The existence of periodic solutions to certain impulsive differential equations

Abstract

For autonomous ordinary differential equations, it is shown that there always exist impulsive forcing patterns which permit the existence of periodic solutions. However, when the differential equation is nonautonomous, proving the existence of impulsive periodic solutions appears more complex. The authors study here an impulsive differential system which appears to have no periodic solutions. Continuous manifolds of initial vectors are found for which it is proven that the trajectory emanating from each initial vector is unstable, hence, nonperiodic. It is also established that a particular type of impulsive periodic solution, the so-called simple periodic motion, cannot exist. By changing the type of impulse applied, some periodic solutions are obtained, one of which, though not physically meaningful, is perhaps mathematically significant.