The Second Part of Hilbert's 16th Problem
Here we are going to consider polynomial vector fields in the real plane, that is a system of differential equations of the form:
where both P and Q are real polynomials of degree n.
These polynomial vector fields were studied by Poincaré, who had the idea of abandoning the search for finding exact solutions to the system, and instead attempted to study the qualitative features of the collection of all possible solutions.
Among many important discoveries, he found that the limit sets of such solutions need not be a stationary point, but could rather be a periodic solution. Such solutions are called limit cycles.
The second part of Hilbert's 16th problem is to decide an upper bound for the number of limit cycles in polynomial vector fields of degree n and, similar to the first part, investigate their relative positions.
Read more about this topic: Hilbert's Sixteenth Problem
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