# Hilbert’s 16th Solution

To solve Hilbert’s 16th problem we investigate relative positions and upper bounds of 2 dimensional polynomial vector fields. Limit cycles are isolated periodic orbits in polynomial planar differential systems — a polynomial vector field possesses an infinite amount of limit cycles.

If a periodic orbit is not isolated, there is a center in which cycles belong to a period annulus. The upper bound for the number of limit cycles is the sum of elements of all cycles belonging to period annulus of the largest element in the set of extended order fields.

Stable periodic orbits corresponding to higher order resonance & quasi-periodic orbits contribute to dynamics in systems with cores. In cases with mixed (regular + chaotic) dynamics, quasi periodicity helps shape bulk of the density distribution.

Investigating orbital structure of galaxies, we’re interested in the birth & disappearance of orbit families + their stability & occupied phase space fraction. The solution to Hilbert’s 16th problem is the largest possible set of Fermi bubbles because limit cycles are their mathematical representation.

When galaxies collide, their Fermi Bubbles coalesce & grow in size. neighboring galaxies’ Fermi bubbles mapped relative to one another is the solution to relative position aspect of Hilbert’s 16th problem — differential equation shifts according to the obtained orientable phase space.

Since Fermi bubbles are dark matter, this differential solution to Hilbert’s 16th problem sets limits on the upper bound of dark matter halos. The Unified Theory of Cosmic Plasma Physics ties this solution to the maximum power of ultra high energy cosmic rays (solution to GZK puzzle). The dominant energy loss process for nuclei is photodisintegration. Tracking the differential relative positions of Fermi bubbles throughout the Baryon ionization history is vital to deriving the fine structure constant from 1st principles.

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