  We consider two types of $p$-centro affine flows on smooth, centrally
symmetric, closed convex planar curves, $p$-contracting, respectively,
$p$-expanding. Here $p$ is an arbitrary real number greater than 1. We show
that, under any $p$-contracting flow, the evolving curves shrink to a point in
finite time and the only homothetic solutions of the flow are ellipses centered
at the origin. Furthermore, the normalized curves with enclosed area $\pi$
converge, in the Hausdorff metric, to the unit circle modulo SL(2). As a
$p$-expanding flow is, in a certain way, dual to a contracting one, we prove
that, under any $p$-expanding flow, curves expand to infinity in finite time,
while the only homothetic solutions of the flow are ellipses centered at the
origin. If the curves are normalized as to enclose constant area $\pi$, they
display the same asymptotic behavior as the first type flow and converge, in
the Hausdorff metric, and up to SL(2) transformations, to the unit circle. At
the end, we present a new proof of $p$-affine isoperimetric inequality, $p\geq
1$, for smooth, centrally symmetric convex bodies in $\mathbb{R}^2$.
