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J. Bifurcation Chaos"],"published-print":{"date-parts":[[2026,3,15]]},"abstract":"<jats:p>This work explores the persistence of periodic orbits under small discontinuous perturbations to a three-dimensional piecewise smooth vector field. We assume that the unperturbed system has an invariant hyperplane that intersects the switching plane transversally and contains a family of crossing periodic orbits around the origin. By developing the Poincar\u00e9 map and defining the displacement function, we derive some sufficient conditions for the existence of periodic solutions. 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