Nanomagnetic particles respond sensitively and nonlinearly to electromagnetic radiation. Many excitation schemes are now well known. However, nonlinear dynamics determinations have not been examined in detail under FMR conditions. The nonlinear dynamics of a simple nanomagnet is studied and the excitation field, H 1 , is varied. We solve numerically the Landau-Lifshitz nonlinear dynamics of M t .( ) There is a special focus on the spherical degrees of freedom: t , q ( ) t . f ( ) We find that the t q ( ) trajectories converge asymptotically to asym q =constant, while t f ( ) is a linear function of time. The combined dynamics of t q¢( ) and t f¢( ) produce a limit cycle for each value of H 1 . The systematic numerical calculations and analysis show that the limit cycle asym q follows a fourth-degree polynomial on H 1 and an inverse law on frequency ν 1 . It is also found that the limit cycles are established after 12-20 nanoseconds. They cause M t ( ) to sweep a constant precession cone that lasts for more than 200 ns independent of initial conditions. These results bring significant novel knowledge for fast information technology.