SYSTEMS OF DIFFERENCE EQUATIONS APPROXIMATING THE LORENZ SYSTEM OF DIFFERENTIAL EQUATIONS

A b s t r a c t: In this paper, starting from the Lorenz system of differential equations, some systems of difference equations are produced. Using some regularities in these systems of difference equations, polynomial approximations of their solutions are found. Taking these approximations as coefficients, three power series are obtained and by computer calculations is examined that these power series are local approximations of the solutions of the starting Lorentz system of differential equations.


INTRODUCTION
The use of power series is one of the oldest methods for examining differential equations.In the literature there are numerous papers concerned with such a use of power series, like the papers [1], [2] and [3].
We consider the well known Lorenz system of differential equations: , we assume that the solutions of the Lorenz system are expanded as Maclaurin series, (1.2).Using consecutive differentiation of (1.1) and the representation (1.2), for every N n ∈ , we obtain the following system of difference equations.
( ) (1.3) Our aim is to express the coefficients n a , n b and n c , as polynomials in variables σ, r, b, a 0 , b 0 and c 0 .Separately, for each of the coefficients n a , n b and n c we transform the system (1.3), by introducing new variables.

THE COEFFICIENT n a
We write the system (1.3) in the form (2.1).
( )  we take the following presentations: For the presentation (2.3) we obtain the system of difference equations (2.6).
For the presentation (2.4) we obtain the system of difference equations: For the presentation (2.5) we obtain the system of difference equations: .
For q k = , directly from the presentations (2.3), (2.4) and (2.5), the , 2.1.and 2.2.imply: With all this, the question of finding a solution for n a is transformed to the question of solving the systems of difference equations (2.6), (2.7) and (2.8).
We start with the system (2.6).By finding expressions for for several small values of n, k, q we found that there are some regularities, and after long calculations, we obtained the following presentations: where: , , satisfy the following system of difference equations: By 2.2. and 2.3., it is enough to find , and since variables σ, r, b, a 0 , b 0 and c 0 .For convenience with the signs, we use the notation: By suitably grouping the parts in these polynomials and calculating the first several of them, we found that: )   and for q>7, the polynomial n n q L , ˆ has the form of the polynomial . So, for q>7 we choose to approximate with the initial values To solve this difference equation, we take the following representation Using the presentation (2.14) we obtain the following system of difference equations: (2.15) The solutions of the system (2.15) are the polynomials: , 7 and from 2.2., for n = q, we obtain: , 7 Next, we continue with the system (2.7).Similarly as above we set: satisfy the difference equations (2.16), L , ˆ for, q from 1 to 6:  By the same argument as above, we arrive to the same difference equations as (2.13) and (2.14), but for q>6, and to the same equation (2.15).With all this we obtain: For the system (2.8), we set: where satisfy the difference equations (2.17), . The system (2.17) is analogous to the system (2.12) (i.e. the system (2.16)), but with different notations, and different initial values.By 2.2. and 2.3., it is enough to find for q from 1 to 5: ( ) ( ) ( ) Again, by the same argument as above, we arrive to the same difference equations as (2.13) and (2.14), but for q>5, and to the same equation (2.15).With all this, we have ( The above discussion and 2.3., produce the following approximation

THE COEFFICIENT n c
For the coefficients n c we consider the system of difference equations (2.1).
For an N n ∈ and for any are new variables.For the new variables we obtain the new systems of difference equations: , and transform the presentation of n c to: . 1 , implies: The systems (3.1) are the same as the systems (2.2), with different letters and different initial values.We use the following presentations as for the coefficient a n , with different symbols for all k q ≤ ≤ 1 , at fixed N k n ∈ , : , and by analogous discussion as for the coefficient a n , we obtain the following approximation for the coefficient c n , n>5, , 4 where the initial values for we calculate from (1.3) the exact values of j c and we set

THE COEFFICIENT n b
For the coefficient b n we write the system (1.3) in the form: ( ) ( are new variables.For the new variables we obtain the new systems of difference equations: . 1 ) , ( , and transform the presentation of n b to: Similarly as (2.3) Again, similarly as (2,6) and (2.7) we obtain the following systems: ( ) ) For , and by 4.1.we obtain: .
By the same discussion as for the coefficient n a we arrived to the following presentations: , have the form of the polynomials: ( ) So, for q>5 we approximate of the difference equation: with the initial values In the same way as for n a , we take the presentation and obtain the system of difference equations of the difference equation: with the initial values Similarly as for n a , all this produces the following approximation:    Next we turn our attention to the following polynomials, (5.2).

1 )
At this moment the following question is open.Question: What conditions would imply the convergence of the power series (5.1).
good approximations for the solutions of the system (1.1).We used the program Mathematica and compared the solutions obtained by the program Mathematica with the functions ) 05; m=20; and the time interval [0,6].

Fig. 1 .
Fig. 1. Results obtained by the program Mathematica The system (2.16) is the system (2.12), but with different notations, and different initial values.
By 2.2. and 2.3., it is enough to find