Nonexistence of periodic solutions for a damped pendulum equation
The author shows the nonexistence of harmonics to the weakly dissipative (\(a>0\) small) pendulum equation under forcing, \[ x''+ax' + b \sin x = p(t), \tag{1} \] for arbitrary positive constants \(a,b\), by constructing a special continuous periodic forcing term \((p,t)\) with a zero mean value. This extends an earlier counter-example of \textit{R. Ortega} [Bull. Cl. Sci., V. Ser., Acad. R. Belg. 73, No. 10, 405-409 (1987; Zbl 0679.70022)] to \textit{J. Mawhin}'s conjecture posed in [Ordinary and partial differential equations, Proc. 7th Conf., Dundee/Scotl. 1982, Lect. Notes Math. 964, 458-476, Springer, Berlin (1982; Zbl 0517.34029)], that (1) admits for arbitrary \(a>0\), \(b>0\) a harmonic, whenever the mean value of a continuous and periodic \(p(t)\) is zero. The reviewer modified in [Boll. Unione Mat. Ital., VII. Ser., A 7, 377-386 (1993; Zbl 0830.70017)] and [Topol. Methods Nonlinear Anal. 6, 163-174 (1995; Zbl 0851.34046)] the original conjecture by extending the class of harmonics to the one involving also subharmonics of the second kind.
scientific article; zbMATH DE number 1450173