The well-known technique of power series is exploited here to find a recurrence algorithm to solve strongly nonlinear oscillators. A wide variety of nonlinearities may be tackled with this approach. Simple recurrence relationship are obtained after some algebra handling. Three illustrations are analytically developed and numerically solved: a parametrically excited nonlinear oscillator, the differential equation of motion of a moored structure with non-truncated mooring nonlinearities and the classical Lorenz oscillator. In the first illustration a strongly nonlinear parametric oscillator that governs a rotor dynamics problem is solved. The results are depicted as trajectories, phase plots and Poincare´ maps. The second illustrations deals with the dynamic behavior of a simplified model of a small floating structure anchored by chains or cables. This type of structure is sometimes known as Catenary Anchor Leg Mooring (CALM) system. The structure is first modeled as a two DOF oscillator with strongly nonlinear stiffness and subjected to a harmonic wave force. A further simplification is introduced by prescribing the vertical motion as a harmonic oscillation. Then, the resulting equation is stated with a single DOF. Here the use of the power series is twofold: an algebraic recurrence algorithm is employed to obtain a nontruncated differential equation and, as in the first illustration, also used as a time integration scheme. When the forcing frequency is chosen as the bifurcation parameter, the system shows diverse type of solutions: 1-, 3-, multi-period, and quasiperiodic behavior. The third illustration deals with the well-known Lorenz system. The strange attractor is obtained without difficulties.

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