UDC 517.917

ON THE AVERAGING OF A CERTAIN CLASS

OF SYSTEMS OF NONLINEAR DIFFERENTIAL EQUATIONS

V. V. Laricheva

For systems in standard form with right-hand sides periodic in time, a theorem is proved on the error of solutions of averaged equations. In contrast to [1], where a more general dependence of the right-hand sides on time is considered, simple estimates of the averaging error, convenient for practical application, are obtained. This made it possible to prove a special theorem on the averaging of equations of celestial mechanics in the form [2]. According to this theorem, depending on the initial conditions, the interval of validity of solutions of the averaged equations can be considerably extended in comparison with the preceding theorem.

Consider the system of differential equations

\[ \frac{dx}{dt}=\varepsilon X(t,x), \tag{1} \]

where \(x, X\) are \(n\)-dimensional vectors; \(\varepsilon\) is a small parameter. Suppose

a) the vector function \(X(t,x)\) is periodic in \(t\) with period \(T=\mathrm{const}\sim 1\);

b) in the domain \(D\) of variation of \(x\) the vector function \(X(t,x)\) is continuously differentiable, and positive constants \(M,\lambda\) can be indicated such that, for all real \(t>0\), in \(D\) the inequalities

\[ |X(t,x)|\leq M,\qquad \left|\frac{\partial X(t,x)}{\partial x}\right|\leq \lambda \tag{2} \]

hold.

According to [1], the form of system (1) is standard. We write the system of averaged equations

\[ \frac{d\xi}{dt}=\varepsilon X_0(\xi). \tag{3} \]

Here

\[ X_0(\xi)=\frac{1}{T}\int_0^T X(t,\xi)\,dt. \]

Obviously, \(|X_0(\xi)|\leq M\) in the domain \(D\).

Let

c) \(\xi(t)\) be defined and lie in the domain \(D\) together with its neighborhood of radius \(\rho\).

The solution (3) \(\xi=\xi(t)\), coinciding with \(x(t)\) at the initial instant, is the first Krylov–Bogoliubov approximation.

Under the listed conditions we shall prove a theorem on the error of the first approximation, setting ourselves the goal of obtaining more precise error estimates than in [1].

Let us represent the solution of (1) in the form

\[ x=\xi+\varepsilon \widetilde X(t,\xi)+\varepsilon r(t), \tag{4} \]

where \(r(t)\) is a new variable; \(x(0)=\xi(0);\ r(0)=0\),

\[ \widetilde X(t,\xi)=\int_0^t [X(t,\xi)-X_0(\xi)]\,dt . \tag{5} \]

Obviously, the integrand in (5) is a vector-function periodic in \(t\) with period \(T\sim 1\), and therefore from inequalities (2) in the domain \(D\) for \(t\geqslant 0\) we have

\[ |\widetilde X(t,\xi)|\leqslant TM,\qquad \left|\frac{\partial \widetilde X(t,\xi)}{\partial \xi}\right|\leqslant T\lambda . \tag{6} \]

Substituting (4) into (1) and taking account of (3), (5), we obtain the equation

\[ \frac{dr}{dt} = -\varepsilon X_0(\xi)\frac{\partial \widetilde X(t,\xi)}{\partial \xi} + X\{t,\xi+\varepsilon \widetilde X(t,\xi)+\varepsilon r\} - X(t,\xi). \tag{7} \]

Introduce the slow time \(\tau=\varepsilon t\) and rewrite (7), taking into account \(r(0)=0\), in the form of an equivalent integral equation

\[ r= \int_0^\tau \left\{ -X_0(\xi)\frac{\partial \widetilde X(t,\xi)}{\partial \xi} + \frac{X(t,\xi+\varepsilon \widetilde X+\varepsilon r)-X(t,\xi)}{\varepsilon} \right\}\,d\tau . \tag{8} \]

We determine \(r\) by successive approximations

\[ r=r_0+(r_1-r_0)+\ldots+(r_n-r_{n-1})+\ldots, \tag{9} \]

where

\[ r_0=r(0)=0, \]

\[ r_n= \int_0^\tau \left\{ -X_0(\xi)\frac{\partial \widetilde X(t,\xi)}{\partial \xi} + \frac{X(t,\xi+\varepsilon \widetilde X+\varepsilon r_{n-1})-X(t,\xi)}{\varepsilon} \right\}\,d\tau, \tag{10} \]

\[ n=1,2,\ldots \]

— the Picard successive approximations.

Using the estimate for \(X_0(\xi)\) and inequalities (2), (6), we obtain, for the terms of the series (9), the estimates

\[ |r_1|\leqslant 2TM\lambda\tau,\ \ldots,\quad |r_n-r_{n-1}|\leqslant \lambda\,2TM\,\frac{\lambda^{\,n-1}\tau^n}{n!},\ \ldots, \tag{11} \]

which are valid under the condition that the successive approximations for \(x=\xi+\varepsilon \widetilde X+\varepsilon r\) do not leave the domain \(D\). Then the series (9) will converge uniformly to \(r\), and \(|r|\leqslant 2TM(\exp \lambda\tau-1)\).

Using condition c), we require that the averaging error \(R=x-\xi\) be less than the prescribed quantity \(\eta\), not exceeding \(\rho\):

\[ |R|=\varepsilon|\widetilde X+r|\leqslant \varepsilon|\widetilde X|+\varepsilon|r| \leqslant \varepsilon TM(2\exp\lambda\tau-1)<\eta . \tag{12} \]

Inequality (12) is satisfied by \(\tau\in[0,L]\), where

\[ L<\frac{1}{\lambda}\ln\frac{1}{2}\left(1+\frac{\eta}{\varepsilon TM}\right); \qquad \eta\leqslant \rho . \tag{13} \]

Since

\[ \varepsilon|\widetilde X+r_n|\leqslant \varepsilon|\widetilde X|+\varepsilon|r_n| \leqslant \varepsilon TM(2\exp\lambda\tau-1)<\eta\leqslant \rho, \]

then for \(\tau \in [0,L]\) the successive approximations for \(x\) will not leave the domain \(D\). Consequently, the estimates (11) will be valid.

By analogy with the theorem on the existence of a solution, one can show that for \(0 \leq \tau < L\) the series (9) converges uniformly to the solution of (8), which is unique.

Thus, on the interval \(0 < t < L/\varepsilon\) the error of the first approximation is less than \(\eta\), where \(\eta \leq \rho\), and consists of two parts: the oscillatory \(\varepsilon \widetilde X \sim \varepsilon T M\) and the systematic \(\varepsilon r \sim 2\varepsilon T M(\exp \lambda t - 1)\).

If \(\varepsilon\) is given and the constants \(T, M, \lambda, L \sim 1\), then inequality (12) makes it possible to establish that \(\eta \sim \varepsilon\). If, conversely, \(\eta\) is fixed, then inequality (12) can be satisfied by decreasing \(\varepsilon\). In this case the \(L\) satisfying inequality (13) will increase.

In [1] for system (1) a theorem is proved on the error of the first approximation under assumptions on \(X(t,x)\) more general than a), b), namely:

a′) uniformly with respect to \(x\) in the domain \(D\) there exists the limit

\[ \lim_{T\to\infty}\frac{1}{T}\int_0^T X(t,x)\,dt = X_0(x); \tag{14} \]

b′) for the domain \(D\) one can indicate such positive constants \(M\) and \(\lambda\) that for all real values \(t \geq 0\) and for any points \(x, x', x''\) from this domain the inequalities

\[ |X(t,x)| \leq M,\quad |X(t,x') - X(t,x'')| \leq \lambda |x' - x''|. \tag{15} \]

are satisfied.

Then to arbitrarily small positive \(\rho, \eta\) and arbitrarily large \(L\) one can assign such a positive \(\varepsilon_0\) that if \(\xi = \xi(t)\) is a solution of the equation

\[ \frac{d\xi}{dt} = \varepsilon X_0(\xi), \tag{16} \]

defined on the interval \(0 < t < \infty\) and lying in the domain \(D\) together with its neighborhood of radius \(\rho\), and \(x(0)=\xi(0)\), then for \(0<\varepsilon<\varepsilon_0\) on the interval \(0 < t < L/\varepsilon\) the inequality

\[ |x(t)-\xi(t)| < \eta \]

is valid.

Let us first consider the case when conditions (15) are fulfilled, but \(X(t,x)\) is periodic in \(t\).

By analogy with [1] we construct the function

\[ \Delta_\varepsilon(x)= \begin{cases} A_\varepsilon \left[1-\left(\dfrac{|x|}{\varepsilon}\right)^2\right]^2, & |x|\leq \varepsilon,\\[6pt] 0, & |x|>\varepsilon, \end{cases} \tag{17} \]

where the positive constant \(A_\varepsilon\) is determined by the relation

\[ \int_{E_n}\Delta_\varepsilon(x)\,dx = 1, \tag{18} \]

in which the integration is performed over the entire \(n\)-dimensional space \(x\).

The function \(\Delta_\varepsilon(x)\) is bounded together with its partial derivatives up to and including the second order.

Using (17), (18) we find that

\[ I_\varepsilon=\int_{E_n}\left|\frac{\partial \Delta_\varepsilon(x)}{\partial x}\right|\,dx\leqslant 4\quad(\varepsilon>0). \tag{19} \]

Instead of \(\widetilde X(t,x)\), in the case under consideration we introduce the function

\[ u(t,x)=\int_D \Delta_\varepsilon(x-x')\left\{\int_0^t [X(t,x')-X_0(x')]\,dt\right\}dx'. \tag{20} \]

Obviously,

\[ \frac{\partial u}{\partial x} = \int_D \frac{\partial \Delta_\varepsilon(x-x')}{\partial x} \left\{\int_0^t [X(t,x')-X_0(x')]\,dt\right\}dx', \]

\[ \frac{\partial u}{\partial t} = \int_D \Delta_\varepsilon(x-x')[X(t,x')-X_0(x')]\,dx'. \tag{21} \]

Since \(\Delta_\varepsilon(x-x')\geqslant 0\) and the domain \(D\) is part of the space \(E_n\) (or coincides with it), taking into account (18), (19), it is easy to obtain

\[ \int_D \Delta_\varepsilon(x-x')\,dx' \leqslant \int_{E_n}\Delta_\varepsilon(x-x')\,dx' = \int_{E_n}\Delta_\varepsilon(x')\,dx'=1, \]

\[ \int_D\left|\frac{\partial \Delta_\varepsilon(x-x')}{\partial x}\right|dx' \leqslant \int_{E_n}\left|\frac{\partial \Delta_\varepsilon(x)}{\partial x}\right|dx = I_\varepsilon\leqslant 4. \tag{22} \]

Using estimate (6) for (5) and taking (22) into account, we obtain

\[ |u(t,x)|\leqslant TM,\qquad \left|\frac{\partial u}{\partial x}\right|\leqslant 4TM. \tag{23} \]

By virtue of (15) we have

\[ |X(t,x')-X_0(x')-X(t,x)+X_0(x)|\leqslant 2\lambda |x'-x|,\quad x,x'\in D. \tag{24} \]

Using (21), (24), we find that

\[ \left| \frac{\partial u}{\partial t} - \left[ X(t,x)-X_0(x)\int_D \Delta_\varepsilon(x-x')\,dx' \right] \right| \leqslant 2\lambda |x'-x|. \tag{25} \]

For any point \(x\) of interest to us whose \(\varepsilon\)-neighborhood belongs to \(D\), with the aid of (17), (18) we obtain

\[ \int_D \Delta_\varepsilon(x-x')\,dx' = \int_{|x-x'|<\varepsilon}\Delta_\varepsilon(x-x')\,dx'=1, \]

and relation (25) for such points gives

\[ \left| \frac{\partial u(t,x)}{\partial t} - X(t,x)+X_0(x) \right| \leqslant 2\lambda\varepsilon. \tag{26} \]

We represent the solution (1) in the form

\[ x=\xi+\varepsilon u(t,\xi)+\varepsilon r, \tag{27} \]

where \(\xi\) is a solution of (3); \(\xi(0)=x(0)\); \(r(t)\) is a new variable; \(r(0)=0\).

Substituting (27) into (1), we obtain for \(r(t)\) the equation

\[ \frac{dr}{dt} = -\left[\frac{\partial u}{\partial t}-X(t,\xi)+X_0(\xi)\right] -\varepsilon\frac{\partial u}{\partial \xi}X_0(\xi) + X(t,\xi+\varepsilon u+\varepsilon r)-X(t,\xi), \tag{28} \]

similar to (7).

Instead of the estimates (11), for the series of successive approximations of the solution of (28) we shall have the following estimates:

\[ |r_1|\leq K\tau,\ \ldots,\ |r_n-r_{n-1}|\leq K\frac{\lambda^{n-1}\tau^n}{n!},\ \ldots, \tag{29} \]

where \(K=2\lambda+4TM+\lambda TM\).

As in the preceding case, if the successive approximations for (27) do not leave the domain \(D\), then the series (29) will converge uniformly to \(r\) and \(|r|\leq K/\lambda(\exp\lambda\tau-1)\). For this it is sufficient to require that

\[ |R|=\varepsilon|u+r|\leq [TM+K/\lambda(\exp\lambda\tau-1)]<\eta\leq \rho. \]

Let us pass to the case [1] when both conditions a′), b′) are satisfied. In the function (17) we replace the index \(\varepsilon\) by \(a\), where \(a\leq\rho\), and take this into account in (20), (21), (26). On the right-hand side of inequality (26), \(a\) will appear instead of \(\varepsilon\); the relations (18), (19), (22) do not change.

The estimates for \(u,\ \dfrac{\partial u}{\partial x}\) will differ essentially from (23). Indeed, by virtue of condition a′) one can construct such a monotonically decreasing function \(f(t)\), tending to zero as \(t\to\infty\), that throughout the whole domain \(D\)

\[ \left|\frac{1}{t}\int_0^t [X(t,x)-X_0(x)]\,dt\right|\leq f(t). \tag{30} \]

With the aid of (22), (30) we find

\[ |u(t,x)|\leq tf(t),\qquad \left|\frac{\partial u}{\partial x}\right|\leq 4tf(t), \tag{31} \]

whence, introducing \(\tau=\varepsilon t\), we obtain

\[ \varepsilon|u|\leq \tau f(\tau/\varepsilon),\qquad \varepsilon\left|\frac{\partial u}{\partial x}\right|\leq 4\tau f(\tau/\varepsilon). \tag{32} \]

For fixed \(\tau\), by decreasing \(\varepsilon\) one can make \(f(\tau/\varepsilon)\) smaller than any prescribed number, in view of the monotone decrease of this function.

Further, by analogy with the preceding cases it is convenient for us to modify somewhat the proof of [1].

Let \(\tau\leq L\). It is obvious that there exists such an \(\varepsilon_0\) that for \(\varepsilon<\varepsilon_0\) one has \(Lf(L/\varepsilon)<a\), however small the number \(a\) may be. Obviously, the smaller \(\varepsilon\) is, the smaller values of \(a\) can be prescribed, i.e. \(a\to 0\) as \(\varepsilon\to 0\).

We represent the solution (1) in the form

\[ x=\xi+\varepsilon u(\xi,t)+ar, \tag{33} \]

where \(\xi\) is a solution of (16), \(x(0)=\xi(0)\); \(r(t)\) is a new variable; \(r(0)=0\).

Substituting (33) into (1), introducing \(\tau=\varepsilon t\), we obtain for \(r\) the integral equation

\[ r=\int_0^\tau \left\{ -\frac{\varepsilon}{a}\cdot \frac{\partial u}{\partial \xi} X_0(\xi) -\frac{1}{a}\left[\frac{\partial u}{\partial t}-X(t,\xi)+X_0(\xi)\right] +X(t,\xi+\varepsilon u+ar)-X(t,\xi) \right\}\,d\tau . \tag{34} \]

Taking into account (26), (32), the estimate \(D\) for \(\tau\uparrow(\tau/\varepsilon)\) when \(\varepsilon<\varepsilon_0\), and conditions \(a')\), \(b')\), for the series of successive approximations of the solution of (34) we obtain the estimates

\[ |r_1|\leq P\tau,\ldots\quad |r_n-r_{n-1}|\leq P\frac{\lambda^{\,n-1}\tau^n}{n!},\ldots , \tag{35} \]

where \(P=4M+3\lambda\).

If the successive approximations for (33) do not leave the domain \(D\), then for \(\varepsilon<\varepsilon_0\) the series (35) will converge uniformly to \(r\), and
\[ |r|\leq P/\lambda\,(\exp\lambda\tau-1). \]
For this it is sufficient to require that
\[ |R|\leq \varepsilon|u|+a|r|\leq a\,[1+P/\lambda\,(\exp\lambda\tau-1)]<\eta . \]
For a given \(\eta\) this is satisfied if
\[ L<\frac{1}{\lambda}\ln\left[1+\frac{\lambda}{P}\left(\frac{\eta}{a}-1\right)\right]. \]
Obviously, \(L\) will increase as \(\varepsilon\) decreases.

Let us note that, since \(a\) is connected with \(\varepsilon\) by a limiting relation, in the case under consideration it is essentially shown that for \(\varepsilon<\varepsilon_0\) majorant estimates for the error of averaging exist in principle, whereas in the case of periodic right-hand sides, for any fixed value of the small parameter \(\varepsilon\), the majorant estimates are written explicitly, and the proof of the theorem on the error of averaging given in this paper for that case does not follow from theorem [1].

Consider an autonomous system which represents equations of celestial mechanics in the form [2]

\[ \frac{dp}{du}=\varepsilon P_1(p,x,y),\qquad \frac{dx}{du}=-y+\varepsilon P_2(p,x,y), \]

\[ \frac{dy}{du}=x+\varepsilon P_3(p,x,y), \tag{36} \]

where \(p, P_1\) are \(n\)-dimensional vectors; \(P_1, P_2, P_3\) are functions of \(p,x,y\), three times continuously differentiable in the domain \(D\) for \(u>0\).

We shall seek the variables \(x,y\) in the form

\[ x=\alpha+A\cos(u-\varphi),\qquad y=\beta+A\sin(u-\varphi), \tag{37} \]

where \(x=\alpha,\ y=\beta\) turn the last two equations (36) into identities; \(A,\varphi\) are new variables, \(A\geq0\).

With the aid of (37), the last equations (36) are rewritten in the form

\[ \frac{dA}{du}=\varepsilon AF(p,x,y),\qquad \frac{d\varphi}{du}=\varepsilon\Phi(p,x,y), \tag{38} \]

where

\[ F=\frac{1}{A}\,[P_2(p,x,y)-P_2(p,\alpha,\beta)]\cos(u-\varphi)+ \]

\[ +\frac{1}{A}\,[P_3(p,x,y)-P_3(p,\alpha,\beta)]\sin(u-\varphi); \tag{39} \]

\[ \Phi=-\frac{1}{A}\,[P_2(p,x,y)-P_2(p,\alpha,\beta)]\sin(u-\varphi)- \]
\[ -\frac{1}{A}\,[P_3(p,x,y)-P_3(p,\alpha,\beta)]\cos(u-\varphi). \tag{39} \]

The functions \(P_1,F,\Phi\) are periodic in \(u\) with period \(T=2\pi\).

If in the domain \(D\) there exist constants \(M,\lambda\) bounding, respectively, \(P_1,P_2,P_3\) and their derivatives with respect to \(p,x,y\), then by these same constants \(F,\Phi\) and their derivatives with respect to \(p,x,y\) are bounded. For \(A\sim 1\) this is obvious. For \(A<\varepsilon\) this follows from the fact that the expansions

\[ P_i(p,x,y)-P_i(p,\alpha,\beta)=(x-\alpha)\frac{\partial P_i}{\partial x}(p,\alpha,\beta)+ \]
\[ +(y-\beta)\frac{\partial P_i}{\partial y}(p,\alpha,\beta)+\ldots, \tag{40} \]

where \(i=2,3\), represent the left-hand sides of (40) with accuracy up to \(A^2\sim \varepsilon^2\). The same applies, for \(A<\varepsilon\), to the expansion

\[ P_1(p,x,y)=P_1(p,\alpha,\beta)+(x-\alpha)\frac{\partial P_1}{\partial x}(p,\alpha,\beta)+ \]
\[ +(y-\beta)\frac{\partial P_1}{\partial y}(p,\alpha,\beta)+\ldots \tag{41} \]

We form the \((n+2)\)-dimensional vector \(X=\{P_1,AF,\Phi\}\). Let us use the first theorem and the notation: the vector \(\xi(u)\) is the solution of the averaged equations; \(X_0(\xi)\) is the vector of the right-hand sides \(X\) averaged over the period; the vector \(\widetilde X(u,\xi)\) is the operator (5). Obviously, the penultimate components of the vectors \(X,X_0\) are bounded by the quantity \(AM\), and the remaining ones by the quantity \(M\). Taking this into account, and for \(A<\varepsilon\) using the expansions (40), (41), one can show that

\[ |\widetilde X|\leq 2\pi AM,\qquad \left|X_0\frac{\partial\widetilde X}{\partial \xi}\right|\leq 2\pi\lambda AM, \tag{42} \]

whence

\[ |X(u,\xi+\varepsilon\widetilde X)-X(u,\xi)|/\varepsilon\leq \lambda|\widetilde X|\leq 2\pi\lambda AM. \tag{43} \]

The averaging error will be \(R=\varepsilon(\widetilde X+r)\), where \(r\) satisfies an integral equation of the form (8), in which \(\tau=\varepsilon u\). Obviously,

\[ \frac{dA}{du}\leq \varepsilon AM, \tag{44} \]

whence

\[ A\leq A_0\exp \varepsilon Mu \tag{45} \]

or

\[ u\geq \frac{1}{\varepsilon M}\ln\frac{A}{A_0}. \tag{46} \]

Thus, the transition from values \(A_0\sim\varepsilon\) to \(A\sim 1\) lasts for

\[ u\geq \frac{1}{\varepsilon M}\ln\frac{1}{\varepsilon}, \]

i.e. on an interval considerably larger than \(u\sim 1/\varepsilon\).

Using (42), (43), (45), we estimate the terms of the series (9) on the interval (46):

\[ r_0=0,\quad |r_1|\leq 4\pi\lambda AM\leq 4\pi\lambda M A_0\exp M\tau, \tag{47} \]

\[ |r_2-r_1|\leqslant \lambda \int_0^\tau |r_1-r_0|\,d\tau \leqslant 4\pi\lambda^2 A_0(\exp M\tau-1) \leqslant 4\pi\lambda^2 A_0 \exp M\tau, \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \tag{47} \]

\[ |r_{n+1}-r_n|\leqslant 4\pi A_0\lambda^{\,n+1}M^{-n+1}\exp M\tau, \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

whence

\[ |r|\leqslant 4\pi\lambda M A_0 \exp M\tau\cdot \left(1+\frac{\lambda}{M}+\cdots+\frac{\lambda^n}{M^n}\cdots\right). \tag{48} \]

If \(\lambda/M<1\), we sum the series (48); then

\[ |r|\leqslant \frac{4\pi\lambda M A}{1-\lambda/M}. \tag{49} \]

Thus, on the interval (46), for the error of averaging we have the estimate

\[ |R|\leqslant \varepsilon\,2\pi\lambda M A\,\frac{3-\lambda/M}{1-\lambda/M}. \tag{50} \]

Let us note that for \(\alpha=\beta=0\), (37) is a solution of the last equations (36) for \(\varepsilon=0\), i.e., the usual unperturbed motion. Then the last equations (36) are not reducible to the form (38), where \(F,\Phi\) possess the property (40); and it is impossible to prove the theorem on the error of averaging on the interval (46).

In [2] a method is proposed for the approximate determination of \(\alpha,\beta\)—corrections to the usual unperturbed motion. The geometric meaning of \(A\) is the deviation from the \(n\)-dimensional manifold \(p=p(u,\alpha,\beta)\). We shall call this manifold stable in the case of decreasing \(A\). In this case, when the trajectory is sufficiently close to \(p=p(u,\alpha,\beta)\), the range of perturbations under consideration can be enlarged. Indeed, \(\varepsilon A\) enters the right-hand side of (50). The smaller \(A\) is, the larger \(\varepsilon\) may be chosen.

References

1. N. N. Bogolyubov, Yu. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations. Fizmatgiz, 1963.

2. V. V. Laricheva, M. V. Rein, journal Kosmicheskie issledovaniya, 3, no. 1, 26–41, 1965.

Received by the editors
February 8, 1965

Central Aerohydrodynamic
Institute named after N. E. Zhukovsky