Periodic solutions of impulse differential systems in critical cases. (Russian) Zbl 0739.34016

The following linear impulse differential system is considered: \[ \dot z=A(t)z+f(t), \;t\neq\tau_ i,\;\Delta z\mid_{t=\tau_ i}=S_ iz+a_ i. \] Here \(z\in \mathbb R^ n\), \(A(t)\) and \(f(t)\) are respectively \(n\times n\) and \(n\times 1\) piecewise continuous and \(T\)-periodic matrix functions, the constant matrices \(S_ i\) and the moments \(\tau_ i\) satisfy \(S_{i+p}=S_ i\), \(a_{i+p}=a_ i\), \(\tau_{i+p}=\tau_ i+T(\tau_ 0<0<\tau_ 1<\cdots<\tau_ p<T)\), the matrices \(I_ n+S_ i\) are non-degenerate (\(I_ n\) is the \(n\times n\) unit matrix). Each solution \(z(t,c)\), \(z(0,c)=c\in\mathbb R^ n\), is of the type \[ z(t,c)=X(t)c+\int^ T_ 0 K(t,\tau)f(\tau)\,d\tau+\sum^ p_{i=1}K(t,\tau_ i)a_ i, \] where \(X(t)\) \((X(0)=I_ n)\) is the matrix fundamental solution of the homogeneous system \[ \dot z=A(t)z,\;t\neq\tau_ i,\;\Delta z\mid_{t=\tau_ i}=S_ iz \] and \[ K(t,\tau)=\begin{cases} X(t)X^{-1}(\tau), & 0\leq\tau\leq t\leq T, \\ 0, & 0\leq t\leq\tau\leq T,\end{cases} \] \(K(t,\tau_ i)=K(t,\tau_ i+0)\), is the Green function for the considered impulse system. The solution \(z(t,c)\) is \(T\)-periodic if and only if \(z(0,c)=z(T,c)\), which leads to the algebraic system \[ Qc=\int^ T_ 0K(t,\tau)f(\tau)\,d\tau+\sum^ p_{i=1}K(t,\tau_ i)a_ i \] with \(Q=I_ n-X(t)\). The paper deals with the critical case \(\det Q=0\). The main result is that if \(\text{rank}\, Q=n_ 1\), then the homogeneous impulse system admits exactly \(r=n-n_ 1\) linearly independent periodic solutions. A condition the non- homogeneous system to be solvable is given, in which case it is shown that it admits an \(r\)-parametric family of periodic solutions. Similar problems for nonlinear differential systems with small parameters of the type \[ \dot z=A(t)z+f(t)+\varepsilon Z(z,t,\varepsilon),\;t\neq\tau_ i,\;\Delta z\mid_{t=t_ i}=S_ iz+a_ i+\varepsilon J_ i(z,\varepsilon) \] are considered.


34A37 Ordinary differential equations with impulses
34C25 Periodic solutions to ordinary differential equations