Separation of motions in multirate discontinuous systems with time delay (Q1281102)

The author considers a system of the form \[ \mu dz/dt = g(z,s,x,u), \quad ds/dt = h_1(z,s,x,u), \quad dx/dt = h_2(z,s,x,u) \] where \(z \in R^m\), \(s \in R\), \(x \in R^n\), \(u(s) = - \text{sign}[s(t-1)]\), \(g\),\(h_1\) and \(h_2\) are smooth functions and \(\mu\) is a small parameter. Sufficient conditions are given under which for sufficiently small \(\mu\) there exists an orbitally asymptotically stable periodic solution. The algorithm of the construction of an asymptotics of a periodic solution is suggested. An example illustrating how the obtained results can be used for the separation of motions in singularly perturbed discontinuous systems with delay is given.