ABSTRACT
The new methods, applied to systems of ordinary differential equations, are known as dynamical systems theory. The state space, with the attractors, basins, and separatrices drawn upon it, is called the portrait of the dynamical system. This portrait comprises the full understanding of the dynamical behavior of the model; at least as far as long-run prediction is concerned. Yet the emerging theory of bifurcations of dynamical schemes is very promising, as it provides the beginnings of an encyclopedia of atomic bifurcations, of which all response diagrams are made. The response diagram of the scheme is a graph showing the dependence of the portrait upon the control parameters. The response diagram is the master map which gives this kind of model great power in applications. Catastrophe theory has provided excellent pedagogic examples of response diagrams for various schemes, establishing their importance as graphic representations in many scientific disciplines.
