The implicit mapping dynamics of a class of time-delay dynamical systems with impulses at fixed times

In this paper, we investigate the periodic flows of a class of time-delay dynamical systems with impulses at fixed times through an implicit mapping method. As the impulsive points coincide with the time-delay points, discrete implicit mappings at impulsive points are obtained based on the impulsive mappings and interpolation techniques according to the given accuracy. As the impulsive points coincide with the regular points, the integral limits of the impulsive points are constructed, and discrete implicit mappings at impulsive points and time-delay points are obtained respectively by using the impulsive functions and integral techniques. The approximate solutions of the periodic flows determined by regular and delay nodes in one period are presented. A second-order delay dynamical system with an impulse at a fixed time is presented as an example. The implicit mapping method provides a plan for the periodic flows of switching delay dynamical systems.