Topologically stable subharmonics of large periods and amplitudes (Q2459761)

The authors consider the equation \[ x''+\alpha^{2}x=b(t)+f(x) \tag{1} \] with a \(2\pi\)-periodic function \(b\), an irrational \(\alpha\), and a bounded continuous function \(f\) satisfying the saturation conditions \(f\to\pm f_{+}\) as \(x\to\pm\infty\). The authors study the existence of topologically stable subharmonics (periodic solutions of multiple periods \(2n\pi\)) with large amplitudes. The proposed condition of the existence of subharmonics involves the three asymptotic relationships characterising 1) how fast \(f\) tends to the limits at infinity (estimate (4) in the manuscript); 2) how slow the Fourier coefficients of \(b\) tend to zero (the smoothness of \(b\)); 3) how fast the denominators of the rational convergents to the number \(\alpha\) tend to infinity. The authors show that under appropriate conditions there exists a sequence \(k_{n}\) of positive integers such that equation (1) has a sequence of periodic solutions \(x_{n}\) with periods \(2k_{n}\pi\) and with the amplitudes satisfying \(\| x_{n}\| _{C}\to\infty\) as \(n\to\infty\).