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• In magnetic films driven by spin-polarized currents, the perpendicular-to-plane anisotropy is equivalent to a parametric pumping. • A space-dependent anisotropy field can excite a family of localized textures. They can be stationary, drifting, or oscillatory patterns. • Localized patterns behave as particles and can interact among them via wave radiation. In magnetic films driven by spin-polarized currents, the perpendicular-to-plane anisotropy is equivalent to breaking the time translation symmetry, i.e., to a parametric pumping. In this work, we numerically study those current-driven magnets via the Landau–Lifshitz–Gilbert–Slonczewski equation in one spatial dimension. We consider a space-dependent anisotropy field in the parametric-like regime. The anisotropy profile is antisymmetric to the middle point of the system. We find several dissipative states and dynamical behavior and focus on localized patterns that undergo oscillatory and phase instabilities. Using numerical simulations, we characterize the localized states' bifurcations and present the corresponding diagram of phases. [ABSTRACT FROM AUTHOR]