Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

Accurate Phase after Slow Passage through Subharmonic Resonance

Jerry D. Brothers and Richard Haberman
SIAM Journal on Applied Mathematics
Vol. 59, No. 1 (Sep. - Oct., 1998), pp. 347-364
Stable URL: http://www.jstor.org/stable/118385
Page Count: 18
  • Subscribe ($19.50)
  • Cite this Item
Accurate Phase after Slow Passage through Subharmonic Resonance
Preview not available

Abstract

A strongly nonlinear oscillator with O(ε ) damping and O(ε ) sinusoidal forcing is considered. The frequency is energy dependent, permitting energy levels corresponding to subharmonic resonance. Before and after subharmonic resonance, equations for the energy and phase of the nonlinear oscillator are derived using multiphase averaging. The average energy and phase are shown to satisfy to sufficiently high order the same differential equations as occur without periodic forcing. The slow passage through a subharmonic resonance is analyzed. By matching the energy and phase to sufficiently high order, an O(ε ) additional jump in the average energy across the subharmonic resonance layer is computed in addition to the previously known O(ε 1/2) jump in the average energy and the previously known O(1) jump in phase. The more accurate jump in energy is used to obtain an asymptotic approximation (whose error is small) of the phas of the nonlinear oscillator after a subharmonic resonance layer. A time shift for the average energy is computed which is equivalent to the entire jump in energy across a subharmonic resonance layer. The time shift accounts for the averaged energy after resonance This time shift is shown to yield the correct phase of the nonlinear oscillator after resonance with an elementary constant phase adjustment chosen to be consistent with the jump in phase. After the subharmonic resonance, the average energy and phase are shown to be the same as the average energy and phase that would occur without the periodic forcing if the time shift (delay) and phase adjustment are included.

Page Thumbnails

  • Thumbnail: Page 
347
    347
  • Thumbnail: Page 
348
    348
  • Thumbnail: Page 
349
    349
  • Thumbnail: Page 
350
    350
  • Thumbnail: Page 
351
    351
  • Thumbnail: Page 
352
    352
  • Thumbnail: Page 
353
    353
  • Thumbnail: Page 
354
    354
  • Thumbnail: Page 
355
    355
  • Thumbnail: Page 
356
    356
  • Thumbnail: Page 
357
    357
  • Thumbnail: Page 
358
    358
  • Thumbnail: Page 
359
    359
  • Thumbnail: Page 
360
    360
  • Thumbnail: Page 
361
    361
  • Thumbnail: Page 
362
    362
  • Thumbnail: Page 
363
    363
  • Thumbnail: Page 
364
    364