Harmonic and subharmonic solutions for superlinear damped Duffing equation. (English) Zbl 1281.34058

Cheng, Zhibo; Ren, Jingli

Nonlinear Anal., Real World Appl. 14, No. 2, 1155-1170 (2013).

The authors claim to prove the existence of harmonic and subharmonic solutions for a damped Duffing equation of the form \[ x''(t) + Cx'(t) + g(x(t)) = 0, \] with \(C\geq0\) and a superlinear condition on the nonlinearity, by making use of a generalized version of the Poincaré-Birkhoff theorem due to W. Y. Ding. In my opinion, the main result should be considered under suspicion, because any version of the Poincaré-Birkhoff theorem requires the area-preserving character of the Poincaré map, a condition which is violated if \(C>0\).

Reviewer: Pedro J. Torres (Granada)

MSC:

34C25

Periodic solutions to ordinary differential equations

Keywords:

harmonic solutions; subharmonic solutions; twisting theorem; time map; damped Duffing equation