Tutorials and Reviews

In the 19th century, Joseph Fourier wrote: “The study of Nature is the most productive source of mathematical discoveries. By offering a specific objective, it provides the advantage of excluding vague problems and unwieldy calculations. It is also a means to form the Mathematical Analysis, and isolate the most important aspects to know and to conserve. These fundamental elements are those which appear in all natural effects”. The important development of the theory of dynamic systems during this century essentially has its origins in the study of the “natural effects” encountered in systems of mechanical, electrical, or electronic engineering, and the rejection of non-essential generalizations. Most of the results obtained in the abstract dynamic systems field have been possible on the foundations of results of the concrete dynamic systems field. It is also worth noting that the majority of scientists (including mathematicians) were not led to their discoveries by a process of deduction from general postulates or general principles, but rather by a thorough examination of properly chosen cases and observation of concrete processes. The generalizations have come later because it is far easier to generalize an established result than to discover a new line of argument. Since Andronov (1932), three different approaches have traditionally been used for the study of dynamical systems: Qualitative methods, analytical methods and numerical methods. To define the “strategy” of qualitative methods, one has to note that the solutions of equations of nonlinear dynamic systems are in general nonclassical, transcendental functions of Mathematical Analysis, which are very complex. This “strategy” is of the same type as the one used for the characterization of a function of the complex variable by its singularities: Zeros, poles, essential singularities. Here, the complex transcendental functions are defined by the singularities of continuous (or discrete) dynamic systems such as: