Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 1003.34041
Zounes, Randolph S.; Rand, Richard H.
Global behavior of a nonlinear quasiperiodic Mathieu equation.
(English)
[J] Nonlinear Dyn. 27, No.2, 87-105 (2002). ISSN 0924-090X; ISSN 1573-269X/e

Here, the interaction of subharmonic resonances in the nonlinear quasiperiodic Mathieu equation $$\ddot x + [\delta + \varepsilon (\cos \omega_1t+\cos \omega_2t)]x + \alpha x^3=0$$ is studied, where $\varepsilon \ll 1$ and the coefficient of the nonlinear term $\alpha$ is positive but not necessarily small. By using Lie transform perturbation theory with elliptic functions, the authors study subharmonic resonances associated with orbits in $2m:1$ resonance with a respective driver. In particular, the authors derive analytic expressions that put conditions on the parameters $(\delta, \varepsilon, \omega_1, \omega_2)$ at which subharmonic resonance bands in a Poincaré section of action space begin. The authors obtain an overview of the $\vartheta(\varepsilon)$ global behavior of above equation as a function of $\delta$ and $\omega_2$ with $\omega_1, \alpha$ and $\varepsilon$ fixed.
[Chen Lan Sun (Beijing)]
MSC 2000:
*34D23 Global stability
70K30 Nonlinear resonances (general mechanics)

Keywords: elliptic functions; Lie transformations; resonance; quasiperiodic Mathieu equation; Poincaré section

Highlights
Master Server

### Zentralblatt MATH Berlin [Germany]

© FIZ Karlsruhe GmbH

Zentralblatt MATH master server is maintained by the Editorial Office in Berlin, Section Mathematics and Computer Science of FIZ Karlsruhe and is updated daily.

Other Mirror Sites

Copyright © 2013 Zentralblatt MATH | European Mathematical Society | FIZ Karlsruhe | Heidelberg Academy of Sciences