A Multi-Point, Boundary-Value Problem, Collocation Toolbox for the Continuation of sets of Constrained Orbit Segments

This paper presents a collocation toolbox for multi-point, boundary-value problems. This toolbox has been recently developed by the authors to support general-purpose parameter continuation of sets of constrained orbit segments, such as i) segmented trajectories in hybrid dynamical systems, for example, mechanical systems with impacts, friction, and switching control, ii) homoclinic orbits represented by an equilibrium point and a finite-time trajectory that starts and ends near this equilibrium point, and iii) collections of trajectories that represent quasi-periodic invariant tori. The collocation algorithm allows for segment-dependent meshing and non-trivial boundary conditions involving internal mesh points and includes a full discretization of the corresponding variational equations. Several examples are chosen to illustrate the formalism and its implementation, including the continuation of families of periodic orbits in a hybrid dynamical system with impacts and friction as well as detection and constrained continuation of selected degeneracies characteristic of such systems, such as grazing and switching-sliding bifurcations.