Introduction to Part Ii

In the following chapters we present the theory of bifurcations of dynamical systems with simple dynamics. It is difficult to over-emphasize the role of bifurcation theory in nonlinear dynamics the reason is quite simple: the methods of the theory of bifurcations comprise a working tool kit for the study of dynamical models. Besides, bifurcation theory provides a universal language to communicate and exchange ideas for researchers from different scientific fields, and to understand each other in interdisciplinary discussions. Bifurcation theory studies the changes in the phase space as we vary the parameters of the system. In essence, this is the authentic notion of bifurcation theory proposed originally by Henry Poincaré when he studied Hamiltonian systems with one degree of freedom. We must, however, note that this intuitively evident definition is not always sufficient at the contemporary stage of the development of the theory. One needs, in fact, to have an appropriate mathematical foundation to define the notions of the structure of the phase space and the changes in the structure. The first attempt at creating such formalization had been made by Andronov and Pontryagin in 1937: they introduced the notion of a rough system. For a system to be rough, it means that any sufficiently close system is to be topologically equivalent to the given one. Moreover, the conjugating homeomorphism must be close to identity. In other words, the two systems must have matching phase portraits and corresponding trajectories can differ only slightly. In the same paper, Andronov and Pontryagin had presented the necessary and sufficient conditions of roughness for systems on the plane. Consequently,