,从而给出了Melnikov方法的新的证明.本文的优点是给出了更明确的解,并把次谐分支的Melnikov函数与稳定流形与不稳定流形横截相交的Melnikov函数统一成为一个公式.     

Algebra retracts and Poincaré-series

Let RA be a local inclusion of noetherian local rings. Assume that R is an algebra retract of A with the local retraction mapping p:AR. Let M be an R-module of finite type. Considering M as A-module via p we get P _M ^A =P _R ^A P _M ^R (Th. 1), where P _N ^S denotes the Poincaréseries of an S-module N. This result is used to give a simple proof of Th. 1 in [2], Also an application to fibre products of rings is given (Th. 2), generalizing slightly a result due to A. Dress and H. Krämer, see [1].     

A quasi-periodic Poincaré's theorem

We study the persistence of invariant tori on resonant surfaces of a nearly integrable Hamiltonian system under the usual Kolmogorov non-degenerate condition. By introducing a quasi-linear iterative scheme to deal with small divisors, we generalize the Poincaré theorem on the maximal resonance case (i.e., the periodic case) to the general resonance case (i.e., the quasi-periodic case) by showing the persistence of majority of invariant tori associated to non-degenerate relative equilibria on any resonant surface.The first author was partially supported by NSFC grant 19971042, the National 973 Project of China: Nonlinearity, and the outstanding young's project of the Ministry of Education of China.The second author was partially supported by NSF grant DMS9803581.Mathematics Subject Classification (2000): Primary 58F05, 58F27, 58F30     

关于Poincaré定理的注记

§1.引言命p,q,n是三个正整数,p+q=n,通常,从考虑n维定向组合同调流形K及其对偶复形K~＊的定向元素之相交指数出发,可以证明(见[2],467-483页).定理1.复形K的p维上同调群~PH~G(K)与复形K~＊的q维同调群~qH_G(K~＊)彼此同构.由于K和K~＊具有同一的重心重分K′,而同调群是重心重分的不变量,所以,从定理     

Inégalités de Poincaré cinétiques

In this Note we prove Poincaré type inequalities for a family of kinetic equations. We apply this inequality to the variational solution of a linear kinetic model by generalizing the STILS method (Azerad, 1996 [1]; Azerad and Pousin, 1996 [2]) to a kinetic setting.