Ivar Otto Bendixson - Scientific Achievements

Scientific Achievements

Bendixson started out very much as a pure mathematician but later in his career he turned to also consider problems from applied mathematics. His first research work was on set theory and the foundations of mathematics, following the ideas which Georg Cantor had introduced. He contributed important results in point set topology. As a young student Bendixson made his name by proving that every uncountable closed set can be partitioned into a perfect set (the Bendixson derivative of the original set) and a countable set. He also gave another important contribution when he gave an example of a perfect set which is totally disconnected.

Concerning solution of a polynomial equation by radicals Bendixson returned to Niels Henrik Abel's original contribution and showed that Abel's methods could be extended to describe precisely which equations could be solved by radicals.

The analysis problem which intrigued Bendixson more than all others was the investigation of integral curves to first order differential equations, in particular he was intrigued by the complicated behaviour of the integral curves in the neighbourhood of singular points. The Poincaré-Bendixson theorem, which says an integral curve which does not end in a singular point has a limit cycle, was first proved by Henri Poincaré but a more rigorous proof with weaker hypotheses was given by Bendixson in 1901.