Periodic solutions to the Duffing equation with impulse action (Q2755266)

The authors deal with the existence of a periodic solution to the Duffing equation \(d^2x/dt^2+\omega^2 x=\sigma x^3, \omega>0,\sigma>0\), with the impulse effect NEWLINE\[NEWLINEdx/dt|_{x=x_{*}}=dx(t_{*}+0)/dt-dx(t_{*}-0)/dt= I(\dot x(t_{*}-0)).NEWLINE\]NEWLINE One of the presented results is the following:NEWLINENEWLINENEWLINELet \(x_{*}\in (-\omega/\sqrt{2\sigma},\omega/\sqrt{2\sigma}).\) Then the solutions to the considered problem with the initial conditions \((x_0,\dot x_0)\in\{(x,y): x\in (-\omega/\sqrt{2\sigma},\omega/\sqrt{2\sigma}), y^2-\sigma(x^2- \omega^2/2\sigma^2)/2<0, y=dx/dt\}\) with \(|x_0|< |x_{*}|\) are periodic with the same period. The behaviour of the phase trajectories is studied.