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In this work we study the local cyclicity of some polynomial vector fields in R 3. In particular, we give a quadratic system with 11 limit cycles, a cubic system with 31 limit cycles, a quartic system with 54 limit cycles, and a quintic system with 92 limit cycles. All limit cycles are small amplitude limit cycles and bifurcate from a Hopf type equilibrium. We introduce how to find Lyapunov constants in R 3 for considering the usual degenerate Hopf bifurcation with a parallelization approach, which enables to prove our results for 4th and 5th degrees. • High local cyclicity values of some polynomial vector fields in R 3. • Improvement of the current lower bound for the quadratic family. • First lower bounds for degrees 3, 4, and 5. • Implementation of a highly efficient parallelization approach. [ABSTRACT FROM AUTHOR]