We introduce Collective Switched Motion (Cosm), a dynamical system-based heuristic algorithm. Cosm combines locally interacting continuous circular variables with novel global coordination rules that facilitate collective dynamics. Pairwise interactions occur sequentially over a set of conflict-free edge partitions, resulting in an interaction network that switches periodically. Unlike conventional gradient-based approaches, Cosm employs structured, non-smooth switching dynamics with finite-magnitude interactions that sustain collective fluctuations and promote exploration beyond local minima. A correlated perturbation mechanism further promotes coordinated cluster motion in the circular phase space. On the three largest Ising problems from the Gset suite, which have 10,000-20,000 variables and represent 2D spin glasses, Cosm attains the optimal solutions (verified with an exact solver) heuristically for the first time. On two large bounded-degree non-lattice graph instances, Cosm reduces the state-of-the-art times-to-target from hundreds of hours to 36-303 s. Results on benchmark problems with tuned hardness suggest favorable scaling relative to previously characterized dynamical solvers. These results suggest that Cosm's synthesis of local interactions, structured switching dynamics, and global coordination provides an effective computational framework for sparse optimization.

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