Numerical Bifurcation Analysis of Electronic Circuits

It is well known that the creation of the modern geometrical theory of dynamical systems by Poincaré at the end of the 19th century was motivated by problems arising in celestial mechanics [41]. Perhaps it is not so widely known that the dynamics of electronic circuits played an important role at the early stages of the development of this theory. In the 1920s Van der Pol [47] described the periodic oscillations of self-sustained circuits in terms of the limit cycles of Poincaré. Heperformed experiments withperiodically excited circuits and measured, for the first time, complex behavior in a nonlinear system. In the 1930s there was pioneering work of Andronov’s Russian school on the theory of oscillations in electronic, mechanical and control systems [11].

It is important to realize that these first applications of Poincaré’s qualitative theory to electronic circuits led to new concepts and theoretical results. Examples of this include Liénard’s theorems [40, Chap. 3], as motivated by the works of Van der Pol, the development of bifurcation theory for planar systems by Andronov and co-workers [9], and the introduction of the concept of structural stability by Andronov and Pontriaguin [10].

The relationship between the mathematical theory of ordinary differential equations (ODEs) and the dynamics of electronic circuits was initially very close, but this did not continue for very long. In fact, one might speak of a divorce between the two fields in the subsequent development, where electronic devices and systems became ever more complex and of greater dimension. Starting with the invention of the transistor in the 1950s, there was a true explosion in the size of electronic circuits, culminating in the ascent of the microchip — a complex circuit with thousands or even millions of components. In theoretical investigations of such electronic circuits one can hardly find a trace of the geometrical theory of dynamical systems. One of the few exceptions is the almost forgotten work of Hayashi’s school in Japan on the global dynamics in experiments with analog computers [46].