Limit cycles in quadratic systems

Abstract

The other type of periodic orbits apart from the type encountered in the previous chapter on quadratic systems with a center point is the limit cycle; its introduction into mathematics goes back to Poincaré [P]. The limit cycle problem is considerably more difficult to solve than the center problem. A limited number of papers over the twentieth century was sufficient to clarify the center problem. However, by the end of that century, despite a multitude of papers, the limit cycle problem for quadratic systems is still left with quite a number of unsolved questions, some of them with poor hope for answers on the short run. This is due to the limitations of available methods of investigation and, in some questions, the pure absence of them. For that reason, it is difficult to give a well structured presentation of what is known about limit cycles in quadratic systems, illustrating at the same time that the underlying structure of this non linear phenomenon is not yet well understood.