Periodic solutions of discontinuous second order differential equations. The porpoising effect (Q6171348)

The authors construct a mathematical model of the form \[ x''+g(x)=\epsilon p(t)\tag{\(\ast\)} \] in order to study the so-called porpoising effect (a bouncing effect) in racing cars. Here, the perturbation \(p:{\mathbb R}\to{\mathbb R}\) is assumed to be continuous and periodic, while \(g:(a,\infty)\to{\mathbb R}\) is continuous except for a discontinuity at some \(\alpha>a\), where both the left limit \(\ell^-\) and the right limit \(\ell^+\) of \(g\) exist as finite numbers \(\ell^-<0<\ell^+\). Under appropriate (further) assumptions on \(g\) it is shown that for sufficiently small \(|\epsilon|\), large-amplitude subharmonic solutions may arise. The proof is based on e.g.\ the Poincaré-Birkhoff theorem. In a conclusion, the authors hint to actual evidence of this phenomenon in recent Formula 1 races.