We consider some dynamical properties of two-dimensional maps having an inverse with vanishing denominator. We put in evidence a link between a fixed point of a map with fractional inverse and a focal point of this inverse.

A specific class of maps, defined by

In this paper, which is a continuation of a former study [

We put in evidence necessary conditions and/or sufficient for a focal point to be a fixed point of the inverse map.

This paper is organized as follows. In Section

In this paper, we use the notation introduced in [

Consider the map (

We can compute prefocal curve and the focal point analytically by the following method (see [

The presence of a vanishing denominator induces important effects on the geometrical and dynamical properties of the map. In this case, we need to locate geometrically the focal point in the phase plane. This concept is stated and proved in the following proposition [

Let

Let

Some of such behaviors can also be observed in maps without a vanishing denominator, but their inverses have a vanishing denominator.

Let

We now describe our main results concerning several focal points in more detail, we confirm and extend some others for the simplest case with one focal point as stated in [

The following proposition shows the link between the structure of the basin of attraction of an attractor of

Let

Let

First, we recall the following implications:

We denote by

( 1) We show that if

Suppose the contrary, in other words there exists

( 2) Conversely, let us prove now that if all

Suppose that all

Let

The proposition does not apply for the case of the periodic attractors of period

The proposition which follows, localizes a focal point of

Let

The following proposition made the link between prefocal curves of

Let

( 1) Let us show that

( 2) Now let us show that

We can give another necessary condition so that a focal point of

Let

Let

Case of a polynomial mapping, having a unique fractional inverse with a unique focal point.

We consider this map

First let us fix

( 1) For

( 2) For the value

( 3) For the value

Case of polynomial mapping, having a unique fractional inverse with two focal points is considered.

We consider the following example:

( I) Let us check that

The result is quite similar for the focal point

(II) Study of the basin bifurcation “connected

Fix

For the value

For the value

For the value

(III) We can obtain basin bifurcations curves, when the focal points

For

By identification of

By identification of

In the parameter plane (

Fractional maps have suggested interesting analysis of the dynamic behavior and showed that they have their own characteristics, and have revealed new types of singularities considered as theoretical tools for analyzing the bifurcation phenomena. In this paper, we have presented complementary results that appear not to have been given in the works cited therein. We have proved some results, the first permits to localize geometrically the focal point in the phase plane. The second result makes the link between the property of the basin of attraction of an attractor of a map

This work has been performed under the activity of the National Research Project B01120060034.