## Abstract

This paper considers (1 + 1) dimensional conservative oscillatory systems with polynomial nonlinearities in the double limit of high amplitude (x_{max} → ∞) and high leading power (x^{2N+1}, N → ∞). In this limit, hitherto not studied in the literature, such systems exhibit behavior akin to boundary layer phenomena: The characteristics of the solution are determined by a vanishingly small segment of the amplitude near ± x_{max}. The oscillating entity, x(t), tends to a periodic saw-tooth shape of linear segments, the velocity, x′(t), tends to a periodic step-function and the x − x′ phase-space plot tends to a rectangle. This is demonstrated by transforming x and t into scaled variables, η and θ, respectively. η(θ) is (2-π) periodic in θ and bounded (|η(θ)|≤ 1). For large x_{max}, the zero-order term in an expansion in powers of a small parameter that is proportional to (1/x_{max}) yields an excellent approximation to the solutions of the scaled equation. The boundary-layer characteristics show up in the double limit by the fact that the deviations of η(θ), η′(θ) and the η − η′ phase-space plot from the sharp asymptotic shapes occur over a narrow range in θ of O(1/N) near the turning points of the oscillations.

Original language | English |
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Pages (from-to) | 2851-2863 |

Number of pages | 13 |

Journal | Nonlinear Dynamics |

Volume | 109 |

Issue number | 4 |

DOIs | |

State | Published - 1 Sep 2022 |

## Keywords

- Boundary-layer characteristics
- Large nonlinearity
- Nonlinear oscillators
- Nonlinearly violent oscillations

## ASJC Scopus subject areas

- Control and Systems Engineering
- Aerospace Engineering
- Ocean Engineering
- Mechanical Engineering
- Applied Mathematics
- Electrical and Electronic Engineering