Synergetics. Etudes 70.
Is Dedicated to Centennial of Burthday
of Outstanding Scientist, Professor
Basin Abram M.
Basina G. I., Basin M. A.
SIC ”Synergetics” of Saint-Petersburg
Association of Scientist and Scholars.
Etude 1.
Bifurcations of Exponent of
Circumference. Numbers of Basin.
21 October 2009
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
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.