Nonlinear Oscillations in Physical Systems |
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12-harmonic oscillation a₁ amplitude analysis applied voltage approximation b₁ b₂ boundary c₁ characteristic exponent circuit consider constant curves of Eq d2x dr2 delta method determined differential equations dr2 dv driving frequency dx dr entrained oscillation equating the coefficients equation d2x equilibrium external force FIGURE fixed point Fourier series function given by Eqs harmonic entrainment harmonic oscillation Hence Hill's equation integral curves investigate Liapunov function limit cycle linear Mathieu functions mentioned in Sec method nonresonant obtained order ½ oscilla oscillation of order periodic oscillation periodic solution phase-plane plane R₁ representative point resonant result roots saddle point second unstable region separately to zero shown in Fig singular point solution curve solution of Eq stability condition Stable focus subharmonic oscillation Substituting Eqs system parameters tion trajectories transient types unstable fixed point values variables variational equation waveforms Y₁