Sliding in a piecewise-smooth dynamical system with a hold-on effect

In this paper, we study a piecewise-smooth dynamical system of a Filippov type that has a specific discontinuity in the form of a hold-on term. The system resembles a driven dry-friction oscillator with the difference that the magnitude of the friction depends on the value of a local maximum of the oscillator displacement. Due to the hold-on term, the definition of the sliding region is not trivial. Along with a sliding region that is found analytically and is valid for any orbits, there exists a sliding region that we name ‘virtual’ because it is specific for each particular orbit. The existence of the virtual sliding region is explained by the specific discontinuity — the hold-on term — and the behavior associated with the sliding region and its boundaries can be considered as a new type of sliding-associated behavior.