Asymptotic Decomposition of Pfaffian Systems with a Small Parameter

Abstract

Consider the following Pfaffian system

$$\eqalign{ & d{x_1} = q_1^{(1)}(t,x,u)d{t_1} + ... + q_1^{(m)}(t,x,u)d{t_m}; \cr & \ldots \ldots \ldots \cr & d{x_k} = q_k^{(1)}(t,x,u)d{t_1} + .... + q_k^{(m)}(t,x,u)d{t_m}, \cr} $$

(7.1)

where coefficients $q_i^{(j)}(t,x,u),i = \overline {1,k} ,j = \overline {1,m} ,$ are functions of a vector $x = \left\| {{x_1},\; \ldots ,\;{x_k}} \right\|$ of dependent variables, vector $t = \left\| {{t_1},\; \ldots ,\;{t_m}} \right\|$ of independent variables, and vector $u = \left\| {{u_1},\; \ldots ,\;{u_h}} \right\|$ of parametric variables. The functions $q_i^{\left( j \right)}\left( {t,x,u} \right)$ are analytical in the domain $G,G = {R^m} \times {R^k} \times {R^h}.$