  We consider a family of $(2,2)$-rational functions given on the set of
complex $p$-adic field $\mathbb{C}_p$. Each such function has a unique fixed
point. We study $p$-adic dynamical systems generated by the $(2,2)$-rational
functions. We show that the fixed point is indifferent and therefore the
convergence of the trajectories is not the typical case for the dynamical
systems. Siegel disks of these dynamical systems are found.
  We obtain an upper bound for the set of limit points of each trajectory,
i.e., we determine a sufficiently small set containing the set of limit points.
For each $(2,2)$-rational function on $\mathbb{C}_p$ there are two points $\hat
x_1=\hat x_1(f)$, $\hat x_2=\hat x_2(f)\in \mathbb{C}_p$ which are zeros of its
denominator. We give explicit formulas of radiuses of spheres (with the center
at the fixed point) containing some points such that the trajectories (under
actions of $f$) of the points after a finite step come to $\hat x_1$ or $\hat
x_2$. Moreover for a class of $(2,2)$-rational functions we study ergodicity
properties of the dynamical systems on the set of $p$-adic numbers $Q_p$. For
each such function we describe all possible invariant spheres. We show that the
$p$-adic dynamical system reduced on each invariant sphere is not ergodic with
respect to Haar measure.
