3. Theoretical basics of the perturbation account

3.1. The double time method (2T)

One of the important goals of this dissertation is the qualitative and quantitative analysis of the errors of the average impulse mechanisms, that is, the frequency disturbances of the clock oscillator caused by their impulse mechanisms. The key results of such an analysis will be described by the specific formulas for calculating the change in the timing of the timer. How is the intention to carry out the analysis using the theory of perturbation, and this, inter alia, by the method of double-scale (scale) of time, will first in this chapter summarize the essence of the method mentioned above. In the third chapter of this dissertation, after explaining the perturbation technique of the double time of time, the essence of the Krylov and Bogoliubov methods is briefly explained. In this way, control can be made of the obtained results and possibly compare both methods and give a critical overview of the benefits and difficulties of their applications.

The double-time technique belongs to one general method of perturbation known under the names: analysis of multiple proportions or scale variables, multiple expansion or multiple-scale methods. It is used for the analysis of phenomena whose states can be described by generalized coordinates of distinctly different dimensions. Poorly suppressed oscillations (linear or nonlinear, free or forced) represented by differential equation

$$𝑥̈ + ω^2𝑥 + ε𝑓(𝑥, 𝑥̇) = 0;\, ε∈(0,1), ε≪1,\tag{3.1}$$

a typical example of such phenomena. In equation (3.1), the free oscillations are poorly disturbed (perturbated) by the member $ε𝑓(𝑥, 𝑥̇)$ which, in general, is non-linear. The constant $ε$ can be chosen in different ways, and in this case, it is convenient that it represents exactly the attenuation factor $ξ$. Namely, in the case of free oscillations with low attenuation, the rate of decrease in the amplitude is significantly lower in relation to the angular oscillation frequency, and in the case of forced oscillations, a very slow change in oscillation phase may occur. The impression is that there are two different scales of time within the oscillatory process. The introduction of multiple proportions of variables into the perturbation account is attributed to mathematician and physicist Poincaré, although he himself claimed that the idea originated from the astronomer Lindstedt. By dealing with the general theory of perturbation, mathematicians Krylov and Bogoliubov, apart from the method of averaging, also developed in detail the technique of double the time.

In order for the double-time process to be perfectly clear, it is necessary first to emphasize the motivation for its application. The purpose and meaning of the double-time technique will become apparent when the crucial deficiency of the method of regular perturbations is discovered. Therefore, an attempt will be made to use the method of regular perturbations to approximately solve the following linear differential equation of the second order with constant coefficients:

$$𝜑̈ + 2ξω_0φ̇ + ω_0^2φ = 0;\,ξ∈(0,1), ξ≪1.\tag{3.2}$$

It describes free oscillations, with its own angular frequency $ω_0$ and small attenuation, if the attenuation factor is $ξ∈(0,1), ξ_1$. The initial conditions are known ($φ(0) = Φ_0, φ̇ = 0$), but they do not matter for this demonstration. This attempt will not lead to a correct solution. It will be precisely this failure to illustrate the disadvantage of regular perturbations that will justify the reason for the introduction of the double-timing method. It should be noted that in equation (3.2), the term $2ξω_0φ̇ $ represents a small disturbance, i.e, a small perturbation of the free unstressed oscillations described by the equation

$$𝜑̈ + ω_0^2φ̇ = 0;\tag{3.3}$$

because the attenuation factor $ξ$ is a very small positive number. The method of regular perturbations requires an approximate solution of the differential equation (3.2) in the form of a perturbation order

$$φ(𝑡) = φ_0(𝑡) + ξ^1φ_1(𝑡) + ξ^2φ_2(𝑡) + ⋯ + ξ^𝑛φ_𝑛(𝑡) + 𝑂(ξ^{𝑛 + 1}),\tag{3.4}$$

which represents the sum of the exact solution $φ_0(𝑡)$ of the simplified differential equation (3.3), in which the disorder $2ξω_0φ̇ $ is ignored, and a number of perturbation members. The symbol $𝑂(ξ^{𝑛 + 1})$ represents the order of the size of the residue of the approximation and is $ξ^{𝑛 + 1}$. These members of the higher order remotely (perturbate) idealized solution $φ_0(𝑡)$, and thus correct it and approach the desired solution $φ(𝑡)$ of the equation (3.2), in which the disorder $2ξω_0φ̇ $ is not neglected. If $ξ∈(0,1)$, the higher order members should decrease successively, which means that the perturbation approximation should of the exact solution (3.4) is an asymptotic functional order. In such a development of an approximate solution to an asymptotic order, each individual function, as a member of the order, grows slower in relation to the previous one at an appropriate time interval. That this does not always have to be and the necessity will just illustrate this example.

The approximation solution of the equation (3.2) is required in the following form of the collection of the initial solution and correction of the first order:

$$φ(𝑡) ≈ φ_0(𝑡) + ξφ_1(𝑡),\tag{3.5}$$

in which all members $ξ^𝑛φ_𝑛(𝑡), 𝑛 > 1$ are ignored. In this case, it is certain that, for the given $ξ$, there is always a finite time interval $𝑡∈[0, 𝑡_𝑀]$, within which the rest of the approximation has an order of magnitude $𝑂(ξ^2)$. The above fact follows directly from the theorem that Murdock (James Murdock) argues in the book "Perturbation - Theory and Methods" [61]. If equation (3.5) to replace the differential equation (3.2) gives the equation:

$$𝜑̈_0 + 2ξω_0φ̇ _0 + ω_0^2φ_0 + ξφ̈_1 + 2ξ^2ω_0φ̇ _1 + ξω_0^2 φ_1 = 0.\tag{3.6}$$

In equation (3.6), the article $2ξ^2ω_0φ̇_1$ is ignored as a small value of higher order. Differential equation (3.6) can be decomposed into the following system of two differential equations:

$$𝜑̈_0 + ω_0^2φ_0 = 0\tag{3.7}$$ $$ξ(𝜑̈_1 + ω_0^2φ_1) = -2ξω_0φ̇ _0.\tag{3.8}$$

The solution of the equation (3.7) describes the free unbroken oscillations and for the initial conditions $φ_0(0) = Φ_0, φ̇_0 = 0$ reads:

$$φ_0(𝑡) = Φ_0 \cos ω_0𝑡.\tag{3.9}$$

If the solution (3.9) is replaced by the equation (3.8), the differential equation is obtained:

$$𝜑̈_1 + ω_0^2φ_1 = 2ω_0^2 Φ_0 \sin ω_0𝑡.\tag{3.10}$$

Equation (3.10) is a non-homogeneous non-homogeneous differential equation of the second order and can be solved by the Lagrange method of variation of the constants. For the initial conditions $φ_1(0) = 0, φ̇_1 = 0$, the equation solution (3.10) is given by:

$$φ_1(𝑡) = Φ_0(\sin ω_0 - ω_0𝑡\cos ω_0𝑡).\tag{3.11}$$

Thus, the approximate solution of the differential equation (3.2) is:

$$𝜑(𝑡)=𝛷_0\cos 𝜔_0𝑡+𝜉(𝛷_0(\sin 𝜔_0𝑡−𝜔_0 𝑡\cos 𝜔_0𝑡))+𝑂(𝜉^2).\tag{3.12}$$

It is immediately noticed that the solution (3.12), as the sum of the expressions (3.9) and (3.11), does not correspond to the solution given by the linear theory of oscillations. It is not correct from the point of view of the perturbation method, as it disrupts the asymptotic approximation to the exact solution, the process on which the theory of perturbation is essentially based. The main cause of this disturbance is the fact that with time $𝑡$, the member $ξ𝑡ω_0Φ_0\cos ω_0𝑡$ grows unlimited and thus loses the character of small corrections of the initial solution $φ_0(𝑡)$. Namely, in expression (3.12), for each $𝑡 = 𝑂(1/ξ)$ the second member grows to such an extent that its order of magnitude becomes equal to the order of the size of the first member, due to which the perturbation approximation of the exact solution ceases to have the character of asymptotic development. The expression of the form 𝑡cosω0𝑡 in the solution (3.12) is called a secular member (secular - from the Latin word saeculum, the one that has a time long, for centuries) and is a formal consequence of the expression:

$$2ω_0^2 Φ_0 \sin ω_0𝑡,\tag{3.13}$$

on the right-hand side of the equation, in the differential equation (3.10). Although there is no active external coercion, the term (3.13) is a fictitious cyclic influence - a coercion, which is described in the resonances with its own oscillations described by the homogeneous part of the differential equation (3.10). Therefore, this term is called a factor of resonant coercion or resonant forcing. And in general, any form of expression $A\cos ω_0𝑡 + 𝐵\sin ω_0𝑡$ in homogeneous differential equation form:

$$𝜑̈_1 + ω_0^2φ_1 = A\cos ω_0𝑡 + 𝐵\sin ω_0𝑡\tag{3.14}$$

represents a factor of resonant coercion and, in its particular solution, causes the appearance of secular members of the form $𝑡\cos ω_0 𝑡$ and $𝑡\sin ω_0𝑡$. The properties of the secular members are examined in detail by Murdock [61] and more precisely explains in what sense their appearance is "harmful". The secularity of such members can be apparent [61], for example in the approximations of periodic functions to infinite degrees. In such a case, secular members may be acceptable if the limited accuracy of an approximate solution is acceptable in a sufficiently short time interval. The same author points out that there are cases where the secular members are inevitable, for example in approximate solutions in the form of asymptotic functional orders, when neither the exact solution is limited, that is, when it also contains secular members [61]. In the case of solving the differential equation (3.2) by the method of regular perturbations, an approximate solution (3.10) containing secular members can be acceptable accuracy, but only in a sufficiently short time interval, which can be estimated by a numerical calculation. In particular, for $𝑡 = 𝑡_𝑆 = 1/(ξω_0)$ the secular member is equated with the first term, and for $𝑡≥𝑡_𝑆$ the asymptotic development ceases to be valid. Time interval $𝑡∈[0, 𝑡_𝑀]$ In which the approximate solution (3.10) has an acceptable accuracy is usually extremely short $𝑡_𝑀≪𝑡_𝑆$ . As already pointed out, at this time interval, for some particular value of the damping factor $ξ∈(0,1)$, the remainder of the approximation has an order of magnitude $𝑂(ξ^2)$. If the members of the higher order $ξ^𝑛φ_𝑛(𝑡), 𝑛>1$ were used in the perturbation calculation , the order of the size of the residual approximation would decrease $𝑂(ξ^{𝑛+1}), 𝑛>1$, but would always be valid at the same time interval $𝑡∈[0, 𝑡_𝑀]$ on which lower-order approximations were also valid. Mardock [61], points out in the commentary of Theorem 3.2.1. that the time interval of the first-order approximation can be extended to a time interval $𝑡∈ [0, 𝑡_𝑀/ ξ ]$ with increasing the order of the size of the approximation residue with $𝑂(ξ^2)$ to $𝑂(ξ^1)$, provided that the functions $φ_0(𝑡)$ and $φ_1 (𝑡)$ perturbation approximations do not contain secular members. Thus, in general, the method of regular perturbations will not be suitable for application whenever differential equations contain resonant coercive factors, or whenever the solutions of these differential equations consequently contain secular members. This fact is a key deficiency of the method of regular perturbations and a motive for applying another technique to a perturbation account.

The double time method is one of the alternative techniques that goes beyond the problem of the secular members present in the method of regular perturbations. In addition, the double-time method extends the time interval of approximation validity, and this is the order of the magnitude of the constant $ξ$, which in the case of regular perturbations is generally impossible to achieve. Namely, precisely by the elimination of secular members, the double time method ensures that the remainder of the approximate approximation of the approximate solution $φ(𝑡)≈φ_0(𝑡)$ has the order of the size $𝑂(ξ)$ at a time interval $𝑡∈[0, 𝑡_𝑀 / ξ]$. This time interval is for the order of magnitude of the damping factor $ξ$ is longer than the time interval $𝑡∈ [0, 𝑡_𝑀 ]$ on which the residue the approximation of the order $𝑛 = 1$, obtained by regular perturbation, has the order of magnitude $𝑂(ξ^2)$. This is a heuristic procedure, the validity of which is proven by post festum. This method will be briefly explained in the example of an approximate solution of the differential equation of free-damped oscillations (3.2). The essence of the dual time scale method is based on replacing independently variable $𝑡$ two variable $𝑡_1$ and $𝑡_2$ :

$$𝑡_1 = 1 ∙ 𝑡; 𝑡_2 = ξ ∙ 𝑡, ξ∈(0,1),\tag{3.15}$$

assuming that they are independent of each other during the performance of the perturbation account. The constant $ξ$ is the attenuation factor. It has already been emphasized that this procedure has a physical meaning, since the rate of falling of the amplitude of the free, poorly suppressed oscillations is considerably lower than the angular frequency of the oscillations. In terms of (3.15), the variable $𝑡_1$ is the flow of "ordinary" time in which harmonic oscillations take place, and the variable $𝑡_2$ represents the "slow" scale of the time during which the dissipation of the oscillator energy takes place and the amplitude of the damped oscillations decreases. The approximate solution $φ(𝑡_1 , 𝑡_2 , ξ)$ is sought in the form of the collection of the initial solution $φ_0(𝑡_1 , 𝑡_2)$ and the first row correction $ξφ_1(𝑡_1 , 𝑡_2)$, as shown by the expression

$$φ(𝑡_1 , 𝑡_2 , ξ) ≈ φ_0(𝑡_1 , 𝑡_2) + ξφ_1(𝑡_1 , 𝑡_2).\tag{3.16}$$

Once again, the most important objective of this method is emphasized, which Murdock in particular emphasizes in [61]. It is a matter of an approximate solution (3.16) in the form of an asymptotic functional order, at some time interval, to be defined. In order to achieve this, it is necessary to ensure that at that time interval the functions of the perturbation order (3.16) are limited, or that they do not contain secular members. The initial conditions $φ(0) = Φ_0, φ(0) = 0$ are formulated for both functions $φ_0(𝑡_1 , 𝑡_2)$ and $φ_1(𝑡_1, 𝑡_2)$, expressions

$$φ_0(0,0) = Φ_0, φ̇ _0(0,0) = 0, φ_1(0,0) = 0, φ̇_1(0,0) = 0.\tag{3.17}$$

In accordance with the formulas (3.15), the first and the second derivative of the angular coordinate $φ$ by time $𝑡$ is given by the following terms, respectively:

$$φ̇ = \frac{𝜕φ}{𝜕𝑡_1} \frac{𝜕𝑡_1}{𝜕𝑡} + \frac{𝜕φ}{𝜕𝑡_2} \frac{𝜕𝑡_2}{𝜕𝑡} = \frac{𝜕φ}{𝜕𝑡_1} + ξ\frac{𝜕φ}{𝜕𝑡_2} ,\tag{3.18}$$ $$𝜑̈ = \frac{𝜕φ̇ }{𝜕𝑡_1} + ξ\frac{𝜕φ̇ }{𝜕𝑡_2} = \frac{𝜕^2φ}{𝜕𝑡_1^2} + 2ξ\frac{𝜕^2φ}{𝜕𝑡_1𝜕𝑡_2} + ξ^2\frac{𝜕^2φ}{𝜕𝑡_1^2}≅\frac{𝜕^2φ}{𝜕𝑡_1^2} + 2ξ\frac{𝜕^2φ}{𝜕𝑡_1𝜕𝑡_2}.\tag{3.19}$$

In the expression (3.19) the term containing $ξ^2$ is neglected as a small value of higher order. Using the expressions (3.16), and by neglecting the members comprising $ξ^2$, the relation (3.18) and (3.19) are, respectively:

$$φ̇ = \frac{𝜕φ_0}{𝜕𝑡_1} + ξ \Big(\frac{𝜕φ_0}{𝜕𝑡_2} + \frac{𝜕φ_1}{𝜕𝑡_1}\Big),\tag{3.20}$$ $$𝜑̈ = \frac{𝜕^2φ_0}{𝜕𝑡_1^2} + 2ξ\frac{𝜕^2φ_0}{𝜕𝑡_1𝜕𝑡_2} + ξ\frac{𝜕^2φ_1}{𝜕𝑡_1^2}.\tag{3.21}$$

When the expressions (3.16), (3.20) and (3.21) are included in the differential equation (3.2), after ignoring the members containing $ξ^2$, the following differential equation is obtained:

$$\frac{𝜕^2φ_0}{𝜕𝑡_1^2} + ω_0^2φ_0 + ξ\Big(\frac{𝜕^2φ_1}{𝜕𝑡_1^2} + ω_0^2φ_1 + 2\frac{𝜕^2φ_0}{𝜕𝑡_1𝜕𝑡_2} + 2ω_0\frac{𝜕φ_0}{𝜕𝑡_1}\Big) = 0.\tag{3.22}$$

Since the attenuation factor $ξ$ is small but different from zero, the left side of the equation (3.22) is equal to zero if the equations are:

$$\frac{𝜕^2φ_0}{𝜕𝑡_1^2} + ω_0^2φ_0 = 0; φ_0 = Φ_0, \frac{𝜕φ_0}{𝜕𝑡_1} = 0, \,\, \text{for}\,\, 𝑡1 = 𝑡2 = 0\tag{3.23}$$ $$\frac{𝜕^2φ_1}{𝜕𝑡_1^2} + ω_0^2φ_1 = -2\frac{𝜕^2φ_0}{𝜕𝑡_1𝜕𝑡_2}-2ω_0\frac{𝜕φ_0}{𝜕𝑡_1}; φ_1 = 0, \frac{𝜕φ_1}{𝜕𝑡_1} = -\frac{𝜕φ_0}{𝜕𝑡_2},\,\, \text{for}\,\, 𝑡1 = 𝑡2 = 0\tag{3.24}$$

simultaneously satisfied. With each of the mentioned equations, specific initial conditions are given, according to their general formulation (3.17), and in accordance with the expressions (3.16) and (3.20).

The system of differential equations (3.23) - (3.24) is solved successively, with the fact that the differential equation (3.23) is only formally partial, and in essence it represents a simple differential equation with constant coefficients with respect to the variable $𝑡_1$. Thus, the equation (3.23) leads to the initial solution $φ_0(𝑡_1, 𝑡_2)$, which is corrected by the correction of the first order $ξφ_1(𝑡_1, 𝑡_2)$. In this corrective term, the function $φ_1(𝑡_1, 𝑡_2)$ represents the solution of the equation (3.24), which depends on the previously obtained function $φ_0(𝑡_1, 𝑡_2)$. One equation solution (3.23) is given by the expression:

$$φ_0(𝑡_1) = Φ\sin(ω_0𝑡_1 + γ), Φ = 𝑐𝑜𝑛𝑠𝑡, γ = 𝑐𝑜𝑛𝑠𝑡,\tag{3.25}$$

describing the free harmonic oscillations by an independent variable $𝑡_1$ and in which the constant $γ$ represents the angle of the phase difference. Therefore, as the solution $φ_0(𝑡_1)$ of the homogeneous differential equation (3.23), the harmonic function (3.25) is variable $𝑡_1$, it is bounded $φ_0(𝑡_1) = 𝑂(1)$ at an arbitrary long time interval $𝑡_1∈[0, ∞)$. It is necessary to provide this property also for the function $φ_1(𝑡_1, 𝑡_2)$ according to the variable $𝑡_1$, so that the approximate solution $φ(𝑡_1, 𝑡_2, ξ)$ would not lose the character of the asymptotic function order, by the variable $𝑡_1$.

In order to solve the differential equation (3.24), the assumption ("Ansatz") is introduced that neither the amplitudes nor the phase difference are of constant size, but that they represent functions of the variable $𝑡_2$:

$$φ_0(𝑡_1, 𝑡_2) = Φ(𝑡_2) \sin(ω_0𝑡_1 + γ(𝑡_2)).\tag{3.26}$$

The partial derivative of the function (3.26) by the variable $𝑡_1$ is given by the function:

$$\frac{𝜕φ_0}{𝜕𝑡_1} = Φ(𝑡_2)ω_0\cos(ω_0𝑡_1 + γ(𝑡_2)) = Φω_0\cos ψ,\tag{3.27}$$

and the partial derivative of the function (3.27) by the variable $𝑡_2$ is given by the formula:

$$\begin{align}\frac{𝜕^2φ_0}{𝜕𝑡_1𝜕𝑡_2} &= \frac{𝜕Φ(𝑡_2)}{𝜕𝑡_2}ω_0\cos(ω_0𝑡_1 + γ (𝑡_2)) - Φ(𝑡_2)ω_0\sin(ω_0𝑡_1 + γ(𝑡_2))\frac{𝜕γ(𝑡_2)}{𝜕𝑡_2}\\ &= \frac{𝜕Φ}{𝜕𝑡^2}ω_0\cos ψ - Φ \frac{𝜕γ}{𝜕𝑡^2}ω_0\sinψ.\tag{3.28}\end{align}$$

For the sake of conciseness, the phase angle (phase) of oscillation was introduced $ψ = ω_0𝑡_1 + γ(𝑡_2)$. When the expressions (3.27) and (3.28) are included in the differential equation (3.24), the following equation is obtained:

$$\frac{𝜕^2φ_1}{𝜕𝑡_1^2} + ω_0^2φ_1 = 2Φ\frac{𝜕γ}{𝜕𝑡_2}ω_0\sin ψ - 2(\frac{𝜕Φ}{𝜕𝑡_2} + Φω_0) ω_0\cos ψ.\tag{3.29}$$

The members on the right-hand side of equation (3.29), which contain the functions $\cos ψ$ and $\sin ψ$, represent members of the resonant coercion, as functions of time $𝑡_1$, and cause the solution of this differential equation to contain secular members. By eliminating the mentioned resonant coercive members in the differential equation (3.29), which are harmonic functions of variable $𝑡_1$, the secular members are eliminated from the solution $φ_1(𝑡_1, 𝑡_2)$ as functions of the same variable $𝑡_1$, thus ensuring that $φ_1(𝑡_1, 𝑡_2)$ is limited by the variable $𝑡_1$. By this procedure, the differential equation (3.29) becomes homogeneous, its solution $φ_1(𝑡_1, 𝑡_2)$ must be a harmonic function of the variable $𝑡_1$, which means that both it and the function $φ_0(𝑡_1, 𝑡_2)$ are limited by the variable $𝑡_1$ at a time interval $𝑡_1 ∈ [0, ∞)$. Annihilation of secular members is done by the following equations:

$$Φ(𝑡_2) \frac{𝜕γ(𝑡_2)}{𝜕𝑡_2} = 0,\tag{3.30}$$ $$\frac{𝜕Φ(𝑡2)}{𝜕𝑡_2} + Φ(𝑡_2)ω_0 = 0.\tag{3.31}$$

As $Φ(𝑡_2) ≠ 0$, from the equation (3.30) it follows immediately that the phase difference does not change with time, ie that it is constant: $γ(𝑡_2) = γ_0 = 𝑐𝑜𝑛𝑠𝑡$. The general solution of the differential equation (3.31) is:

$$Φ(𝑡_2) = 𝐶𝑒^{-ω_0𝑡^2}\tag{3.32}$$

in which the constant $𝐶 = 𝑐𝑜𝑛𝑠𝑡$ is determined from the initial conditions. As the equation (3.30) describes the exponential decrease in the amplitude of the free-damped oscillations with time $𝑡_2$, and it is limited by $𝑡_2∈[0, ∞)$. Replacement of the solution (3.32) in (3.26) gives the zero approximation of the differential equation (3.2), that is, the formula of the first member of the perturbation order:

$$φ_0(𝑡_1, 𝑡_2) = 𝐶𝑒^{-ω_0𝑡^2} ∙ \sin (ω_0𝑡_1 + γ_0).\tag{3.33}$$

From the initial conditions given in (3.23), the constants $𝐶 = Φ_0$ and $γ_0 = π/2$ are determined and the final expression for the zero approximation of the solution of the differential equation (3.2) in the function of time coordinates $𝑡_1$ and $𝑡_2$ is:

$$φ_0(𝑡_1, 𝑡_2) = Φ_0𝑒^{-ω0𝑡2} ∙ \cos ω_0𝑡_1.\tag{3.34}$$

These are the exhausted possibilities for further solving the differential equation (3.24), so that the function $φ_1(𝑡_1, 𝑡_2)$ remains indeterminate. The approximate solution of the equation (3.2) is defined only as the zero "improved" approximation (3.34) of the perturbation order. By passing to the "regular" time coordinate $𝑡$, and in accordance with the equations $𝑡_1 = 𝑡$ and $𝑡_2 = ξ ∙ 𝑡$, formula (3.34) becomes:

$$φ_0(𝑡) = Φ_0𝑒^{-ξω0𝑡} ∙ \cos ω_0𝑡.\tag{3.35}$$

However, in spite of the fact that no analytical expression for the function $φ_1 (𝑡_1, 𝑡_2)$ has been defined, some of its essential characteristics are defined, on the axes where it is possible to estimate the order of the size of the remainder of the zero approximation $φ_0(𝑡)$, as well as the order of the time interval in which this approximation is valid. This assessment will also be proof of the validity of the implemented procedure for solving the differential equation (3.2) by the perturbation method of the double time scale. To evaluate the magnitude of the residue of the approximation and the time interval at which the scale can be used, it is necessary to discuss the properties of the asymptotic functional order,

$$φ(𝑡_1, 𝑡_2, ξ) = φ_0(𝑡_1, 𝑡_2) + ξφ_1 (𝑡_1, 𝑡_2) + O(ξ^2) = φ_0(𝑡_1, 𝑡_2) + O(ξ), ξ∈(0,1),\tag{3.36}$$

which approximates the exact solution of the differential equation (3.2). First of all, the characteristic of asymptotic development (3.36) is that the order of the size of the approximation residue is equal to the order of the size of the first excluded member $ξφ_1(𝑡_1, 𝑡_2) = O (ξ)$, at some interval of variables $𝑡_1$ and $𝑡_2$. In the case of solving the differential equation (3.2) by the double time method, it is ensured that the function $φ_0(𝑡_1, 𝑡_2)$ is bounded $φ_0(𝑡_1, 𝑡_2) = O(1)$ by both variables $𝑡_1$ and $𝑡_2$, at intervals $𝑡_1∈ [0, ∞)$ and $𝑡_2∈ [0, ∞)$. By passing to the regular time coordinate $𝑡$, it follows immediately that the function $φ_0(𝑡)$ is bounded $φ_0(𝑡) = 𝑂(1)$, for $𝑡∈[0, ∞)$. By eliminating secular members from the function $φ_1(𝑡_1, 𝑡_2)$ according to the variable $𝑡_1$, its limit on the interval $𝑡_1∈[0, ∞)$ is also ensured. However, since the process of solving the differential equation (3.29), the elimination of secular members from the function $φ_1(𝑡_1, 𝑡_2)$ according to the variable $𝑡_2$ is not ensured, for the function $φ_1(𝑡_1, 𝑡_2)$ it can only be claimed [61] that $φ_1(𝑡_1, 𝑡_2) = O (1)$, at some final time interval $𝑡_2∈ [0, 𝑡_𝑀]$. By passing to the regular time coordinate, $𝑡_2 = ξ ∙ 𝑡$, it follows that the function $φ_1(𝑡)$ is bounded $φ_1(𝑡) = 𝑂(1)$ at the final time interval $𝑡 ∈ [0, 𝑡_𝑀/ξ]$ which is the order of the dimension factor $ξ$ is longer than the interval $𝑡∈ [0, 𝑡_𝑀]$. From here it follows implicitly that $ξφ_1 (𝑡_1, 𝑡_2) = O (ξ)$, at $𝑡∈ [0, 𝑡_𝑀/ξ], 𝑡 = 𝑂 (1/ξ)$. For the fixed value of the damping factor $ξ$, the conclusions are correct for each other value of the number $ξ_1≤ξ$, that is, the approximation is more accurate and valid for a longer time interval, if the attenuation factor is lower $𝑡_2$, for the function $φ_1(𝑡_1, 𝑡_2)$ can only be claimed [61] that $φ_1(𝑡_1, 𝑡_2) = O (1)$ is bounded according to the variable $𝑡_2$, at some final time interval $𝑡_2∈ [0, 𝑡_𝑀]$. By passing to the regular time coordinate, $𝑡_2 = ξ ∙ 𝑡$, it follows that the function $φ_1 (𝑡)$ is bounded $φ_1 (𝑡) = 𝑂(1)$ at the final time interval $𝑡∈ [0, 𝑡_𝑀/ξ]$ which is the order of the dimension factor ξ is longer than the interval $𝑡∈ [0, 𝑡_𝑀]$. From here it follows implicitly that $ξφ_1 (𝑡_1, 𝑡_2) = O (ξ)$, at $𝑡∈ [0, 𝑡_𝑀/ξ], 𝑡 = 𝑂(1/ξ)$. For the fixed value of the damping factor $ξ$, the conclusions are correct for each other value of the number $ξ_1≤ξ$, that is, the approximation is more accurate and valid for a longer time interval, if the attenuation factor is lower $𝑡_2$ for the function $φ_1(𝑡_1, 𝑡_2)$ can only be claimed [61] that $φ_1(𝑡_1, 𝑡_2) = O(1)$ is bounded according to the variable $𝑡_2$, at some final time interval $𝑡_2∈[0,𝑡_𝑀 ]$. By passing to the regular time coordinate, $𝑡_2 = ξ ∙ 𝑡$, it follows that the function $φ_1(𝑡)$ is bounded $φ_1(𝑡) = 𝑂(1)$ at the final time interval $𝑡∈ [0, 𝑡_𝑀/ξ]$ which is the order of the dimension factor $ξ$ is longer than the interval $𝑡∈ [0, 𝑡_𝑀]$. From here it follows implicitly that $ξφ_1(𝑡_1, 𝑡_2) = O(ξ)$, at $𝑡∈[0, 𝑡_𝑀/ξ], 𝑡 = 𝑂(1/ξ)$. For the fixed value of the damping factor $ξ$, the conclusions are correct for each other value of the number $ξ_1≤ξ$, that is, the approximation is more accurate and valid for a longer time interval, if the attenuation factor is lower.at some final time interval $𝑡_2∈ [0, 𝑡_𝑀]$. By passing to the regular time coordinate, $𝑡_2 = ξ ∙ 𝑡$, it follows that the function $φ_1(𝑡)$ is bounded $φ_1(𝑡) = 𝑂(1)$ at the final time interval $𝑡∈ [0, 𝑡_𝑀/ξ]$ which is the order of the dimension factor $ξ$ is longer than the interval $𝑡∈ [0, 𝑡_𝑀]$. From here it follows implicitly that $ξφ_1(𝑡_1, 𝑡_2) = O(ξ)$, at $𝑡∈[0, 𝑡_𝑀/ξ], 𝑡 = 𝑂(1/ξ)$. For the fixed value of the damping factor $ξ$, the conclusions are correct for each other value of the number $ξ_1≤ξ$, that is, the approximation is more accurate and valid for a longer time interval, if the attenuation factor is lower at some final time interval $𝑡_2∈ [0, 𝑡_𝑀]$. By passing to the regular time coordinate, $𝑡_2 = ξ ∙ 𝑡$, it follows that the function $φ_1(𝑡)$ is bounded $φ_1(𝑡) = 𝑂(1)$ at the final time interval $𝑡∈[0, 𝑡_𝑀/ξ]$ which is the order of the dimension factor $ξ$ is longer than the interval $𝑡∈ [0, 𝑡_𝑀]$. From here it follows implicitly that $ξφ_1 (𝑡_1, 𝑡_2) = O(ξ)$, at $𝑡∈ [0, 𝑡_𝑀/ξ], 𝑡 = 𝑂(1/ξ)$. For the fixed value of the damping factor $ξ$, the conclusions are correct for each other value of the number $ξ_1≤ξ$, that is, the approximation is more accurate and valid for a longer time interval, if the attenuation factor is lower $𝑡 = [0, 𝑡_𝑀/ξ]$ which for the order of the dimension factor $ξ$ is longer than the interval $𝑡∈ [0, 𝑡_𝑀]$. For the fixed value of the damping factor $ξ$, the conclusions are correct for each other value of the number $ξ_1≤ξ$, that is, the approximation is more accurate and valid for a longer time interval, if the attenuation factor is lower.

On the basis of the above, it follows that

$$φ_0(𝑡) = 𝑂(1),\,\,\text{for}\,\, 𝑡∈ [0, 𝑡_𝑀/ξ_1]\,\,\text{and for each}\,\,ξ_1≤ξ, ξ∈ (0,1).\tag{3.37}$$

As for the error of approximation, which is calculated as the difference of the exact solution and its zero approximation, $|φ(𝑡, ξ)-φ_0(𝑡, ξ𝑡)| = |ξφ_1(𝑡, ξ𝑡)| = O(ξ)$ then it follows immediately that there are constants $𝑐$, $ξ_1$ and $𝑡_𝑀$ such that the exact solution $φ(𝑡, ξ)$ satisfies the following condition:

$$|φ(𝑡, ξ) - φ_0(𝑡, ξ𝑡) | ≤ 𝑐ξ,\,\,\text{for}\,\, 𝑡∈ [0 , 𝑡_𝑀/ξ_1]\,\,\text{and for each}\,\, ξ_1≤ξ, ξ∈(0,1).\tag{3.38}$$

This statement confirms that the approximate solution $φ_0(𝑡)$ asymptotically approaches the exact solution $φ(𝑡)$ at a time interval $𝑡∈ [0, 𝑡_𝑀/ξ]$. By the proof of the theorem 5.2.1. In [61], Murdock confirms that the perturbation method of double time conditions can be successfully applied in solving both non-homogeneous differential equations that describe oscillatory processes. Theorem 5.2.1. has the same formal record as paragraph (3.38), as well as the same essence of the evidence presented here.

3.2. Perturbation method of averaging over Krylov and Bogoliubov

It has already been pointed out that the errors of the average impulse mechanisms are analyzed using the theory of perturbations, and not only by the method of the double time, but also by the technique of averaging by the Krilov method (Nikolai Mitrofanovich Krylov, 1879-1955) and Bogoliubov (Nikolay Nikolaevich Bogoliubov, 1909-1992). In this chapter, the essence of the mentioned method will be summarized, as well as its similarity to the double-timing method. The perturbation technique of Krylov and Bogoliubov is a mathematical procedure for the approximate analysis of nonlinear oscillatory processes, by replacing the exact differential equation with an approximate, simpler equation, formed by a special technique of averaging. Except in the analysis of nonlinear oscillations, it is used in heavenly mechanics, for example in the determination of slow planetary precession of periapsis.

Let the differential equation of the oscillations be given in the form:

$$\frac{𝑑^2𝑢}{𝑑𝑡^2} + ω_0^2𝑢 = 𝑎 + ε ∙ 𝑓(𝑢, \frac{𝑑𝑢}{𝑑𝑡});\, ε∈ (0,1),\, 𝜀≪1.\tag{3.39}$$

If $ε=0$, the equation (3.39) describes a harmonic oscillator with constant force (force or momentum), and its solution is:

$$𝑢_0(𝑡) = aω_0^2 + A\sin (ω_0𝑡 + 𝐵).\tag{3.40}$$

Constants $A$ and $B$ are determined from the initial conditions. If $ε≠0$, the solution (3.40) of the equation (3.39) has a small disorder, perturbation and, according to the Krylov and Bogoliubov method, is assumed in the same form, but so that $A$ and $B$ are no longer constants but represent the functions of time $t$ and the parameter $ε$. Therefore, the equation solution (3.39) when $ε≠0$ is sought in the form:

$$𝑢_0(𝑡) = aω_0^2 + A(𝑡) \sin (ω_0𝑡 + 𝐵(𝑡)).\tag{3.41}$$

Since two unknown functions of time $t$ have been introduced, it is necessary to introduce some suitable restriction, that is, an additional condition. For this method, the following condition is introduced:

$$\frac{𝑑𝑢_0}{𝑑𝑡} = A(𝑡) ω_0\cos (ω_0𝑡 + 𝐵(𝑡)).\tag{3.42}$$

As $A$ and $B$ are functions of time $t$, the first derivative of the equation (3.41) is:

$$\frac{𝑑𝑢_0}{𝑑𝑡} = \frac{𝑑𝐴(𝑡)}{𝑑𝑡}\sin(ω_0𝑡 + 𝐵(𝑡)) + 𝐴(𝑡)\cos(ω_0𝑡 + 𝐵(𝑡)) ∙ (ω_0 + \frac{𝑑𝐵(𝑡)}{𝑑𝑡}).\tag{3.43}$$

From the equations (3.42) and (3.43), the additional condition can be expressed concisely by the equation:

$$ \frac{𝑑𝐴(𝑡)}{𝑑𝑡}\sin(ω_0𝑡 + 𝐵(𝑡)) + 𝐴(𝑡)\frac{𝑑𝐵(𝑡)}{𝑑𝑡}\cos (ω_0𝑡 + 𝐵)) = 0.\tag{3.44}$$

In accordance with the expressions (3.41), (3.43) and (3.44), the differential equation (3.39) is equivalent to the following equation system:

$$ \frac{𝑑𝐴(𝑡)}{𝑑𝑡} = εω_0 𝑓(aω_0^2 + A(𝑡)\sin(ψ), \cos(ψ)) ∙ \cos (ψ),\tag{3.45}$$ $$\frac{𝑑𝐵(𝑡)}{𝑑𝑡} = 𝜀𝜔_0𝑓(а𝜔_0^2 + А(𝑡)\sin(𝜓), \cos(𝜓))∙(−1𝐴(𝑡))\sin(𝜓),\tag{3.46}$$

where $ψ = ω_0𝑡 + 𝐵(𝑡)$ phase angle (phase) oscillation.

The equations (3.45) and (3.46) are exact, because so far no approximation has been applied. Since the functions $𝐴(𝑡)$ and $𝐵(𝑡)$ change slowly with time (since $ε≪1$), the right sides of equations (3.45) and (3.46) can be approximated to their mean values ​​over a time interval $𝑡∈⌈0, 2π/ω_0⌉$ ie, $ψ∈⌈0,2π⌉$. The approximation is performed by assuming that the functions $A$ and $B$ at the interval are constant and independent of time. In this way the equations

$$\frac{𝑑𝐴_{𝑆𝑅}(𝑡)}{𝑑𝑡} = \frac{ε}{2πω_0} \int_{0}^{2π} 𝑓 \big(aω_0^2 + A\sin (ψ), cos (ψ)\big) ∙ cos (ψ) 𝑑ψ,\tag{3.47}$$ $$\frac{𝑑𝐵_{𝑆𝑅}(𝑡)}{𝑑𝑡} = \frac{ε}{2πω0} \int_{0}^{2π} 𝑓 \big(aω_0^2 + A\sin (ψ), cos (ψ)\big) ∙ (-1𝐴) sin (ψ) 𝑑ψ,\tag{3.48}$$

which represent the very essence of the Krylov and Bogoliubov method. We point out once again that the functions $A$ and $B$, on the right-hand sides of equations (3.9) and (3.10), are inside the integral, the constants. After solving this system of differential equations (3.47-3.48), we can obtain approximately the solution of the differential equation (3.39), as the zero approximation, in the form:

$$𝑢_0(𝑡) = aω_0^2 + 𝐴_{𝑆𝑅}(𝑡) \sin (ω_0𝑡 + 𝐵_{𝑆𝑅}(𝑡)).\tag{3.49}$$

The purpose of the described procedure is based on the assumption that the system of equations (3.47-3.48) is simpler to solve from the initial differential equation (3.39).

The error of the approximate solution is calculated as the difference of the exact solution of the differential equation (3.39) and its zero approximation $𝑢_0(𝑡)$ of the $𝑢(𝑡)$. An order of magnitude of this error, which assesses the accuracy of approximation

$$\big| 𝑢(𝑡, ε) - 𝑢_0(𝑡, ε) \big| = O(ε),\tag{3.50}$$

is equal to the order of magnitude $O(ε)$ omitted the first member of the perturbation asymptotic development of accurate solutions. Krylov and Bogoliubov proved the position that there are constants $𝑐$, $ε_1$ and $𝑡_𝑀$, such that the error of approximation (3.50) fulfills the following requirements:

$$\big| 𝑢(𝑡, ε) - 𝑢_0(𝑡, ε) | ≤ 𝑐 ∙ ε\,\, \text{for}\,\, 𝑡∈ [0, 𝑡_𝑀/ε] \,\, \text{and for each}\,\, ε_1 ≤ ε, ε∈ (0,1).\tag{3.51}$$

It should be noted that the interval of approximation validity, as in the double time scale method, is extended to the order of magnitude of the parameter of the disorder $ε$. For the fixed value of the parameter $ε$, the conclusions are also valid for each other value of the parameter $ε_1 ≤ ε$ ie, the approximation is more accurate and valid for a longer time interval, if the number $ε$ is smaller. The position (3.51) confirms that the approximate solution $𝑢_0(𝑡,ε)$ asymptotically approaches the exact solution $𝑢(𝑡,ε)$ at the time interval $𝑡∈[0, 𝑡_𝑀/ε]$. Enclosing the proof of this paragraph in [61], Murdock confirms the correctness of the perturbation method of Krylov and Bogoliubov as one possible heuristic procedure in the approximate solution of differential equations of the form (3.39). Since the differential equation (3.39) and the equations (3.47) and (3.48) represent only the mathematical models of a group of physical phenomena,the proof of the practical usability of this, as well as of other perturbation procedures, can be verified only by experiment or by computer simulation of these physical phenomena.

In this analysis, the equation (3.47) describes the expected slow change ($ε≪1$) of the amplitude of the oscillatory process and is coupled with the equation (3.48). For the purposes of our analysis, the equation (3.48) is particularly significant and interesting. It reveals that the phase difference $B$ of the oscillatory process is not constant, but that it changes slowly over time and describes the speed of that change, that is, the disturbance of the frequency of the oscillator $ω_0$ caused by external coercion. If it is a clock oscillator, the equation (3.48) is just the formula for calculating the error of the average-impulse mechanism.

It is necessary to notice and discuss the similarity of the perturbation method of Krylov and Bogoliubov and the methods of two time conditions. First of all, a full consensus is found on the estimation of the order of the size of the error of approximation and the order of the size of the time interval to which this approximation is valid by both methods (3.51) and (3.52). The dual-time method introduces the coordinates of the "regular" time $𝑡$ and the coordinate of the "slow" time $ε𝑡$, thus, in general terms, explicitly separates the "fast" from the "slow" changes within the same dynamic process. The method of Krylov and Bogoliubov does not explicitly do so, but introduces the assumption that the functions $𝐴(𝑡)$ and $𝐵(𝑡)$ are changed so slowly in relation to the oscillation phase $ψ(𝑡)$ at the time interval $𝑡∈⌈0, 2π/ω_0⌉$, that it can be assumed that those at that time interval are approximately constant and independent of time.