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26. CMB 2007 (vol 50 pp. 377)

Gutierrez, C.; Jarque, X.; Llibre, J.; Teixeira, M. A.
 Global Injectivity of $C^1$ Maps of the Real Plane, Inseparable Leaves and the Palais--Smale Condition We study two sufficient conditions that imply global injectivity for a $C^1$ map $X\colon \R^2\to \R^2$ such that its Jacobian at any point of $\R^2$ is not zero. One is based on the notion of half-Reeb component and the other on the Palais--Smale condition. We improve the first condition using the notion of inseparable leaves. We provide a new proof of the sufficiency of the second condition. We prove that both conditions are not equivalent, more precisely we show that the Palais--Smale condition implies the nonexistence of inseparable leaves, but the converse is not true. Finally, we show that the Palais--Smale condition it is not a necessary condition for the global injectivity of the map $X$. Categories:34C35, 34H05

27. CMB 2006 (vol 49 pp. 170)

Atkins, Richard
 The Geometry of $d^{2}y^{1}/dt^{2} = f(y, \dot{y},t) \; \text{and} \; d^{2}y^{2}/dt^{2} = g(y,\dot{y},t)$, and Euclidean Spaces This paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler--Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations $d^{2}y^{1}/dt^{2} = f(y,\dot{y},t)$ and $d^{2}y^{2}/dt^{2} = g(y,\dot{y},t)$ be reparameterized by $t\rightarrow T(y,t)$ so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions. Category:34A26

28. CMB 2005 (vol 48 pp. 405)

Froese, Richard
 Liouville's Theorem in the Radially Symmetric Case We present a very short proof of Liouville's theorem for solutions to a non-uniformly elliptic radially symmetric equation. The proof uses the Ricatti equation satisfied by the Dirichlet to Neumann map. Categories:35B05, 34A30

29. CMB 2002 (vol 45 pp. 355)

Cresson, Jacky
 Obstruction Ã  la linÃ©arisation des champs de vecteurs polynomiaux On explicite une classe de champ de vecteurs polynomiaux non analytiquement lin\'earisables \a l'aide de la correction introduite par \'Ecalle-Vallet. Notamment, on \'etend des r\'esultats de Schuman sur la trivialit\'e des hamiltoniens homog\enes isochrones. We characterize a class of polynomial vector fields which are not analytically linearizable using the correction introduced by \'Ecalle-Vallet. Then, we extend Schuman's result about non existence of isochronous homogenous Hamiltonian systems. Keywords:linÃ©arisation-problÃ¨me du centre-hamiltonien-darboux-champs polynomiauxCategories:34D10, 34D30

30. CMB 2001 (vol 44 pp. 323)

Schuman, Bertrand
 Une classe d'hamiltoniens polynomiaux isochrones Soit $H_0 = \frac{x^2+y^2}{2}$ un hamiltonien isochrone du plan $\Rset^2$. On met en \'evidence une classe d'hamiltoniens isochrones qui sont des perturbations polynomiales de $H_0$. On obtient alors une condition n\'ecessaire d'isochronisme, et un crit\ere de choix pour les hamiltoniens isochrones. On voit ce r\'esultat comme \'etant une g\'en\'eralisation du caract\ere isochrone des perturbations hamiltoniennes homog\`enes consid\'er\'ees dans [L], [P], [S]. Let $H_0 = \frac{x^2+y^2}{2}$ be an isochronous Hamiltonian of the plane $\Rset^2$. We obtain a necessary condition for a system to be isochronous. We can think of this result as a generalization of the isochronous behaviour of the homogeneous polynomial perturbation of the Hamiltonian $H_0$ considered in [L], [P], [S]. Keywords:Hamiltonian system, normal forms, resonance, linearizationCategories:34C20, 58F05, 58F22, 58F30

31. CMB 1998 (vol 41 pp. 207)

Philos, Ch. G.; Sficas, Y. G.
 An oscillation criterion for first order linear delay differential equations A new oscillation criterion is given for the delay differential equation $x'(t)+p(t)x \left(t-\tau(t)\right)=0$, where $p$, $\tau \in \C \left([0,\infty),[0,\infty)\right)$ and the function $T$ defined by $T(t)=t-\tau(t)$, $t\ge 0$ is increasing and such that $\lim_{t\to\infty}T(t)=\infty$. This criterion concerns the case where $\liminf_{t\to\infty} \int_{T(t)}^{t}p(s)\,ds\le \frac{1}{e}$. Keywords:Delay differential equation, oscillationCategory:34K15

32. CMB 1998 (vol 41 pp. 214)

Shackell, John
 On a problem of Rubel concerning the set of functions satisfying all the algebraic differential equations satisfied by a given function For two functions $f$ and $g$, define $g\ll f$ to mean that $g$ satisfies every algebraic differential equation over the constants satisfied by $f$. The order $\ll$ was introduced in one of a set of problems on algebraic differential equations given by the late Lee Rubel. Here we characterise the set of $g$ such that $g\ll f$, when $f$ is a given Liouvillian function. Categories:34A34, 12H05

33. CMB 1998 (vol 41 pp. 23)

Clemence, Dominic P.
 Subordinacy analysis and absolutely continuous spectra for Sturm-Liouville equations with two singular endpoints The Gilbert-Pearson characterization of the spectrum is established for a generalized Sturm-Liouville equation with two singular endpoints. It is also shown that strong absolute continuity for the one singular endpoint problem guarantees absolute continuity for the two singular endpoint problem. As a consequence, we obtain the result that strong nonsubordinacy, at one singular endpoint, of a particular solution guarantees the nonexistence of subordinate solutions at both singular endpoints. Categories:34L05, 34B20, 34B24

34. CMB 1997 (vol 40 pp. 416)

Clemence, Dominic P.
 On the singular behaviour of the Titchmarsh-Weyl $m$-function for the perturbed Hill's equation on the line For the perturbed Hill's equation on the line, $$-\frac{d^2y}{dx^2}+ [P (x) +V (x )] y=Ey,\quad -\infty Categories:34L05, 34B20, 34B24 35. CMB 1997 (vol 40 pp. 448) Kaczynski, Tomasz; Mrozek, Marian  Stable index pairs for discrete dynamical systems A new shorter proof of the existence of index pairs for discrete dynamical systems is given. Moreover, the index pairs defined in that proof are stable with respect to small perturbations of the generating map. The existence of stable index pairs was previously known in the case of diffeomorphisms and flows generated by smooth vector fields but it was an open question in the general discrete case. Categories:54H20, 54C60, 34C35 36. CMB 1997 (vol 40 pp. 276) Chouikha, Raouf  Fonctions elliptiques et Ã©quations diffÃ©rentielles ordinaires In this paper, we detail some results of a previous note concerning a trigonometric expansion of the Weierstrass elliptic function \{\wp(z);\, 2\omega, 2\omega'\}. In particular, this implies its classical Fourier expansion. We use a direct integration method of the ODE$$(E)\left\{\matrix{{d^2u \over dt^2} = P(u, \lambda)\hfill \cr u(0) = \sigma\hfill \cr {du \over dt}(0) = \tau\hfill \cr}\right. where $P(u)$ is a polynomial of degree $n = 2$ or $3$. In this case, the bifurcations of $(E)$ depend on one parameter only. Moreover, this global method seems not to apply to the cases $n > 3$. Categories:33E05, 34A05, 33E20, 33E30, 34A20, 34C23
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