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
$$π₯̈ + Ο^2π₯ + Ξ΅π(π₯, π₯̇) = 0;\, Ξ΅∈(0,1), Ξ΅≪1,\tag{3.1}$$
a typical example of such phenomena. In equation (3.1), the free oscillations are poorly disturbed (perturbated) by the member $Ξ΅π(π₯, π₯̇)$ which, in general, is non-linear. The constant $Ξ΅$ can be chosen in different ways, and in this case, it is convenient that it represents exactly the attenuation factor $ΞΎ$. 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 PoincarΓ©, 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:
$$π̈ + 2ΞΎΟ_0Ο̇ + Ο_0^2Ο = 0;\,ΞΎ∈(0,1), ΞΎ≪1.\tag{3.2}$$
It describes free oscillations, with its own angular frequency $Ο_0$ and small attenuation, if the attenuation factor is $ΞΎ∈(0,1), ΞΎ_1$. The initial conditions are known ($Ο(0) = Ξ¦_0, Ο̇ = 0$), 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 $2ΞΎΟ_0Ο̇ $ represents a small disturbance, i.e, a small perturbation of the free unstressed oscillations described by the equation
$$π̈ + Ο_0^2Ο̇ = 0;\tag{3.3}$$
because the attenuation factor $ΞΎ$ 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
$$Ο(π‘) = Ο_0(π‘) + ΞΎ^1Ο_1(π‘) + ΞΎ^2Ο_2(π‘) + ⋯ + ΞΎ^πΟ_π(π‘) + π(ΞΎ^{π + 1}),\tag{3.4}$$
which represents the sum of the exact solution $Ο_0(π‘)$ of the simplified differential equation (3.3), in which the disorder $2ΞΎΟ_0Ο̇ $ is ignored, and a number of perturbation members. The symbol $π(ΞΎ^{π + 1})$ represents the order of the size of the residue of the approximation and is $ΞΎ^{π + 1}$. These members of the higher order remotely (perturbate) idealized solution $Ο_0(π‘)$, and thus correct it and approach the desired solution $Ο(π‘)$ of the equation (3.2), in which the disorder $2ΞΎΟ_0Ο̇ $ is not neglected. If $ΞΎ∈(0,1)$, 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:
$$Ο(π‘) ≈ Ο_0(π‘) + ΞΎΟ_1(π‘),\tag{3.5}$$
in which all members $ΞΎ^πΟ_π(π‘), π > 1$ are ignored. In this case, it is certain that, for the given $ΞΎ$, there is always a finite time interval $π‘∈[0, π‘_π]$, within which the rest of the approximation has an order of magnitude $π(ΞΎ^2)$. 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:
$$π̈_0 + 2ΞΎΟ_0Ο̇ _0 + Ο_0^2Ο_0 + ΞΎΟ̈_1 + 2ΞΎ^2Ο_0Ο̇ _1 + ΞΎΟ_0^2 Ο_1 = 0.\tag{3.6}$$
In equation (3.6), the article $2ΞΎ^2Ο_0Ο̇_1$ is ignored as a small value of higher order. Differential equation (3.6) can be decomposed into the following system of two differential equations:
$$π̈_0 + Ο_0^2Ο_0 = 0\tag{3.7}$$
$$ΞΎ(π̈_1 + Ο_0^2Ο_1) = -2ΞΎΟ_0Ο̇ _0.\tag{3.8}$$
The solution of the equation (3.7) describes the free unbroken oscillations and for the initial conditions $Ο_0(0) = Ξ¦_0, Ο̇_0 = 0$ reads:
$$Ο_0(π‘) = Ξ¦_0 \cos Ο_0π‘.\tag{3.9}$$
If the solution (3.9) is replaced by the equation (3.8), the differential equation is obtained:
$$π̈_1 + Ο_0^2Ο_1 = 2Ο_0^2 Ξ¦_0 \sin Ο_0π‘.\tag{3.10}$$
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 $Ο_1(0) = 0, Ο̇_1 = 0$, the equation solution (3.10) is given by:
$$Ο_1(π‘) = Ξ¦_0(\sin Ο_0 - Ο_0π‘\cos Ο_0π‘).\tag{3.11}$$
Thus, the approximate solution of the differential equation (3.2) is:
$$π(π‘)=π·_0\cos π_0π‘+π(π·_0(\sin π_0π‘−π_0 π‘\cos π_0π‘))+π(π^2).\tag{3.12}$$
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 $π‘$, the member $ΞΎπ‘Ο_0Ξ¦_0\cos Ο_0π‘$ grows unlimited and thus loses the character of small corrections of the initial solution $Ο_0(π‘)$. Namely, in expression (3.12), for each $π‘ = π(1/ΞΎ)$ 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 π‘cosΟ0π‘ 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:
$$2Ο_0^2 Ξ¦_0 \sin Ο_0π‘,\tag{3.13}$$
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 $A\cos Ο_0π‘ + π΅\sin Ο_0π‘$ in homogeneous differential equation form:
$$π̈_1 + Ο_0^2Ο_1 = A\cos Ο_0π‘ + π΅\sin Ο_0π‘\tag{3.14}$$
represents a factor of resonant coercion and, in its particular solution, causes the appearance of secular members of the form $π‘\cos Ο_0 π‘$ and $π‘\sin Ο_0π‘$. 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 $π‘ = π‘_π = 1/(ΞΎΟ_0)$ the secular member is equated with the first term, and for $π‘≥π‘_π$ the asymptotic development ceases to be valid. Time interval $π‘∈[0, π‘_π]$ In which the approximate solution (3.10) has an acceptable accuracy is usually extremely short $π‘_π≪π‘_π$ . As already pointed out, at this time interval, for some particular value of the damping factor $ΞΎ∈(0,1)$, the remainder of the approximation has an order of magnitude $π(ΞΎ^2)$. If the members of the higher order $ΞΎ^πΟ_π(π‘), π>1$ were used in the perturbation calculation , the order of the size of the residual approximation would decrease $π(ΞΎ^{π+1}), π>1$, but would always be valid at the same time interval $π‘∈[0, π‘_π]$ 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 $π‘∈ [0, π‘_π/ ΞΎ ]$ with increasing the order of the size of the approximation residue with $π(ΞΎ^2)$ to $π(ΞΎ^1)$, provided that the functions $Ο_0(π‘)$ and $Ο_1 (π‘)$ 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 $ΞΎ$, 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 $Ο(π‘)≈Ο_0(π‘)$ has the order of the size $π(ΞΎ)$ at a time interval $π‘∈[0, π‘_π / ΞΎ]$. This time interval is for the order of magnitude of the damping factor $ΞΎ$ is longer than the time interval $π‘∈ [0, π‘_π ]$ on which the residue
the approximation of the order $π = 1$, obtained by regular perturbation, has the order of magnitude $π(ΞΎ^2)$. 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 $π‘$ two variable $π‘_1$ and $π‘_2$ :
$$π‘_1 = 1 ∙ π‘; π‘_2 = ΞΎ ∙ π‘, ΞΎ∈(0,1),\tag{3.15}$$
assuming that they are independent of each other during the performance of the perturbation account. The constant $ΞΎ$ 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 $π‘_1$ is the flow of "ordinary" time in which harmonic oscillations take place, and the variable $π‘_2$ 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 $Ο(π‘_1 , π‘_2 , ΞΎ)$ is sought in the form of the collection of the initial solution $Ο_0(π‘_1 , π‘_2)$ and the first row correction $ΞΎΟ_1(π‘_1 , π‘_2)$, as shown by the expression
$$Ο(π‘_1 , π‘_2 , ΞΎ) ≈ Ο_0(π‘_1 , π‘_2) + ΞΎΟ_1(π‘_1 , π‘_2).\tag{3.16}$$
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 $Ο(0) = Ξ¦_0, Ο(0) = 0$ are formulated for both functions $Ο_0(π‘_1 , π‘_2)$ and $Ο_1(π‘_1, π‘_2)$, expressions
$$Ο_0(0,0) = Ξ¦_0, Ο̇ _0(0,0) = 0, Ο_1(0,0) = 0, Ο̇_1(0,0) = 0.\tag{3.17}$$
In
accordance with the formulas (3.15), the first and the second derivative of the angular coordinate $Ο$ by time $π‘$ is given by the following terms, respectively:
$$Ο̇ = \frac{πΟ}{ππ‘_1} \frac{ππ‘_1}{ππ‘} + \frac{πΟ}{ππ‘_2} \frac{ππ‘_2}{ππ‘} = \frac{πΟ}{ππ‘_1} + ΞΎ\frac{πΟ}{ππ‘_2} ,\tag{3.18}$$
$$π̈ = \frac{πΟ̇ }{ππ‘_1} + ΞΎ\frac{πΟ̇ }{ππ‘_2} = \frac{π^2Ο}{ππ‘_1^2} + 2ΞΎ\frac{π^2Ο}{ππ‘_1ππ‘_2} + ΞΎ^2\frac{π^2Ο}{ππ‘_1^2}≅\frac{π^2Ο}{ππ‘_1^2} + 2ΞΎ\frac{π^2Ο}{ππ‘_1ππ‘_2}.\tag{3.19}$$
In the expression (3.19) the term containing $ΞΎ^2$ is neglected as a small value of higher order. Using the expressions (3.16), and by neglecting the members comprising $ΞΎ^2$, the relation (3.18) and (3.19) are, respectively:
$$Ο̇ = \frac{πΟ_0}{ππ‘_1} + ΞΎ \Big(\frac{πΟ_0}{ππ‘_2} + \frac{πΟ_1}{ππ‘_1}\Big),\tag{3.20}$$
$$π̈ = \frac{π^2Ο_0}{ππ‘_1^2} + 2ΞΎ\frac{π^2Ο_0}{ππ‘_1ππ‘_2} + ΞΎ\frac{π^2Ο_1}{ππ‘_1^2}.\tag{3.21}$$
When the expressions (3.16), (3.20) and (3.21) are included in the differential equation (3.2), after ignoring the members containing $ΞΎ^2$, the following differential equation is obtained:
$$\frac{π^2Ο_0}{ππ‘_1^2} + Ο_0^2Ο_0 + ΞΎ\Big(\frac{π^2Ο_1}{ππ‘_1^2} + Ο_0^2Ο_1 + 2\frac{π^2Ο_0}{ππ‘_1ππ‘_2} + 2Ο_0\frac{πΟ_0}{ππ‘_1}\Big) = 0.\tag{3.22}$$
Since the attenuation factor $ΞΎ$ is small but different from zero, the left side of the equation (3.22) is equal to zero if the equations are:
$$\frac{π^2Ο_0}{ππ‘_1^2} + Ο_0^2Ο_0 = 0; Ο_0 = Ξ¦_0, \frac{πΟ_0}{ππ‘_1} = 0, \,\, \text{for}\,\, π‘1 = π‘2 = 0\tag{3.23}$$
$$\frac{π^2Ο_1}{ππ‘_1^2} + Ο_0^2Ο_1 = -2\frac{π^2Ο_0}{ππ‘_1ππ‘_2}-2Ο_0\frac{πΟ_0}{ππ‘_1}; Ο_1 = 0, \frac{πΟ_1}{ππ‘_1} = -\frac{πΟ_0}{ππ‘_2},\,\, \text{for}\,\, π‘1 = π‘2 = 0\tag{3.24}$$
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 $π‘_1$. Thus, the equation (3.23) leads to the initial solution $Ο_0(π‘_1, π‘_2)$, which is corrected by the correction of the first order $ΞΎΟ_1(π‘_1, π‘_2)$. In this corrective term, the function $Ο_1(π‘_1, π‘_2)$ represents the solution of the equation (3.24), which depends on the previously obtained function $Ο_0(π‘_1, π‘_2)$. One equation solution (3.23) is given by the expression:
$$Ο_0(π‘_1) = Ξ¦\sin(Ο_0π‘_1 + Ξ³), Ξ¦ = ππππ π‘, Ξ³ = ππππ π‘,\tag{3.25}$$
describing the free harmonic oscillations by an independent variable $π‘_1$ and in which the constant $Ξ³$ represents the angle of the phase difference. Therefore, as the solution $Ο_0(π‘_1)$ of the homogeneous differential equation (3.23), the harmonic function (3.25) is variable $π‘_1$, it is bounded $Ο_0(π‘_1) = π(1)$ at an arbitrary long time interval $π‘_1∈[0, ∞)$. It is necessary to provide this property also for the function $Ο_1(π‘_1, π‘_2)$ according to the variable $π‘_1$, so that the approximate solution $Ο(π‘_1, π‘_2, ΞΎ)$ would not lose the character of the asymptotic function order, by the variable $π‘_1$.
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 $π‘_2$:
$$Ο_0(π‘_1, π‘_2) = Ξ¦(π‘_2) \sin(Ο_0π‘_1 + Ξ³(π‘_2)).\tag{3.26}$$
The partial derivative of the function (3.26) by the variable $π‘_1$ is given by the function:
$$\frac{πΟ_0}{ππ‘_1} = Ξ¦(π‘_2)Ο_0\cos(Ο_0π‘_1 + Ξ³(π‘_2)) = Ξ¦Ο_0\cos Ο,\tag{3.27}$$
and the partial derivative of the function (3.27) by the variable $π‘_2$ is given by the formula:
$$\begin{align}\frac{π^2Ο_0}{ππ‘_1ππ‘_2} &= \frac{πΞ¦(π‘_2)}{ππ‘_2}Ο_0\cos(Ο_0π‘_1 + Ξ³ (π‘_2)) - Ξ¦(π‘_2)Ο_0\sin(Ο_0π‘_1 + Ξ³(π‘_2))\frac{πΞ³(π‘_2)}{ππ‘_2}\\ &= \frac{πΞ¦}{ππ‘^2}Ο_0\cos Ο - Ξ¦ \frac{πΞ³}{ππ‘^2}Ο_0\sinΟ.\tag{3.28}\end{align}$$
For the sake of conciseness, the phase angle (phase) of oscillation was introduced $Ο = Ο_0π‘_1 + Ξ³(π‘_2)$. When the expressions (3.27) and (3.28) are included in the differential equation (3.24), the following equation is obtained:
$$\frac{π^2Ο_1}{ππ‘_1^2} + Ο_0^2Ο_1 = 2Ξ¦\frac{πΞ³}{ππ‘_2}Ο_0\sin Ο - 2(\frac{πΞ¦}{ππ‘_2} + Ξ¦Ο_0) Ο_0\cos Ο.\tag{3.29}$$
The members on the right-hand side of equation (3.29), which contain the functions $\cos Ο$ and $\sin Ο$, represent members of the resonant coercion, as functions of time $π‘_1$, 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 $π‘_1$, the
secular members are eliminated from the solution $Ο_1(π‘_1, π‘_2)$ as functions of the same variable $π‘_1$, thus ensuring that $Ο_1(π‘_1, π‘_2)$ is limited by the variable $π‘_1$. By this procedure, the differential equation (3.29) becomes homogeneous, its solution $Ο_1(π‘_1, π‘_2)$ must be a harmonic function of the variable $π‘_1$, which means that both it and the function $Ο_0(π‘_1, π‘_2)$ are limited by the variable $π‘_1$ at a time interval $π‘_1 ∈ [0, ∞)$. Annihilation of secular members is done by the following equations:
$$Ξ¦(π‘_2) \frac{πΞ³(π‘_2)}{ππ‘_2} = 0,\tag{3.30}$$
$$\frac{πΞ¦(π‘2)}{ππ‘_2} + Ξ¦(π‘_2)Ο_0 = 0.\tag{3.31}$$
As $Ξ¦(π‘_2) ≠ 0$, from the equation (3.30) it follows immediately that the phase difference does not change with time, ie that it is constant: $Ξ³(π‘_2) = Ξ³_0 = ππππ π‘$. The general solution of the differential equation (3.31) is:
$$Ξ¦(π‘_2) = πΆπ^{-Ο_0π‘^2}\tag{3.32}$$
in which the constant $πΆ = ππππ π‘$ 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 $π‘_2$, and it is limited by $π‘_2∈[0, ∞)$. 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:
$$Ο_0(π‘_1, π‘_2) = πΆπ^{-Ο_0π‘^2} ∙ \sin (Ο_0π‘_1 + Ξ³_0).\tag{3.33}$$
From the initial conditions given in (3.23), the constants $πΆ = Ξ¦_0$ and $Ξ³_0 = Ο/2$ 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 $π‘_1$ and $π‘_2$ is:
$$Ο_0(π‘_1, π‘_2) = Ξ¦_0π^{-Ο0π‘2} ∙ \cos Ο_0π‘_1.\tag{3.34}$$
These are the exhausted possibilities for further solving the differential equation (3.24), so that the function $Ο_1(π‘_1, π‘_2)$ 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 $π‘$, and in accordance with the equations $π‘_1 = π‘$ and $π‘_2 = ΞΎ ∙ π‘$, formula (3.34) becomes:
$$Ο_0(π‘) = Ξ¦_0π^{-ΞΎΟ0π‘} ∙ \cos Ο_0π‘.\tag{3.35}$$
However, in spite of the fact that no analytical expression for the function $Ο_1 (π‘_1, π‘_2)$ 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 $Ο_0(π‘)$, 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,
$$Ο(π‘_1, π‘_2, ΞΎ) = Ο_0(π‘_1, π‘_2) + ΞΎΟ_1 (π‘_1, π‘_2) + O(ΞΎ^2) = Ο_0(π‘_1, π‘_2) + O(ΞΎ), ΞΎ∈(0,1),\tag{3.36}$$
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 $ΞΎΟ_1(π‘_1, π‘_2) = O (ΞΎ)$, at some interval of variables $π‘_1$ and $π‘_2$.
In the case of solving the differential equation (3.2) by the double time method, it is ensured that the function $Ο_0(π‘_1, π‘_2)$ is bounded $Ο_0(π‘_1, π‘_2) = O(1)$ by both variables $π‘_1$ and $π‘_2$, at intervals $π‘_1∈ [0, ∞)$ and $π‘_2∈ [0, ∞)$.
By passing to the regular time coordinate $π‘$, it follows immediately that the function $Ο_0(π‘)$ is bounded $Ο_0(π‘) = π(1)$, for $π‘∈[0, ∞)$.
By eliminating secular members from the function $Ο_1(π‘_1, π‘_2)$ according to the variable $π‘_1$, its limit on the interval $π‘_1∈[0, ∞)$ is also ensured.
However, since the process of solving the differential equation (3.29), the elimination of secular members from the function $Ο_1(π‘_1, π‘_2)$ according to the variable $π‘_2$ is not ensured, for the function $Ο_1(π‘_1, π‘_2)$ it can only be claimed [61] that $Ο_1(π‘_1, π‘_2) = O (1)$, at some final time interval $π‘_2∈ [0, π‘_π]$.
By passing to the regular time coordinate, $π‘_2 = ΞΎ ∙ π‘$, it follows that the function $Ο_1(π‘)$ is bounded $Ο_1(π‘) = π(1)$ at the final time interval $π‘ ∈ [0, π‘_π/ΞΎ]$ which is the order of the dimension factor $ΞΎ$ is longer than the interval $π‘∈ [0, π‘_π]$.
From here it follows implicitly that $ΞΎΟ_1 (π‘_1, π‘_2) = O (ΞΎ)$, at $π‘∈ [0, π‘_π/ΞΎ], π‘ = π (1/ΞΎ)$.
For the fixed value of the damping factor $ΞΎ$, the conclusions are correct for each other value of the number $ΞΎ_1≤ΞΎ$, that is, the approximation is more accurate and valid for a longer time interval, if the attenuation factor is lower $π‘_2$, for the function $Ο_1(π‘_1, π‘_2)$ can only be claimed [61] that $Ο_1(π‘_1, π‘_2) = O (1)$ is bounded according to the variable $π‘_2$, at some final time interval $π‘_2∈ [0, π‘_π]$.
By passing to the regular time coordinate, $π‘_2 = ΞΎ ∙ π‘$, it follows that the function $Ο_1 (π‘)$ is bounded $Ο_1 (π‘) = π(1)$ at the final time interval $π‘∈ [0, π‘_π/ΞΎ]$ which is the order of the dimension factor ΞΎ is longer than the interval $π‘∈ [0, π‘_π]$.
From here it follows implicitly that $ΞΎΟ_1 (π‘_1, π‘_2) = O (ΞΎ)$, at $π‘∈ [0, π‘_π/ΞΎ], π‘ = π(1/ΞΎ)$.
For the fixed value of the damping factor $ΞΎ$, the conclusions are correct for each other value of the number $ΞΎ_1≤ΞΎ$, that is, the approximation is more accurate and valid for a longer time interval, if the attenuation factor is lower $π‘_2$ for the function $Ο_1(π‘_1, π‘_2)$ can only be claimed [61] that $Ο_1(π‘_1, π‘_2) = O(1)$ is bounded according to the variable $π‘_2$, at some final time interval $π‘_2∈[0,π‘_π ]$.
By passing to the regular time coordinate, $π‘_2 = ΞΎ ∙ π‘$, it follows that the function $Ο_1(π‘)$ is bounded $Ο_1(π‘) = π(1)$ at the final time interval $π‘∈ [0, π‘_π/ΞΎ]$ which is the order of the dimension factor $ΞΎ$ is longer than the interval $π‘∈ [0, π‘_π]$.
From here it follows implicitly that $ΞΎΟ_1(π‘_1, π‘_2) = O(ΞΎ)$, at $π‘∈[0, π‘_π/ΞΎ], π‘ = π(1/ΞΎ)$.
For the fixed value of the damping factor $ΞΎ$, the conclusions are correct for each other value of the number $ΞΎ_1≤ΞΎ$, 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 $π‘_2∈ [0, π‘_π]$.
By passing to the regular time coordinate, $π‘_2 = ΞΎ ∙ π‘$, it follows that the function $Ο_1(π‘)$ is bounded $Ο_1(π‘) = π(1)$ at the final time interval $π‘∈ [0, π‘_π/ΞΎ]$ which is the order of the dimension factor $ΞΎ$ is longer than the interval $π‘∈ [0, π‘_π]$.
From here it follows implicitly that $ΞΎΟ_1(π‘_1, π‘_2) = O(ΞΎ)$, at $π‘∈[0, π‘_π/ΞΎ], π‘ = π(1/ΞΎ)$.
For the fixed value of the damping factor $ΞΎ$, the conclusions are correct for each other value of the number $ΞΎ_1≤ΞΎ$, 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 $π‘_2∈ [0, π‘_π]$.
By passing to the regular time coordinate, $π‘_2 = ΞΎ ∙ π‘$, it follows that the function $Ο_1(π‘)$ is bounded $Ο_1(π‘) = π(1)$ at the final time interval $π‘∈[0, π‘_π/ΞΎ]$ which is the order of the dimension factor $ΞΎ$ is longer than the interval $π‘∈ [0, π‘_π]$.
From here it follows implicitly that $ΞΎΟ_1 (π‘_1, π‘_2) = O(ΞΎ)$, at $π‘∈ [0, π‘_π/ΞΎ], π‘ = π(1/ΞΎ)$.
For the fixed value of the damping factor $ΞΎ$, the conclusions are correct for each other value of the number $ΞΎ_1≤ΞΎ$, that is, the approximation is more accurate and valid for a longer time interval, if the attenuation factor is lower $π‘ = [0, π‘_π/ΞΎ]$ which for the order of the dimension factor $ΞΎ$ is longer than the interval $π‘∈ [0, π‘_π]$.
For the fixed value of the damping factor $ΞΎ$, the conclusions are correct for each other value of the number $ΞΎ_1≤ΞΎ$, 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
$$Ο_0(π‘) = π(1),\,\,\text{for}\,\, π‘∈ [0, π‘_π/ΞΎ_1]\,\,\text{and for each}\,\,ΞΎ_1≤ΞΎ, ΞΎ∈ (0,1).\tag{3.37}$$
As for the error of approximation, which is calculated as the difference of the exact solution and its zero approximation, $|Ο(π‘, ΞΎ)-Ο_0(π‘, ΞΎπ‘)| = |ΞΎΟ_1(π‘, ΞΎπ‘)| = O(ΞΎ)$ then it follows immediately that there are constants $π$, $ΞΎ_1$ and $π‘_π$ such that the exact solution $Ο(π‘, ΞΎ)$ satisfies the following condition:
$$|Ο(π‘, ΞΎ) - Ο_0(π‘, ΞΎπ‘) | ≤ πΞΎ,\,\,\text{for}\,\, π‘∈ [0 , π‘_π/ΞΎ_1]\,\,\text{and for each}\,\, ΞΎ_1≤ΞΎ, ΞΎ∈(0,1).\tag{3.38}$$
This statement confirms that the approximate solution $Ο_0(π‘)$ asymptotically approaches the exact solution $Ο(π‘)$ at a time interval $π‘∈ [0, π‘_π/ΞΎ]$.
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:
$$\frac{π^2π’}{ππ‘^2} + Ο_0^2π’ = π + Ξ΅ ∙ π(π’, \frac{ππ’}{ππ‘});\, Ξ΅∈ (0,1),\, π≪1.\tag{3.39}$$
If $Ξ΅=0$, the equation (3.39) describes a harmonic oscillator with constant force (force or momentum), and its solution is:
$$π’_0(π‘) = aΟ_0^2 + A\sin (Ο_0π‘ + π΅).\tag{3.40}$$
Constants $A$ and $B$ are determined from the initial conditions. If $Ξ΅≠0$, 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 $Ξ΅$. Therefore, the equation solution (3.39) when $Ξ΅≠0$ is sought in the form:
$$π’_0(π‘) = aΟ_0^2 + A(π‘) \sin (Ο_0π‘ + π΅(π‘)).\tag{3.41}$$
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:
$$\frac{ππ’_0}{ππ‘} = A(π‘) Ο_0\cos (Ο_0π‘ + π΅(π‘)).\tag{3.42}$$
As $A$ and $B$ are functions of time $t$, the first derivative of the equation (3.41) is:
$$\frac{ππ’_0}{ππ‘} = \frac{ππ΄(π‘)}{ππ‘}\sin(Ο_0π‘ + π΅(π‘)) + π΄(π‘)\cos(Ο_0π‘ + π΅(π‘)) ∙ (Ο_0 + \frac{ππ΅(π‘)}{ππ‘}).\tag{3.43}$$
From the equations (3.42) and (3.43), the additional condition can be expressed concisely by the equation:
$$ \frac{ππ΄(π‘)}{ππ‘}\sin(Ο_0π‘ + π΅(π‘)) + π΄(π‘)\frac{ππ΅(π‘)}{ππ‘}\cos (Ο_0π‘ + π΅)) = 0.\tag{3.44}$$
In accordance with the expressions (3.41), (3.43) and (3.44), the differential equation (3.39) is equivalent to the following equation system:
$$ \frac{ππ΄(π‘)}{ππ‘} = Ξ΅Ο_0 π(aΟ_0^2 + A(π‘)\sin(Ο), \cos(Ο)) ∙ \cos (Ο),\tag{3.45}$$
$$\frac{ππ΅(π‘)}{ππ‘} = ππ_0π(Π°π_0^2 + Π(π‘)\sin(π), \cos(π))∙(−1π΄(π‘))\sin(π),\tag{3.46}$$
where $Ο = Ο_0π‘ + π΅(π‘)$ phase angle (phase) oscillation.
The equations (3.45) and (3.46) are exact, because so far no approximation has been applied. Since the functions $π΄(π‘)$ and $π΅(π‘)$ change slowly with time (since $Ξ΅≪1$), the right sides of equations (3.45) and (3.46) can be approximated to their mean values over a time interval $π‘∈⌈0, 2Ο/Ο_0⌉$ ie, $Ο∈⌈0,2Ο⌉$. 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
$$\frac{ππ΄_{ππ
}(π‘)}{ππ‘} = \frac{Ξ΅}{2ΟΟ_0} \int_{0}^{2Ο} π \big(aΟ_0^2 + A\sin (Ο), cos (Ο)\big) ∙ cos (Ο) πΟ,\tag{3.47}$$
$$\frac{ππ΅_{ππ
}(π‘)}{ππ‘} = \frac{Ξ΅}{2ΟΟ0} \int_{0}^{2Ο} π \big(aΟ_0^2 + A\sin (Ο), cos (Ο)\big) ∙ (-1π΄) sin (Ο) πΟ,\tag{3.48}$$
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:
$$π’_0(π‘) = aΟ_0^2 + π΄_{ππ
}(π‘) \sin (Ο_0π‘ + π΅_{ππ
}(π‘)).\tag{3.49}$$
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 $π’_0(π‘)$ of the $π’(π‘)$. An order of magnitude of this error, which assesses the accuracy of approximation
$$\big| π’(π‘, Ξ΅) - π’_0(π‘, Ξ΅) \big| = O(Ξ΅),\tag{3.50}$$
is equal to the order of magnitude $O(Ξ΅)$ omitted the first member of the perturbation asymptotic development of accurate solutions. Krylov and Bogoliubov proved the position that there are constants $π$, $Ξ΅_1$ and $π‘_π$, such that the error of approximation (3.50) fulfills the following requirements:
$$\big| π’(π‘, Ξ΅) - π’_0(π‘, Ξ΅) | ≤ π ∙ Ξ΅\,\, \text{for}\,\, π‘∈ [0, π‘_π/Ξ΅] \,\, \text{and for each}\,\, Ξ΅_1 ≤ Ξ΅, Ξ΅∈ (0,1).\tag{3.51}$$
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 $Ξ΅$. For the fixed value of the parameter $Ξ΅$, the conclusions are also valid for each other value of the parameter $Ξ΅_1 ≤ Ξ΅$ ie, the approximation is more accurate and valid for a longer time interval, if the number $Ξ΅$ is smaller. The position (3.51) confirms that the approximate solution $π’_0(π‘,Ξ΅)$ asymptotically approaches the exact solution $π’(π‘,Ξ΅)$ at the time interval $π‘∈[0, π‘_π/Ξ΅]$. 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 ($Ξ΅≪1$) 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 $Ο_0$ 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 $π‘$ and the coordinate of the "slow" time $Ξ΅π‘$, 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 $π΄(π‘)$ and $π΅(π‘)$ are changed so slowly in relation to the oscillation phase $Ο(π‘)$ at the time interval $π‘∈⌈0, 2Ο/Ο_0⌉$, that it can be assumed that those at that time interval are approximately constant and independent of time.
