To verify the approximate analytical solutions of periodic motion in such a quadratic nonlinear oscillator, numerical simulations are carried out by the symplectic scheme. The initial conditions for numerical simulation are computed from the approximate analytical solutions. The numerical results are depicted by solid curves, but the analytical solutions are given by red circular symbols. The big filled circular symbols are initial conditions.

The displacement, velocity, trajectory, and amplitude spectrum of stable period-4 motion are presented in Figure 5.14 for with initial conditions . This analytical solution is based on 48 harmonic terms (HB48) in the Fourier series solution of period-4 motion. In Figure 5.14(a),(b), for over 40 periods, the analytical and numerical solutions match very well. In Figure 5.14(c), analytical and numerical trajectories match very well. It is obviously observed that the period-4 motion is generated just off the Hopf bifurcation of period-2 motion. This special Hopf bifurcation of the dynamical systems coefficients of period-2 motion gives the so called period-doubling bifurcation of the period-2 motion. The doubled trajectories of period-2 motion are very close to the trajectory of period-4 motion. In Figure 5.14(d), ...

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