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We present a systematic methodology to determine and locate analyticallyisolated periodic points of discrete and continuous dynamical systems withalgebraic nature. We apply this method to a wide range of examples, including aone-parameter family of counterexamples to the discrete Markus-Yamabeconjecture (La Salle conjecture); the study of the low periods of aLotka-Volterra-type map; the existence of three limit cycles for a piece-wiselinear planar vector field; a new counterexample of Kouchnirenko's conjecture;and an alternative proof of the existence of a class of symmetric centralconfiguration of the $(1+4)$-body problem.

Periodic orbits of discrete and continuous dynamical systems via Poincar\'{e}-Miranda theorem