A multiflow is the set-valued analogue of a flow: an upper-semicontinuous, compact-valued map on R+ x X obeying the monoid identities, used to describe the many non-unique solutions of a differential inclusion x' in F(x), and allowed to take the empty value once a solution leaves the compact domain. Working only under the Filippov conditions (F upper-semicontinuous with compact, convex, non-empty values, and no growth bound), you learn to define omega- and alpha-limit sets relative to an invariant set S and to build from them the attractor, its dual repeller, and the connecting region whose points run forward to the attractor and backward to the repeller — a decomposition in which the duality is symmetric, so the dual repeller is genuinely a repeller and the attractor is its dual. You also learn why this decomposition continues: the isolating neighbourhoods of the attractor, the repeller and the invariant set stay isolating under perturbation of the set-valued map, so the whole pair persists in nearby inclusions, including smooth families limiting to a discontinuous Filippov system.
A differential inclusion replaces an ordinary differential equation's single prescribed velocity with a whole set of admissible ones, written x' in F(x), so one initial condition can have many soluti…