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Articles and Publication  Mathematics, calculation, statistic EXPONENTS OF CIRCUMFERENCE AS PHASE CURVES OF NONLINEAR DYNAMICALLY SYSTEMS.
EXPONENTS OF
CIRCUMFERENCE AS PHASE CURVES OF NONLINEAR DYNAMICALLY SYSTEMS.
Etude 2.
© 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
At the investigation of dynamically systems for
the determination of their integral parameters we recommend the introduction of
complex parameter of wholeness and also finding and decision of complex
differential equation, to which it satisfies.
 ,
(1)
where 
may be complex algebraic variable 
or complex spiral variablle  .
Phase trajectory which is the decision of equation (1), may be written in the
form
 ,
(2)
where  is
the meaning of algebraic complex coordinate of phase trajectory, passing through
the point 
at the moment of time  .
The same decision may be written and in spiral varieties.
 .
(3)
Shall we introduce new function  .
Then the equation (1) will transform in the
form  ,
or
 .
(4)
Corresponding phase trajectory for variable
quantity 
has the form:
 So,
having the decision of one equation (1), we may build the class of derivative
nonlinear differential equations, connected with given equation, which decisions
are building on the base of decision of given equation. It is interesting one
important particular case of fulfilled transformations. As the main equation we
take the linear complex differential equation
 .
(5)
Its decision may be received with such a
matter.:  .
Integrating both parts of received equation, we shall receive:  .
Exhibiting both parts of received equation, we receive:  .
If  ,
then  ,
and phase trajectories of decision of this equation have the form
 .
(6).
Shall we represent:  .
Then the decision of the equation (5) shall receive the form:
 .
(7)
Phase curves represent the developing or
twisting spirals, or coming to zero, or going away on the infinity. Further us
will be interesting the particular case, corresponding to  .
In this case we have  .
If we shall consider, not losing the generality, that the primary meaning of the
unknown quality is the real number  ,
then the equation of phase trajectory has the form:
 .
(8)
The point of phase trajectory of the
equation (5) in considerable particular case moves on the circumference with
radius  -
on the cyclic trajectory. Further shall we take that the connection between
complex variables
and 
has the form
(9).
Then the equation, describing the dynamics
of variable
receives the form:  ,
or
 .
(10)
This equation we met at the investigation of
dynamics of populations.
Cyclic phase trajectories of dynamically
system, describing with this equation, have the form 
The last equality describes the trajectory of the
point, moving on the exponent of circumference (See Etude ¹1). So, the
exponent of circumference is a cyclic phase trajectory of any dynamically system,
describing with complex differential equation (10).
But with this not exhaust the class of
differential equations, having phase trajectories with the type of exponent of
circumference. It is known that in dynamically systems of different type occurs
the bifurcation of cycles birth. If we shall consider the variables of these
systems as the logarithms of any new variables, then for these new variables we
receive the equations, for which the bifurcation of cycles birth shall transform
in the bifurcation of the birth of any new trajectory which is the exponent of
circumference.
The undergoing from cyclic phase trajectory to
the phase trajectory in the form of the exponent of circumference must be
standard for self organizing information- transport systems, which have
different but strongly connected dimensions of variables, describing them.
.
Publishing date: January 5, 2010
Source: SciTecLibrary.ru
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