Existence of periodic solutions to differential equations with pulse action in a neighborhood of composite singular points (Q2740284)

The authors study the phenomenon of appearance of discontinuous periodic solutions in nonlinear second-order differential equations under the action of impulses. The following equations with composite equilibrium are considered NEWLINE\[NEWLINE(a)\quad \ddot x-\frac{2\dot x^2}{x}+x^5=0; \qquad (b)\quad \ddot x-\frac{2\dot x+6x^3}{x}\dot x-x^5.NEWLINE\]NEWLINE A phase point \((x(t),\dot x(t))\) undergoes pulse influence each time it intersects the line \(x=x_*\), where \(x_*\) is a given number. The corresponding action is defined by the mappingNEWLINE\[NEWLINE(x,\dot x)|_{x=x_*} \mapsto (x,\dot x+I(\dot x))|_{x=x_*}NEWLINE\]NEWLINE where \(I(y)\) is a continuous function. The existence of discontinuous periodic solutions is reduced to the existence of periodic points to a one-dimensional Poincaré mapping of the line \(x=x_*\) into itself.NEWLINENEWLINENEWLINEIt should be noted that the phase portrait of the second equation seems to contain a bug.NEWLINENEWLINENEWLINESee also the paper of \textit{A. M. Samojlenko, V. G. Samojlenko} and \textit{V. V. Sobchuk} [Ukr. Math. J. 51, No. 6, 926-933 (1999); translation from Ukr. Mat. Zh. 51, No. 6, 827-834 (1999; Zbl 0941.34030)].