Invariant manifolds for a singular ordinary differential equation
Abstract
We study the singular ordinary differential equation $$ \frac{d U}{d t} = f (U) / z (U) + g (U), $$ where $U \in R^N$, the functions $f \in R^N $ and $g \in R^N $ are of class $C^2$ and $z $ is a real valued $C^2$ function. The equation is singular in the sense that $z (U)$ can attain the value 0. We focus on the solutions of the singular ODE that belong to a small neighborhood of a point $\bar U$ such that $f (\bar U) = g (\bar U) = \vec 0$, $z (\bar U) =0$. We investigate the existence of manifolds that are locally invariant for the singular ODE and that contain orbits with a suitable prescribed asymptotic behaviour. Under suitable hypotheses on the set $\{U: z (U) = 0 \}$, we extend to the case of the singular ODE the definitions of center manifold, center stable manifold and of uniformly stable manifold. An application of our analysis concerns the study of the viscous profiles with small total variation for a class of mixed hyperbolicparabolic systems in one space variable. Such a class includes the compressible Navier Stokes equation.
 Publication:

arXiv eprints
 Pub Date:
 January 2008
 arXiv:
 arXiv:0801.4425
 Bibcode:
 2008arXiv0801.4425B
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Analysis of PDEs;
 34C30;
 34E15
 EPrint:
 35 pages, more general case considered