Study of piecewise linear second order differential equations

The behavior of trajectories of solutions to piecewise linear second order differential equations is being studied. These equations are widely used in mechanics, electrical engineering and automatic control theory. Of particular interest are the conditions for the emergence of limit cycles in the vicinity of the rest region of a second order piecewise linear differential equation with a discontinuous switching line. It has been established that if a region of rest (consisting of rest points) exists, then it remains inside the limit cycle. One of the primary tasks is to determine the region of rest that appears on the line of stitching solutions. In the course of the work, new relations were obtained that provide limited solutions to piecewise linear equations. Using these new conditions, phase portraits are constructed that take into account the coefficients of the equations. Conditions have also been found under which there is no rest region. To solve these problems, the method of stitching solutions from two half-planes was used.