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Spin-torque-driven vortex dynamics are studied by micromagnetic modeling in a spin-valve pillar which contains a perpendicular polarizer and a vortex free layer. Two kinds of transient oscillations mediated by the vortex-core motion are observed. The oscillations are treated as the competition among the spin torque, gyroforce, Gilbert damping, and the restoring force, governed by the generalized Thiele equation [A. A. Thiele, J. Appl. Phys. 45, 377 (1974)]. The fundamental frequency is dominated by the gyrotropic motion, while the high-frequency oscillation is triggered by the balance of the spin torque and demagnetizing field. The polarity of the vortex core can be switched through a vortex-antivortex pair creation and annihilation process.