  It has been known for almost $40$ years that general planar quadratic
polynomial systems can have four limit cycles. Recently, four limit cycles were
also found in near-integrable quadratic polynomial systems. To help more people
to understand limit cycles theory, the visualization of such four numerically
simulated limit cycles in quadratic systems has attracted researchers'
attention. However, for near integral systems, such visualization becomes much
more difficult due to limitation on choosing parameter values. In this paper,
we start from the simulation of the well-known quadratic systems constructed
around the end of 1979, then reconsider the simulation of a recently published
quadratic system which exhibits four big size limit cycles, and finally provide
a concrete near-integral quadratic polynomial system to show four normal size
limit cycles.
