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Singularly perturbed equations with pulse effect. (English. Ukrainian original) Zbl 1041.34041
Ukr. Math. J. 54, No. 8, 1309-1323 (2002); translation from Ukr. Mat. Zh. 54, No. 8, 1089-1099 (2002).
The authors study the problem of finding asymptotic solutions to a singularly perturbed equation with impulses \[ \varepsilon \dot x=g(t,x),\quad \Delta x| _{t=t_i}=I_i(x),\quad i\in \mathbb Z,\tag{1} \] where \(g:\mathbb R^2 \mapsto \mathbb R,\) and \(\inf_{i}(t_{i+1}-t_i)>0\). It is assumed that the corresponding generating problem \((\epsilon =0)\) has a solution \(\bar x_0(t)\in C^\infty(\mathbb R\setminus \cup\{t_i\} \mapsto \mathbb R)\). A formal solution to (1) is searched in the form \(x(t,\varepsilon)=\bar x(t,\varepsilon)+\sum_{k=0}^{\infty}\varepsilon^k\Pi_{k}x(\tau)\), where \(\bar x(t,\varepsilon)=\sum_{k=0}^{\infty}\varepsilon^k\bar x_k(t)\) is the regular part, \(\tau =\bigcup_{i}\{\tau_i:=(-1)^i(t-t^i)/\varepsilon \}\), and \(\Pi_k(\tau)\) are boundary layer functions. Under certain natural conditions, it is shown that this solution exists, has an asymptotic property, and that in the case where \(\bar x_0(t)\) is \(T\) periodic there exists a formal \(T\)-periodic solution. The equation of form (1), where \(I_i(x)\) is replaced by \(\epsilon I_i(x)\), is considered, too.
34E15 Singular perturbations, general theory for ordinary differential equations
34A37 Ordinary differential equations with impulses
34E05 Asymptotic expansions of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations