Articles and Publication  Mathematics, calculation, statistic BIFURCATIONS OF EXPONENT OF CIRCUMFERENCE. NUMBERS OF BASIN.
BIFURCATIONS OF
EXPONENT OF CIRCUMFERENCE. NUMBERS OF BASIN.
Etude 1.
© Basina G.
I., Basin M. A.
Contact to the author: basinm@yandex.ru
SIC
”Synergetics” of Saint-Petersburg Association of Scientist and Scholars.
Synergetics. Etudes 70.
Is Dedicated to Centennial of
Burthday of Outstanding Scientist, Professor
Basin Abram M.
Phase trajectories of linear
dynamically systems are often represent closed cycles, near to the
circumferences. In complex field 
circumference with radius 
may be described with formula  ,
where 
Most typical nonlinear problem, yielding to analytically solving, is the problem
of determining of dynamics of system, logarithms of which dynamic parameters are
subordinated to lineal regularities. Shall we call as exponent of circumference
the closed curve in complex plane, which ensues as reflection  .
Shall we divide in the last formula the real part from imaginary part  .
Shall we find the points of
crossing of exponents of circumference with real axis. Condition of there
finding may be written in the form:  .
From here follows, that at 
 .
Last equation has counting multitude of decisions  .
From here we receive:  .
Decisions of last equation subordinate to the restriction 
or  .
If  ,
then  .
Real coordinate of exponent of circumference at the points of crossing with real
axis determines for the next formulas:
Åñëè 
òî  .
If 
then  .
If  then
 .
If 
then  .
If we shall consider parametric
family of exponents of circumference, depending from parameter  ,
then meanings of 
are bifurcation meanings. If  ,
then at the achieving of
appear “from nothing” two negative meanings of points of crossing of
exponent of circumference with abscissa (radicals) in the area of spiral numbers,
which may be identify in the area of algebraic complex numbers with -1. Later at
the increasing of 
these radicals split and go away from -1, one to the right, another –to the
left. So, appear and then slide four (two double) radicals. This process
continues till then
achieves the value  .
Then appears new bifurcation: at the area of spiral complex numbers appear two
new radicals, having at the area of algebraic complex numbers the same meaning  .
Then occurs splitting of each of these radicals on two, which go away from 1 at
different sides. Explanations of these “tricks” may be found, if we shall
deduce the character of changing of exponent of circumstance depending from
parameter 
in the complex plane. Shall we consider the dynamics of the point, corresponding
 .
We have  .
That means that this point moves with the changing of
on the circumference with the radius equal to 1, rotating then more quickly as
more is the value of .
This circumference crosses abscissa at the points -1 and +1. That why new
radicals form namely in these points. So the meanings of abscissas of exponents
of circumference at the points of crossing with real axis determine for the next
formulas:
If
then  .
If
then  .
If then
 .
If
then  .
The numbers  ,
which are the combinations of transcendent and algebraic irrational numbers, are
the universal no dimensionally numbers, which characterize bifurcation dynamics
of system of exponents of circumference.
On the honor of Centennial of
Birthday of outstanding scientist, professor Abram M. Basin we called them
Numbers of Basin.
As the universality of fulfilled
analysis, we consider, that these numbers in one or another form must be found
at the investigation of nonlinear dynamically systems.
Publishing date: December 10, 2009
Source: SciTecLibrary.ru
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