### Abstract

The slow passage problem through a resonance is considered. As a model problem, we consider a damped harmonically forced oscillator whose forcing frequency is slowly ramped linearly in time. The setup is similar to the familiar slow passage through a Hopf bifurcation problem, where for slow variations of the control parameter, oscillations are delayed until the parameter has exceeded the critical value of the static-parameter problem by an amount that is the difference between the Hopf value and the initial value of the parameter. In sharp contrast, in the resonance problem there is an early onset of resonance, setting in when the ramped forcing frequency is midway between its initial value and the natural frequency for resonance in the unforced problem; we term this value the jump frequency. Numerically, we find that the jump frequency is independent of the system's damping coefficient, and so we also consider the undamped problem, which is analytically tractable. The analysis of the undamped problem confirms the numerical results found in the damped problem that the maximal amplitude obtained at the jump frequency scales as A∼ε^{-1/2}, ε being the ramp rate, and that the jump frequency is midway between the initial frequency at the start of the ramp and the natural frequency of the unforced problem.

Original language | English (US) |
---|---|

Article number | 056604 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 84 |

Issue number | 5 |

DOIs | |

State | Published - Nov 11 2011 |

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### ASJC Scopus subject areas

- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Statistics and Probability

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*84*(5), [056604]. https://doi.org/10.1103/PhysRevE.84.056604

**Slow passage through resonance.** / Park, Youngyong; Do, Younghae; Lopez, Juan.

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 84, no. 5, 056604. https://doi.org/10.1103/PhysRevE.84.056604

}

TY - JOUR

T1 - Slow passage through resonance

AU - Park, Youngyong

AU - Do, Younghae

AU - Lopez, Juan

PY - 2011/11/11

Y1 - 2011/11/11

N2 - The slow passage problem through a resonance is considered. As a model problem, we consider a damped harmonically forced oscillator whose forcing frequency is slowly ramped linearly in time. The setup is similar to the familiar slow passage through a Hopf bifurcation problem, where for slow variations of the control parameter, oscillations are delayed until the parameter has exceeded the critical value of the static-parameter problem by an amount that is the difference between the Hopf value and the initial value of the parameter. In sharp contrast, in the resonance problem there is an early onset of resonance, setting in when the ramped forcing frequency is midway between its initial value and the natural frequency for resonance in the unforced problem; we term this value the jump frequency. Numerically, we find that the jump frequency is independent of the system's damping coefficient, and so we also consider the undamped problem, which is analytically tractable. The analysis of the undamped problem confirms the numerical results found in the damped problem that the maximal amplitude obtained at the jump frequency scales as A∼ε-1/2, ε being the ramp rate, and that the jump frequency is midway between the initial frequency at the start of the ramp and the natural frequency of the unforced problem.

AB - The slow passage problem through a resonance is considered. As a model problem, we consider a damped harmonically forced oscillator whose forcing frequency is slowly ramped linearly in time. The setup is similar to the familiar slow passage through a Hopf bifurcation problem, where for slow variations of the control parameter, oscillations are delayed until the parameter has exceeded the critical value of the static-parameter problem by an amount that is the difference between the Hopf value and the initial value of the parameter. In sharp contrast, in the resonance problem there is an early onset of resonance, setting in when the ramped forcing frequency is midway between its initial value and the natural frequency for resonance in the unforced problem; we term this value the jump frequency. Numerically, we find that the jump frequency is independent of the system's damping coefficient, and so we also consider the undamped problem, which is analytically tractable. The analysis of the undamped problem confirms the numerical results found in the damped problem that the maximal amplitude obtained at the jump frequency scales as A∼ε-1/2, ε being the ramp rate, and that the jump frequency is midway between the initial frequency at the start of the ramp and the natural frequency of the unforced problem.

UR - http://www.scopus.com/inward/record.url?scp=81555216837&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=81555216837&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.84.056604

DO - 10.1103/PhysRevE.84.056604

M3 - Article

C2 - 22181533

AN - SCOPUS:81555216837

VL - 84

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

SN - 1539-3755

IS - 5

M1 - 056604

ER -