Linearization of class C1 for contractions on Banach spaces

AbstractIn this work we prove a C1-linearization result for contraction diffeomorphisms, near a fixed point, valid in infinite-dimensional Banach spaces. As an intermediate step, we prove a specific result of existence of invariant manifolds, which can be interesting by itself and that was needed on the proof of our main theorem. Our results essentially generalize some classical results by P. Hartman in finite dimensions, and a result of Mora–Sola-Morales in the infinite-dimensional case. It is shown that the result can be applied to some abstract systems of semilinear damped wave equations

Known Results and Open Problems on C1 linearization in Banach Spaces

nuloThe purpose of this paper is to review the results obtained by the authors on linearization of dynamical systems in infinite dimensional Banach spaces, especially in the C1 case, and also to present some open problems that we believe that are still important for the understanding of the theory

Microstructure and morphoquantitative study of pineal gland in santa ines sheep

O presente estudo avaliou os aspectos morfoquantitativos e qualitativos da glândula pineal em fêmeas de ovinos da raça Santa Inês em atividade reprodutiva e anestro fisiológico. Foram utilizados sete ovinos, fêmeas e adultas, coletados em frigoríficos da região de Brasília-DF. Estes não apresentavam sinais clínicos relacionados a afecções do sistema nervoso. Os encéfalos foram retirados e suas dimensões mensuradas. Posteriormente foram seccionados para expor a glândula pineal e mensurar suas dimensões. Os fragmentos do diencéfalo, contendo a glândula pineal, foram submersos em solução aquosa a 20% de formaldeído e posteriormente submetidos a técnicas histológicas convencionais. Para a avaliação morfoquantitativa desta glândula, foram empregadas as colorações Hematoxilia-Eosina, Azul de Toluidina e Tricrômio de Gomori. O encéfalo teve um comprimento médio de 67,25 ± 1,75 mm e a largura média de 58,97 ± 4,0 mm. Já a glândula pineal apresentou o comprimento médio de 6,98 ± 0,79 mm e a largura média de 6,40 ± 1,35 mm. Diante da análise microscópica, foi obtida uma média de 86,27 ± 30,44 pinealócitos por campo. Ao ser aplicado o teste de correlação de Pearson, o número de pinealócitos apresentou fraca correlação linear negativa (r = -0,11) em relação ao comprimento da glândula pineal e uma fraca correlação linear positiva (r = 0,39) em relação à largura da mesma. Portanto o número de pinealócitos apresenta uma maior correlação com a largura da glândula do que com o seu comprimento. Mastócitos estiveram presentes em apenas um animal (14,28%) e as concreções calcareas foram observadas em dois animais (28,57%). A glândula se mostrou envolta por uma cápsula de tecido conjuntivo com ausência de projeções para o parênquima. _________________________________________________________________________________ ABSTRACTThis study examined morphologic aspects of the pineal gland in Santa Ines sheep in anoestrus physiological and reproductive activity. Seven female adult sheep were collected at slaughterhouses in the region of Brasilia-DF, with no clinical signs related to diseases of the nervous system were used. The brains were removed and their dimensions were measured. The brains were sectioned to expose and measure the pineal gland. The fragments of the diencephalon containing the pineal gland were immersed in 20% formaldehyde solution. Subsequently, the fragments were submitted to conventional histological techniques. The Hematoxylin-eosin, Toluidine blue and Gomori trichrome staining were used for a morphoquantitative analysis. The brain had an average length of 67.25 ± 1.75 mm and average width of 58.97 ± 4.0 mm. The pineal gland had an average length of 6.98 ± 0.79 mm and average width of 6.40 ± 1.35 mm. The quantitative analysis showed an average 86.27 ± 30.44 of pinealocytes per field. According to the Pearson correlation test, the number of pinealocytes showed a weak negative linear correlation (r = -0.11) with the length of the pineal gland and a weak positive linear correlation (r = 0.39) with the width. Therefore, the number of pinealocytes has a stronger correlation with the width of the gland compared to the length. Mastocytes were present in only one animal (14.28%). Calcareous concretions were observed in two animals (28.57%). The connective tissue formed the capsule surrounding the gland with no projections into the parenchyma

Invertible Contractions and Asymptotically Stable ODE’S that are not C1-Linearizable

We present an example of a contraction diffeomorphism in infinite dimensions that is not C1-linearizable, and we construct a regular ordinary differential equation in a Hilbert space whose time-one map is that diffeomorphism. With this we have an example of an asymptotically stable ODE that is not C1-conjugate to its linear part

Uniform Ultimate Boundedness and Synchronization for Nonautonomous Equations

In this work we present some mathematical methods for the study of synchronization of two similar nonautonomous systems of differential equations. We give sufficient conditions to control the coupling devices in order to accomplish the synchronization, even when each of the systems moves chaotically. Some examples that include systems of coupled lasers and coupled mechanical systems are discussed.Partially supported by ARO DAAH-0493-G-0199, U.S.A. and by FAPESP and CNPq, Brazil

On the Hartman-Grobman Theorem with Parameters

The Hartman-Grobman Theorem of linearization is extended to families of dynamical systems in a Banach space X, depending continuously on parameters. We prove that the conjugacy also changes continuously. The cases of nonlinear maps and flows are considered, and both in global and local versions, but global in the parameters. To use a special version of the Banach-Caccioppoli Theorem we introduce equivalent norms on X depending on the parameters. The functional setting is suitable for applications to some nonlinear evolution partial differential equations like the nonlinear beam equation.CNPq[301994/85-4]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Programa Pronex-Projeto Tematico CNPq-FAPESP[2003/10042-0]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)CAPES[DGU 127/07]MEC-SpainMEC, Spain[MTM2005-07660-C02-01]MEC-SpainMEC, Spain[PHB2006-006

Differentiability with respect to parameters in global smooth linearization

Let XX be a Banach space and T¿:X¿XT¿:X¿X a family of invertible contractions, T¿=L¿+f¿T¿=L¿+f¿, where L¿L¿ is linear and f¿f¿ is nonlinear with f¿(0)=0f¿(0)=0. We give conditions for the existence of a family of global linearization maps H¿H¿, such that View the MathML sourceH¿°T¿°H¿-1=L¿, with a smooth dependence on ¿. The results depend strongly on the choice of some appropriate spaces of maps, adapted norms and the use of a specific fixed point theorem with smooth dependence on parametersPeer Reviewe