Point Mapping

Abstract

In system analysis a dynamical system of finite degrees of freedom is often modeled in the form of an ordinary differential equation

$$\dot x = F\left( {x,t,\mu } \right);x \in {\Bbb {R}^N},t \in \Bbb {R},\mu \in {\Bbb {R}^K}, $$

(2.1.1)

where x is an N-dimensional state vector, t the time variable, μ a K-dimensional parameter vector, and F a vector-valued function of x, t,and μ. A motion of the system with a given μ defines a trajectory in the N-dimensional state space of the system which will be denoted by X ^N. We assume that F(x, t, μ) satisfies the Lipschitz condition so that uniqueness of solutions is assured. For cases where F(x, t,μ) may be such that the state variables of the solution suffer discontinuities at discrete instants of time, we assume that sufficient information is provided and the physical laws governing the discontinuities are known so that the magnitudes of the discontinuities at these instants can be deter­mined uniquely without ambiguity.