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The Effect of Dissipation on Linearly Coupled, Slowly Varying Oscillators
Author(s) -
Grimshaw R.,
Allen J. S.
Publication year - 1982
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1982673169
Subject(s) - dissipation , dissipative system , coincidence , coupling (piping) , action (physics) , physics , dynamical systems theory , mode (computer interface) , normal mode , mode coupling , cylinder , classical mechanics , dynamical system (definition) , mathematics , mathematical analysis , quantum mechanics , geometry , medicine , mechanical engineering , alternative medicine , pathology , computer science , engineering , vibration , operating system
A dynamical system is considered whose normal frequencies and normal modes vary slowly with time in such a way that two frequencies come into close coincidence. When this occurs the corresponding normal modes undergo a drastic change in their physical description. A previous paper by us (1979) considered a conservative dynamical system, and showed that in general action is exchanged between modes at coincidence, but that except for very strong coupling the amount of action exchanged is quite small. The present paper extends this analysis to dissipative, or nonconservative, dynamical systems. Using a multiple‐time‐scale asymptotic procedure, the equations which describe the mode coupling at coincidence are derived and solved exactly using parabolic cylinder functions. The solutions show that while action is exchanged between modes at coincidence in a manner similar to that described above for conservative dynamical systems, the effect of dissipation is to ensure that the mode which suffers the smaller dissipation will dominate after mode coupling.