On a nonlinear p-adic dynamical system

We investigate the behavior of trajectories of a (3, 2)-rational p-adic dynamical system in the complex p-adic field ? _p , when there exists a unique fixed point x ₀. We study this p-adic dynamical system by dynamics of real radiuses of balls (with the center at the fixed point x ₀). We show that there exists a radius r depending on parameters of the rational function such that: when x ₀ is an attracting point then the trajectory of an inner point from the ball U _r (x ₀) goes to x ₀ and each sphere with a radius > r (with the center at x ₀) is invariant; When x ₀ is a repeller point then the trajectory of an inner point from a ball U _r (x ₀) goes forward to the sphere S _r (x ₀). Once the trajectory reaches the sphere, in the next step it either goes back to the interior of U _r (x ₀) or stays in S _r (x ₀) for some time and then goes back to the interior of the ball. As soon as the trajectory goes outside of U _r(x ₀) it will stay (for all the rest of time) in the sphere (outside of U _r(x ₀)) that it reached first.