Oscillatory escape in a Duffing equation with a polynomial potential
A large class of systems \(x''+V_x(t,x)=0\) with periodic potential turns out to be near-integrable at infinity; as a consequence, ''most'' large-amplitude solutions are quasiperiodic, and furthermore, all solutions are bounded for all time. The main distinguishing feature of this class of systems is the super-quadratic behavior of the potential \(V\) for large \(| x| \). All proofs are based on the application of Moser's theorem [\textit{J. Moser}, Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1962, 1-20 (1962; Zbl 0107.29301)]. In 1966 \textit{J. E. Littlewood} [J. Lond. Math. Soc. 41, 497-507 (1966; Zbl 0144.10603)] considered a system \(x''+V_x(x)=p(t)\), where \(p\) is piecewise continuous, with an unbounded solution. In this paper the special polynomial equations \(x''+x^{2n+1} + a(t)x^{2l+1}=0\) with periodic and jump discontinuity in \(a(t)\) causes ''chaos'': no bounding KAM circles exit near infinity in the Poincaré phase plane. Moreover, existence of solutions which oscillate chaotically between infinity and a bounded disk is shown. As the simplest case the function \(a(t)=k^{[t]mod2}\) is considered, but the proof extends to general case.
scientific article; zbMATH DE number 1109525
scientific article