Synthesis of nonlinear systems with prescribed singularities

This paper presents an affirmative answer to the heretofore unresolved question as to whether an autonomous nonlinear system can always be synthesized to possess a finite number of prescribed singular points, as well as a prescribed set of eigenvalues associated with each singular point. The answer is given in the form of a new canonic nth-order nonlinear system containing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(3n-2)</tex> single-valued functions. In the case <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n=2</tex> the canonic system is shown to possess many nice general properties which are useful for determining aspects of the system's global qualitative behavior, such as global stability.