Asymptotics of the solution to a differential equation with a small parameter in the case of two limit solutions. (English) Zbl 1163.34036

Maksimov, V. I. (ed.), Dynamical systems: modeling, optimization, and control. Transl. from the Russian. Moscow: Maik Nauka/Interperiodica, Pleiades Publishing/distrib. by Springer. Proc. Steklov Inst. Math. 2006, Suppl. 1, S105-S116 (2006); translation from Tr. Inst. Mat. Mekh. 12, No. 1, 98-108 (2006).
The authors consider the initial value problem
\[ \varepsilon\frac{dx}{dt}=f(x,t,\varepsilon),\quad x(t_0)=A,\quad t_0\leq t\leq t_1 \]
with a small parameter \(\varepsilon>0\), in the case when the reduced equation \(f(x,t,0)=0\) has two intersecting roots \(\varphi_1(t)\) and \(\varphi_2(t)\). Assuming that the latter intersection is transversal and imposing some additional restrictions on the behaviour of the function \(f\) at the intersection point, they construct the uniform asymptotics of the solution to the problem under consideration. Main attention during the constructing procedure is devoted to matching of the inner and outer expansions in vicinity of the intersection point.
For the entire collection see [Zbl 1116.37003].


34E05 Asymptotic expansions of solutions to ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
34E13 Multiple scale methods for ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
Full Text: DOI MNR