A class of kick-excited self-adaptive dynamical systems is formed and proposed. The class is characterized by a nonlinear (inhomogeneous) external periodic excitation (as regards the coordinates of the excited system) and is remarkable for the occurrence of the following objective regularities: the phenomenon of ''discrete'' (''quantized'') oscillation excitation in macro-dynamical systems having multiple branch attractors and strong self-adaptive stability. The main features of the class of systems are studied both numerically and analytically on the basis of the general model of a pendulum under inhomogeneous action of a periodic force (referred to as a kicked pendulum). A diagram involving multiple bifurcations for the attractor set of the system under consideration is obtained and analyzed. The complex dynamics, evolution and the fractal boundaries of the multiple attractor basins in state space corresponding to energy and initial phase variables are obtained, traced and discussed. An analytic proof is presented showing the existence of ''quantized'' oscillations for the kick-excited pendulum. An analytic approach is given applicable to the cases of small and large amplitudes (small and large non-linearity). The spectrum of possible oscillation amplitudes for the pendulum is studied as well as its motion in a rotational regime under the influence of an external non-homogeneous periodic force. Generalized conditions for the excitation of pendulum oscillations under the influence of an external non-linear force are derived. A wide spectrum of applications of the formed class of systems is presented.