Analysis demonstrates the role of the Poincaré sphere in identifying invariant manifolds in linear control systems.
For linear control systems with bounded control range, the state space is compactified using the Poincar\'e sphere. The linearization of the induced control flow allows the construction of invariant manifolds on the sphere and of corresponding manifolds in the state space of the linear control system.
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Colonius et al. (2025) studied this question.
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