Bifurcation analysis of piecewise-smooth engineering systems with delays through numeric continuation of periodic orbits

Abstract

This paper presents a numeric continuation framework for periodic orbits of piecewise-smooth and hybrid dynamical systems with fixed point delays. For the numeric solution of the corresponding infinite dimensional multi-point boundary value problem, a novel discretization and interpolation scheme is developed employing Chebyshev polynomial based spectral collocation techniques. The same approach is employed for the formulation of the corresponding monodromy matrix enabling stability analysis on the found periodic orbits. Special care is attributed to the accurate detection of discontinuity induced bifurcations such as grazing and sliding, and the implemented pseudo-arclength framework is adapted to allow two parameter continuation of these critical points. The capabilities of the developed algorithms are demonstrated on a set of delayed piecewise-smooth and hybrid dynamical systems, showcasing potential engineering applications from the fields of control theory, traffic dynamics modelling, and machine tool vibrations. Finally, a detailed tutorial is attached in the appendix to accompany the open-source release of the developed codebase.