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Perturbed linear rough differential equations

Laure Coutin, Antoine Lejay (2014)

Annales mathématiques Blaise Pascal

We study linear rough differential equations and we solve perturbed linear rough differential equations using the Duhamel principle. These results provide us with a key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebras more general than tensor algebras and to consider extensions of classical results such as the Magnus...

Polynomial algebra of constants of the Lotka-Volterra system

Jean Moulin Ollagnier, Andrzej Nowicki (1999)

Colloquium Mathematicae

Let k be a field of characteristic zero. We describe the kernel of any quadratic homogeneous derivation d:k[x,y,z] → k[x,y,z] of the form d = x ( C y + z ) ∂ ∂ x + y ( A z + x ) ∂ ∂ y + z ( B x + y ) ∂ ∂ z , called the Lotka-Volterra derivation, where A,B,C ∈ k.

Principal solutions and transformations of linear Hamiltonian systems

Ondřej Došlý (1992)

Archivum Mathematicum

Sufficient conditions are given which guarantee that the linear transformation converting a given linear Hamiltonian system into another system of the same form transforms principal (antiprincipal) solutions into principal (antiprincipal) solutions.

Psi-series of quadratic vector fields on the plane.

Amadeu Delshams, Arnau Mir (1997)

Publicacions Matemàtiques

Psi-series (i.e., logarithmic series) for the solutions of quadratic vector fields on the plane are considered. Its existence and convergence is studied, and an algorithm for the location of logarithmic singularities is developed. Moreover, the relationship between psi-series and non-integrability is stressed and in particular it is proved that quadratic systems with psi-series that are not Laurent series do not have an algebraic first integral. Besides, a criterion about non-existence of an analytic...

Réversibilité et classification des centres nilpotents

Michel Berthier, Robert Moussu (1994)

Annales de l'institut Fourier

Nous considérons un germe ω de 1-forme analytique dans ℝ 2 , 0 dont le 1-jet est y d y . Nous montrons que si l’équation ω = 0 définit un centre (i.e toutes les courbes solutions sont des cycles) il existe une involution analytique de ℝ 2 , 0 préservant le portrait de phase du système. Géométriquement ceci signifie que les centres analytiques nilpotents sont obtenus par image réciproque par des applications pli. Un théorème de conjugaison équivariante permet d’obtenir une classification complète de ces centres.

Smoothness Properties of Solutions of Caputo-Type Fractional Differential Equations

Diethelm, Kai (2007)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 26A33, 34A25, 45D05, 45E10We consider ordinary fractional differential equations with Caputo-type differential operators with smooth right-hand sides. In various places in the literature one can find the statement that such equations cannot have smooth solutions. We prove that this is wrong, and we give a full characterization of the situations where smooth solutions exist. The results can be extended to a class of weakly singular Volterra integral equations.

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