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Semi-Topological Horseshoes and Incomplete Crossings for Chaos Detection

An extension of topological horseshoe theory that uses finite subshift theory from symbolic dynamics to handle incomplete crossings, defining semi-topological horseshoes. The framework detects chaos in partial-crossing cases where classical complete-crossing horseshoes fail and yields sharper lower-bound estimates of topological entropy, demonstrated on a periodically perturbed Duffing system and a Chen polynomial system via their Poincare maps.