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Autor: =Cabot, Alexandre
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Publicación seriada
Referencias AnalíticasReferencias Analíticas
Autor: Cabot, Alexandre ; Engler, Hans ; Gadat, Sebastien
Título: On the long time behavior of second order differential equations with asymptotically samll dissipation
Páginas/Colación: pp. 5983-6017
Fecha: November 2009
Transactions of the American Mathematical Society Vol. 361, no.11 November 2009
Información de existenciaInformación de existencia

Palabras Claves: Palabras: ASYMPTOTIC BEHAVIOR ASYMPTOTIC BEHAVIOR, Palabras: AVERAGED GRADIENT SYSTEM AVERAGED GRADIENT SYSTEM, Palabras: BESSEL EQUATION BESSEL EQUATION, Palabras: DIFFERENTIAL EQUATION DIFFERENTIAL EQUATION, Palabras: DISSIPATIVE DYNAMICAL SYSTEM DISSIPATIVE DYNAMICAL SYSTEM, Palabras: VANISHING DAMPING VANISHING DAMPING

Resumen
In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds

We investigate the asymptotic properties as $ t\to \infty$of the following differential equation in the Hilbert space $ H$:

$\displaystyle (\mathcal{S})\qquad\qquad\qquad\ddot{x}(t)+a(t)\dot{x}(t)+ \nabla G(x(t))=0, \quad t\geq 0,\qquad\qquad\qquad\qquad\quad$

where the map $ a:\mathbb{R}_+\to \mathbb{R}_+$is nonincreasing and the potential $ G:H\to \mathbb{R}$is of class $ \mathcal{C}^1$. If the coefficient $ a(t)$is constant and positive, we recover the so-called ``Heavy Ball with Friction'' system. On the other hand, when $ a(t)=1/(t+1)$we obtain the trajectories associated to some averaged gradient system. Our analysis is mainly based on the existence of some suitable energy function. When the function $ G$is convex, the condition $ \int_0^\infty a(t) dt =\infty$guarantees that the energy function converges toward its minimum. The more stringent condition $ \int_0^{\infty} e^{-\int_0^t a(s) ds}dt<\infty$is necessary to obtain the convergence of the trajectories of $ (\mathcal{S})$toward some minimum point of $ G$. In the one-dimensional setting, a precise description of the convergence of solutions is given for a general nonconvex function $ G$. We show that in this case the set of initial conditions for which solutions converge to a local minimum is open and dense.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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