## Truly Nonlinear Oscillations: Harmonic Balance, Parameter Expansions, Iteration, and Averaging MethodsThis unique book provides a concise presentation of many of the fundamental strategies for calculating approximations to the oscillatory solutions of truly nonlinear (TNL) oscillator equations. The volume gives a general overview of the author''s work on harmonic balance, iteration and combined linearization-averaging methods. However, full discussions are also presented on parameter expansion procedures and a first-order averaging technique for TNL oscillators. The calculational basis of each method is clarified by applying them to a set of standard TNL oscillator equations. This allows a direct comparison to be made among the various methods. The book is self-contained and therefore suitable for both classroom use and self-study by students and professionals who desire to learn, understand, and apply these technique to the field of nonlinear oscillations. |

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### Contents

1 Background and General Comments | 1 |

2 Establishing Periodicity | 23 |

3 Harmonic Balance | 43 |

4 Parameter Expansions | 75 |

5 Iteration Methods | 89 |

6 Averaging Methods | 123 |

7 Comparative Analysis | 155 |

Appendix A Mathematical Relations | 183 |

Appendix C Fourier Series | 193 |

Appendix D Basic Theorems of the Theory of SecondOrder Differential Equations | 199 |

Appendix E Linear SecondOrder Differential Equations | 203 |

Appendix F LindstedtPoincare Perturbation Method | 213 |

Appendix G A Standard Averaging Method | 217 |

Appendix H Discrete Models of Two TNL Oscillators | 221 |

227 | |

237 | |

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