Finite-dimensional monodromy operators for periodic systems of differential equations with aftereffect (Q2755755)

The authors deal with the system of linear periodic functional-differential equations NEWLINE\[NEWLINE\frac{dx(t)}{dt} = \int_{-\omega}^0 d_s \eta (t,s) x(t+s), \tag{1}NEWLINE\]NEWLINE with \(t \in (0, \infty)\), \(x: [-\omega, +\infty) \to \mathbb{R}^n\), \(\omega > 0,\) the matrix-function \(\eta\) is \(\omega\)-periodic in \(t.\) The monodromy operator for (1) is defined by the linear completely continuous operator \(U: C([-\omega,0], \mathbb{R}^n) \to C([-\omega,0], \mathbb{R}^n)\), \((U\phi)(\theta) = x(\omega + \theta, \phi)\), \(\theta \in [-\omega, 0],\) where \(x(t, \phi)\) is a solution to (1) with initial function \(\phi,\) i.e. \(x(t, \phi) = \phi (t)\), \(-\omega \leq t \leq 0.\) Necessary and sufficient conditions are obtained for the monodromy operator to be finite-dimensional. An effective method for the construction of a finite-dimensional monodromy operator is given.