Discontinuous Cucker–Smale model: Achieving flocking behavior through Filippov’s differential inclusion

Abstract

We introduce a modified multi-agent system inspired by the Cucker–Smale model [1], incorporating a discontinuous interaction law to better capture abrupt behavioral transitions in collective dynamics. To rigorously address the discontinuities, we employ Filippov’s differential inclusion framework, which enables a systematic analysis of solutions in the presence of vector field discontinuities. Within this framework, we identify appropriate sufficient conditions on the initial data and system parameters under which the agents are guaranteed to achieve flocking in finite time. This model proposes a new analytical framework that explains discontinuous flocking behavior, which classical continuous models often fail to capture.