Asymptotic behavior of solutions of a system of nonlinear differential equations with small parameter

. The present paper addresses a qualitative pattern of the behavior of solutions of a system of ordinary differential equations when small parameter tends to zero at a finite amount of time where slow variable passes through a certain point that corresponds to a bifurcation in the system of fast motions: stable limit cycle merges with unstable one and disappears. The problem of generation of an asymptotic approximation of the solution of a perturbed differential equation system is considered in the case where a bifurcation occurs in the “fast motions” equation when the parameter changes: two equilibrium positions merge, followed by a change in stability. The results of the article are used to determine second-order perturbations in rectangular coordinates and components of the velocity of the body under study. The coefficients at the projections of the perturbing acceleration are integral functions of the independent regularizing variable. Singular points are used to reduce the degree of approximating polynomials and to choose regularizing variables.


Introduction
One of the essential problems related to trajectory measurements is to approximate the rectangular coordinates, which constitute velocity components of the moving body, with low-degree algebraic polynomials with respect to an artificial variable with a predefined degree of accuracy.General principles for developing the perturbation theory in a coordinate system are addressed in works [1][2].Various types of generic variables such as Sk and b are considered in work [2], and some ways of using generic variables in a number of problems in mechanics are outlined, in particular, for the purpose of determining perturbations using the method of variation of constants.In works [3][4][5], artificial variables Sk are introduced in the form of power series with respect to variable b, and issues of using regularizing variables to calculate passive flight trajectories are addressed.When calculating perturbations in a coordinate system, the small parameter method was used, which allowed for obtaining asymptotic decomposition of the solution in the neighborhood of the initial point.
In work [11], a basic procedure is described to derive a specific system of differential equations of perturbed motion of the body.The main advantage of the obtained system of differential equations of perturbed motion is that the coefficients at X,Y.Z components of the perturbing acceleration are integer functions of an independent regularizing variable ѱ.This make it possible to efficiently calculate perturbations not only of first order but also of higher orders.

Problem specification and decision
Let us consider a system of differential equations with small parameter at derivatives in the vector form.This type of systems is utilized in many problems of the oscillation theory.Let us investigate the behavior of the solutions of the system (1) when 0   depending on the properties of the stationary solutions of the system of fast motions.
( , ), ( , ), where х and f are k -dimensional vectors, y and g are l -dimensional vectors, 0   is a small parameter.This type of systems is utilized in many problems of the oscillation theory.Let us investigate the behavior of the solutions of the system (1) when 0   depending on the properties of the stationary solutions of the system of fast motions.
( , ); dx f x y d  where , t y Let us consider a case when stable and unstable limit cycles merge in the system (2) with a change of parameter y , as a result of which two new cycles emerge with the change of stability.Let us assume that the system (2) when 0 yy  has a degenerate limit cycle 0 ()


with a period 0 1.T  Now we consider an analog of first return function for the system (2), ( , ) uy   is self-mapping of ( 1)   k  -dimensional space of variables 11 ,..., k uu  , and 0 u  is a fixed point of this mapping at 0 , yy  which corresponds to the limit cycle 0 ( ).


Let us assume that one of the eigenvalues of matrix 00 0 ( , ) , ( , 1,..., 1) is equal to 1 and the absolute value of all other eigenvalues are less than 1.

Let
In this case, with additional constraints imposed to the right parts of the system (1), there exists a smooth surface  in the form of 12 ( ,..., ),

.
Let us consider averaged system of differential equations:

(
), i dy gy dt The main result may be summarized as follows.Let us put ( , ), ( , ) x t y t  as a solution of the system (1), the initial point of which is in the neighborhood of  -order of the curve   where,i=1 at 12 Using transformation of coordinates, the system (1) may be reduced to where, In short, the system (9) may be written as ( , , ), d Fs ds where the functions ( , , ), Fs ( , , ) Gs are periodic functions with respect to variable s with a period of 2 and ( , 0, 0) 0 Fs  Equation of the surface  is written as 1 0   .  , and the second one will be stable at 1 0   .The initial values for these limit cycles will be fixed points of first return mapping 12 ( ), ( ) UU  Taking into account the specific form of the right parts in the system of differential equations ( 9) and the dependance of its solutions on the initial conditions, let us introduce the following auxiliary formulars:   If the initial point of the solution ( , ), ( , ) ss     of the system ( 9) is in the region 1 D , then the solution remains in the region We may find the derivative 1 V from the system of differential equations ( 9) in those points of the region 1 D which satisfy the equation ( 19) By making transformations and using equalities ( 11)-( 15), the derivative Where from, while assuming: Similarly, it may be demonstrated that 1 0 V  at the points satisfying the equation ( 21)

 
For the derivative 2 V , from the system of differential equations (9) at the points satisfying the equation ( 22) the following inequality is true: Considering the conditions that define the region 1

D
with sufficiently high M , we obtain the derivative 2 0 V  .
For the case when , we define a region 2 D using inequalities (24). ,

   
To illustrate the developed research methodology for the convergence of successive approximations, calculations [9] were carried out with specified initial conditions, parabolic motion conditions of the body, considering assumptions made, and this resulted in the numerical estimate of the convergence domain.Several variants were calculated with different values of the variable u ; the dependence of the variable u on С* was established, and it was proved that this dependence has the extremum.Dependence of the parameter C* on the quantity of the scale factors i  (i=1,2,3,4,5,7) was analyzed.In the case of degeneracy the dependence of the quantity C* on the variable u has no extremum with 0  i  (i=1,2,3,4,5,7), since the quantities A and B do not depend on the C*.When increasing the factors i  , this dependence has an extremum, which indicates an extension of the convergence domain.The quantity C* is limited, so it can be assumed that the scale factors fall within a certain range of variation.
When giving the value of the parameter C*, it was defined more precisely using the developed technique.For the given initial conditions, we found the extremal value of the quantity C*=4,987 and the corresponding value of the independent variable 1 u =0.24.Thus, if the mismatches in the rectangular coordinates of a body perturbed motion at the same instants of time at different stages of approximation do not exceed the quantity C*=4,987 the successive approximations which represent the solution of the perturbed motion differential equations system converge absolutely.
Figure 1 shows the dependence of the parameter C* on the standardized regularizing variable u (the dotted line shows the dependence in the case of degeneracy with 0 

Results
The main paper conclusions are as follows:  The problem of generation of an asymptotic approximation of the solution of a disturbed differential equation system is considered in the case where a bifurcation occurs in the "fast motions" equation when the parameter changes: two equilibrium positions merge, followed by a change in stability.
 The qualitative nature of behavior of solutions to the system of ordinary differential equations in which a small parameter tends to zero on a finite time interval when the slow variable passes through a point corresponding to a bifurcation in the "fast motions" system, where a stable limit cycle merges with an unstable one and disappears, is explored. The results of the article were used to determine the second-order perturbations in rectangular coordinates and the components of the velocity of the body under study.The coefficients at the projections of the perturbing acceleration are integral functions of the independent regularizing variable.
 Singular points are used to reduce the degree of approximating polynomials.
one degenerate limit cycle of the same type as 0 () the same sign as parameter por have the sign opposite to that of parameter p the surface  and is directed to the region of stability of the nondegenerate limit cycle is stable.Let us put the solution of the averaged system of differential equations (5) as () ytat 1 i  and with an initial condition 11 () y t y  .At a certain moment of time 2 t , the solution of the averaged system of differential equations () ytfalls on the surface  .Let us continuously extrapolate the solution ()

1 0
Let us provide   evidence for this result.