On the analysis of internally resonant oscillators

1988 ◽  
Vol 35 (2) ◽  
pp. 221-229 ◽  
Author(s):  
A.M. Sommariva
Keyword(s):  
2013 ◽  
Author(s):  
Y. Fainman ◽  
C. Tu ◽  
V. Lomakin ◽  
B. Slutsky ◽  
M. Nezhad
Keyword(s):  

2020 ◽  
Vol 102 (3) ◽  
Author(s):  
Y. N. Zhang ◽  
J. Shen ◽  
H. D. Liu ◽  
X. X. Yi

1990 ◽  
Vol 45 (7) ◽  
pp. 839-846 ◽  
Author(s):  
D. Pfirsch

AbstractIn 1925 Cherry [1] discussed two oscillators of positive and negative energy that are nonlinearly coupled in a special way, and presented a class of exact solutions of the nonlinear equations showing explosive instability independent of the strength of the nonlinearity and the initial amplitudes. In this paper Cherry's Hamiltonian is transformed into a form which allows a simple physical interpretation. The new Hamiltonian is generalized to three nonlinearly coupled oscillators; it corresponds to three-wave interaction in a continuum theory, like the Vlasov-Maxwell theory, if there exist linear negative energy waves [2-4, 5, 6], Cherry was able to present a two-parameter solution set for his example which would, however, allow a four-parameter solution set, and, as a first result, an analogous three-parameter solution set for the resonant three-oscillator case is obtained here which, however, would allow a six-parameter solution set. Nonlinear instability is therefore proven so far only for a very small part of the phase space of the oscillators. This paper gives in addition the complete solution for the three-oscillator case and shows that, except for a singular case, all initial conditions, especially those with arbitrarily small amplitudes, lead to explosive behaviour. This is true of the resonant case. The non-resonant oscillators can sometimes also become explosively unstable, but the initial amplitudes must not be infinitesimally small. A few examples are presented for illustration.


Author(s):  
A.K. Armenta Marquez ◽  
L. Hernandez Martinez ◽  
M. Linares Aranda ◽  
W. Calleja Arriaga

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