^{1}

We construct the family of Poisson nonlinear transformations defined on the countable sample space of nonnegative integers and investigate their trajectory behavior. We have proved that these nonlinear transformations are regular.

Let

Let

We consider a nonlinear transformation called quadratic stochastic operator (qso)

If a state space

In this case, the probabilistic measure

In this paper, we consider nonlinear transformations defined on countable state space and investigate their limit behavior of trajectories.

Let

Let

In this case, a qso (

In this paper, we consider a Poisson qso which is a Poisson distribution

Let

A quadratic stochastic operator

Assume that

In this paper, we will study limit behavior of trajectories of Poisson qso.

Let us consider a qso

A measure

Let

A qso

In measure theory, there are various notions of the convergence of measures: weak convergence, strong convergence, and total variation convergence. Below we consider strong convergence.

For

If

In statistical mechanics, the ergodic hypothesis proposes a connection between dynamics and statistics. In the classical theory, the assumption was made that the average time spent in any region of phase space is proportional to the volume of the region in terms of the invariant measure. More generally, such time averages may be replaced by space averages.

For nonlinear dynamical systems, Ulam [

A nonlinear operator

On the ground of numerical calculations for quadratic stochastic operators defined on

In 1977, Zakharevich [

In the next section, we will show that Ulam’s conjecture is true for some class of Poisson qso.

Let

We call a Poisson quadratic stochastic operator

Thus

A homogenious Poisson qso is a regular transformation.

We consider a Poisson qso such that

For any initial measure

By simple calculations, we have

Thus, by using induction on the sequence

It is obvious that the limit behavior of the recurrent equation (

Since

Solving the following quadratic equation

Graph of the function (

When

When

Thus, for any initial measure

Thus, for any initial measure

As corollary we have following statement.

A Poisson qso with two different parameters is a regular and, respectively, ergodic transformation with respect to strong convergence.

We consider a Poisson qso such that

Thus, by using induction on sequence

It is obvious that the limit behavior of the recurrent equation (

Since

Starting from arbitrary initial data, we iterate the recurrence equations (

Limit behavior of the dynamical system (

Diagram when

Diagram when

One can prove that for any values of parameters

If these parameters are very small, for instance,

As above, from (

As corollary we have following statement.

A Poisson qso with three different parameters is a regular and, respectively, ergodic transformation with respect to strong convergence.

In this paper, we present a construction of Poisson quadratic stochastic operators and prove their regularity when the number of different parameters

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by Ministry of Higher Education Malaysia (MOHE) under Grant FRGS14-116-0357.