Articles and Publication  Mathematics, calculation, statistic EXPONENT OF CIRCUMFERENCE AS THE PHASE TRAJECTORY OF NONLINEAR ITERATION RELATION. Etude 3.
EXPONENT OF
CIRCUMFERENCE AS THE PHASE TRAJECTORY OF NONLINEAR ITERATION RELATION. Etude
3.
Synergetics.
Etudes 70.
© Basina G. I.,
Basin M. A.
Contact to the
author:
basinm@yandex.ru
SIC
”Synergetics” of Saint-Petersburg Association of Scientist and Scholars.
Is Dedicated to Centennial of
Birthday of Outstanding Scientist, Professor
Basin Abram M.
In this etude are determined criteria of
equivalence of complex differential equation of first order and nonlinear
iteration. Conditions, at which the points, corresponding to the decision of
iterative relation, lay on the exponent of circumference, are determined.
___________________________________________________________________
At the investigation of dynamically systems may
be introduced the complex parameter of wholeness, found and solved complex usual
differential equation, to which it satisfies:
 .
(1)
Here 
may be complex algebraic variable
or complex spiral variable .
Family of phase trajectories, which are the aggregate of decisions of equation
(1), may be represented in the form:
 ,
(2)
where  -
the meaning of complex coordinate of phase trajectory passing through the point 
at the moment of time 0. This decision
may be
written and
in spiral
variables:
 .
(3)
Shall we build iteration relation equivalent to
the described dynamically system å.
Shall we solve equations (2, 3) to
the relation of  :
 .
(4)
Suppose, that we know the condition of
one-dimensional complex dynamically system in moment  ,
corresponding to the point 
and want to determine the condition of the same system  at
the moment  .
Then, using former formulas, we shall receive:
 .
(5)
We shall introduce the operator  :
 .
(6)
Operator generates
the iteration process and shows transformation of condition of dynamically
systemat
the moment of time
in its condition at
the moment
 .
(7)
The last equation describes discrete system,
equivalent to the continuous dynamically system.
It is another method of undergoing from
continuous model to the discrete model. Shall we write instead of system (1) the
approximate system
 ,
(8)
which, after series of transformations, becomes
the form:
 .
(9)
Operators and
 are
not equivalent: one aspirates to another at the attention of  to
zero. Shall we consider one practically important particular case. As the base
equation shall we take linear complex equation
 .
(10)
Decision of this equation has the form:
 .
(11)
Finally we receive iterative relation:
 .
(12).
So, in the case of constant time interval, the
iterative process of the first type, satisfying the equalities (5-7), represents
geometric series at the area of complex numbers. If the value 
is imaginary, then all members of series (12) lay on the circumference with
radius
.
Iterative process (9) in our particular case
becomes the form:
 .
(13)
Formulas (12) and (13) coincide only in limiting
case at  .
In the case of finite meanings of  these
iterative processes distinguish one from another then more as greater is the
modulus of value.
But in the second case we also have the geometric series. Shall we introduce in
consideration new complex variable:
 .
(14)
Then the equation (1) transforms in the form:
 ,-
(15)
and corresponding iterative processes have the
next form:
 .
(16)
Last equation describes iterative process, which
is equivalent to the equation (15).
Another, approximate variant of iterative process,
corresponding to the equation (15), has the form:
 .
(1
7)
In considered earlier particular case of linear
differential equation shall we take, that the relation of new variable from old
variable has the form  .
Then the iterative relation (12) after the series
of transformations shall have the form
 .
(18)
Iterative process (18) in the right part of
equation has the power function with complex power coefficient. For the adequate
description of such functions must be introduced the representation of the field
of spiral complex numbers. If the value
is the imaginary number, then all points of iteration process (18) lay on one of
exponents of circumference, which parameters determine by the meaning of  .
Second type of iteration process gives the next
equality:
 .
(19)
In this case we also have in the right part of
equation (19) the power function with complex index of power, but the points of
this iteration process lay on the exponent of circumference only in the case of
imaginary meaning of and
aspirating of its modulus to zero.
Publishing date: June 20, 2010
Source: SciTecLibrary.ru
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