Exponentially accurate balance dynamics. (English) Zbl 1120.34040

Wirosoetisno, D.

Adv. Differ. Equ. 9, No. 1-2, 177-196 (2004).

By explicitly bounding the growth of terms in the expansion of the solution of the singularly perturbed system of the form
\[ \dot x + \frac{1}{\varepsilon} Lx+F(x,y), \qquad \dot y = G(x,y) \]
with \(x\in \mathbb R^p\), \(y\in \mathbb R^q\), \(L\) is a skew-Hermitian nonsingular matrix, \(F\) and \(G\) are \(p\)- and \(q\)-vector valued functions of their arguments, the author derives conditions for the existence of the solution of the corresponding balance equation up to an error that scales exponentially in \(\varepsilon\) as \(\varepsilon \to 0\). This solution defines the slow manifold of the system. It is done for a finite-dimensional system with polynomial nonlinearity. An example from the continuous fluid dynamics is given. In addition, for the finite-dimensional system it is shown that the solution of the full model stays within an exponential distance to the the solution of the balance equation over a timescale of order one, independent of \(\varepsilon\).

Reviewer: E. V. Shchetinina (Samara)

MSC:

34E15	Singular perturbations for ordinary differential equations
34E05	Asymptotic expansions of solutions to ordinary differential equations
86A10	Meteorology and atmospheric physics

Keywords:

singular perturbation; expansions; polynomial system