"Discrete" oscillations and multiple attractors in kick-excited systems

A class of kick-excited self-adaptive dynamical systems is formed and propo sed, The class is characterized by nonlinear (inhomogeneous) external perio dic excitation (as regards to the coordinates of excited systems) and is re markable for its objective regularities: the phenomenon of "discrete" ("qua ntized") oscillation excitation and strong self-adaptive stability, The mai n features of these systems are studied both numerically and analytically o n the basis of a general model: a pendulum under inhomogeneous action of a periodic force which is referred to as a kicked pendulum, Multiple bifurcat ion diagram for the attractor set of the system under consideration is obta ined and analyzed, The complex dynamics, evolution and the fractal boundari es of the multiple attractor basins in state space corresponding to energy and phase variables are obtained, traced and discussed. A two-dimensional d iscrete map is derived for this case, A general treatment of the class of h ick-excited self-adaptive dynamical systems is made by putting it in corres pondence to a general class of dissipative twist maps and shelving that the latter is an immanent tool for general description of its behavior.