V. M. Volosov

On Higher Approximations in Averaging

(Presented by Academician I. G. Petrovskii, November 17, 1960)

§ 1. Statement of the problem

In \((^3)\) an averaging method was developed for systems

\[ \dot{x}=\varepsilon X(x,y,t,\varepsilon),\qquad \dot{y}=Y(x,y,t,\varepsilon) \tag{1} \]

(\(x,X\) are \(n\)-dimensional, \(y,Y\) are \(m\)-dimensional vectors, \(\varepsilon>0\) is a small parameter). For \(\varepsilon=0\), (1) passes into the degenerate system

\[ \dot{y}=Y_0(x,y,t)\equiv Y(x,y,t,0),\qquad x=\mathrm{const}, \tag{2} \]

whose general solution is assumed known:

\[ y=\varphi(x,y_0,t_0,t)\qquad \bigl(\varphi(x,y_0,t_0,t_0)\equiv y_0\bigr). \tag{3} \]

It is assumed that the right-hand sides of (1) and certain other functions have mean values along the trajectories (3): the mean value of the function \(F(x,y_0,t_0,y,t)\) is taken to be

\[ \bar{F}(x)=\lim_{T\to\infty}\frac{1}{T}\int_{t_0}^{t_0+T} \bigl(F\mid_{y=\varphi(x,y_0,t_0,t)}\bigr)\,dt \]

(here and below it is assumed everywhere that \(\bar{F}(x)\) is continuous and satisfies the Lipschitz condition), and the limit exists uniformly with respect to \(x,y_0,t_0\) in the domain under consideration and does not depend on \(y_0,t_0\). (It is shown in \((^3)\) that the assumption of independence of \(y_0,t_0\) does not restrict generality.) In \((^3)\) it was proved that, under certain conditions, the solution \(x\) of system (1) is approximated with error \(\alpha(\varepsilon)\) (\(\alpha(\varepsilon)\) here and below denotes quantities for which \(\lim_{\varepsilon\to0}\alpha(\varepsilon)=0\)) on the interval \(t\sim 1/\varepsilon\) by the solution of the averaged system of the first approximation

\[ \dot{x}=\varepsilon \bar{X}_1(x) \]

\[ \bigl(X_1(x,y,t)\equiv X(x,y,t,0)\bigr). \]

In the present article the averaged system of the second approximation is considered,

\[ \dot{\bar{x}}=\varepsilon \bar{X}_1(\bar{x})+\varepsilon^2 A_2(\bar{x}),\qquad \dot{\bar{y}}=Y_0(\bar{x},\bar{y},t)+\varepsilon B_1(\bar{x}), \tag{4} \]

derived in \((^3)\) (\(A_2,B_1\) are indicated below). It is proved that, under certain conditions, the solutions of (4) approximate the solutions of (1) with error \(\varepsilon\alpha(\varepsilon)\) for \(x\) and \(\alpha(\varepsilon)\) for \(y\) on the interval \(t\sim 1/\varepsilon\).

§ 2. Theorems on higher approximations

The domain of definition of (1): \(0\leq\varepsilon\leq\varepsilon_0;\ x,y,t\in G\), where \(G\) is an open domain.

Let:

1) \(X,Y\) be bounded, continuous, and have continuous uniformly bounded derivatives with respect to \(x,y,t\), while with respect to \(\varepsilon\) they have derivatives up to the second order inclusive, with \(X_{\varepsilon^2}^{\prime\prime},Y_{\varepsilon^2}^{\prime\prime}\) uniformly bounded.

2) Through each point of the domain \(G\) there passes a unique integral curve (3) of system (2), lying in \(G\) for \(t_0 \leqslant t<\infty\), continued for \(t\leqslant t_0\) to the boundary of \(G\) or to \(t\to -\infty\). The function (3) is continuous, and with respect to \(y_0,t_0\) has continuous bounded derivatives up to the second order inclusive. \(0<c_1\leqslant |\operatorname{Det}D|\leqslant c_2<\infty\), where \(D\equiv \partial\varphi/\partial y_0\) (here and below \(\partial\varphi/\partial y_0,\ \partial Y/\partial y\), etc. denote the matrices \(\|\partial\varphi_i/\partial y_{0k}\|,\ \|\partial Y_i/\partial y_k\|\), etc.).

3) In \(G\) there lies an \((n+m)\)-dimensional manifold \(M\), given parametrically:
\(x=a(\lambda),\ y=b(\lambda),\ t=c(\lambda)\)
\((\lambda=\{\lambda_1,\ldots,\lambda_{n+m}\}\in \Lambda,\ \Lambda\) an open domain); \(a,b,c\) are continuous and have continuous bounded derivatives;

\[ \sum_{i=1}^{n+m+1} A_i^2 \geqslant \operatorname{const}>0, \]

where \(A_i\) are the minors of order \((n+m)\) of the matrix \(\|\partial a/\partial\lambda,\ \partial b/\partial\lambda,\ \partial c/\partial\lambda\|\). The absolute values of the angles of intersection of (3) with \(M\) are bounded below by a positive constant. Every curve (3) from \(G\) intersects \(M\) once.

4) There exists the mean value \(\overline{X}_1\) of the function \(X_1\) (here and below mean values are understood in the sense of the definition of § 1). The function \(S\equiv X_1-\overline{X}_1\) is uniformly bounded.

5) For \(0<\varepsilon\leqslant\varepsilon_0\) there exist open bounded subdomains \(G_0(\varepsilon)\subseteq G\), containing the fixed initial point \(x_0,y_0,t_0\) together with some \(\rho\)-neighborhood \((\rho=\operatorname{const}>0)\). The time of passage of curve (3) from any point of \(G_0\) to the intersection with \(M\) does not exceed, in absolute value, \(K/\varepsilon\) \((K=\operatorname{const}>0)\). For \(0<\varepsilon\leqslant\varepsilon_0\) there exist open subdomains \(G_1(\varepsilon)\subset G_0\), containing \(x_0,y_0,t_0\); the distances from the points of \(G_1\) to the boundary of \(G_0\) are bounded below by a positive constant.

Under conditions 1)—5), in \(G\) there exists a continuously differentiable solution of the equation

\[ \frac{\partial u_1}{\partial t}+\frac{\partial u_1}{\partial y}Y_0=S,\quad u_1(x,y,t)\big|_{x,y,t\in M}=0, \]

and \(u_1\) is easily constructed from the characteristics (3) and is therefore regarded as known.

6) Let \(u_1\) be bounded in \(G\) and have bounded continuous derivatives with respect to \(x,y,t\).

7) \(\dfrac{\partial X_1}{\partial x}u_1\) has a mean value.

8) For \(\partial u_1/\partial x,\ \partial u_1/\partial y\) there exist mean values.

9) \(X_2\equiv X'_{\varepsilon}\big|_{\varepsilon=0}\) also has mean values.

10) There exists the mean value \(H(x)\) of the matrix \(D^{-1}\), \(0<c_1\leqslant |\operatorname{Det}H|\leqslant c_2<\infty\).

11) There exists the mean value \(R(x)\) of the function \(D^{-1}\left(Y_1+\dfrac{\partial Y_0}{\partial x}u_1\right)\)
\((Y_1\equiv Y'_{\varepsilon}\big|_{\varepsilon=0})\).

12) There exists the mean value of the expression

\[ \frac{\partial X_1}{\partial y}\, D\int_{t_0}^{t} \left[ D^{-1}\left( Y_1+ \frac{\partial Y_0}{\partial x}u_1-B_1 \right) \right]\bigg|_{y=\varphi(x,y_0,t_0,t)}dt, \]

where \(B_1(x)\equiv H^{-1}R\) (\(B_1\) enters system (4)).

From conditions 1)—12) it follows that in \(G\) there is defined a continuously differentiable solution of the equation

\[ \frac{\partial v_1}{\partial t} +\frac{\partial v_1}{\partial y}Y_0 -\frac{\partial Y_0}{\partial y}v_1 = Y_1+ \frac{\partial Y_0}{\partial x}u_1-B_1, \quad v_1\big|_{x,y,t\in M}=0, \]

and moreover \(\dfrac{\partial X_1}{\partial y}v_1\) has a mean value,—this is easily verified, since \(v_1\) can be constructed from the characteristics, as can \(u_1\). From what has been said it follows that the function

\[ P\equiv \frac{\partial X_1}{\partial x}u_1+ \frac{\partial X_1}{\partial y}v_1 -\frac{\partial u_1}{\partial x}\overline{X}_1 -\frac{\partial u_1}{\partial y}B_1 +X_2 \]

has a mean value \(\equiv A_2(x)\) (\(A_2\) enters system (4)).

Let:

13) The function \(P-A_2\) is uniformly bounded.

14) The expressions

\[ \frac{\partial}{\partial y_0}\left(\int_{t_0}^{t_0+T} (P-A_2)\,dt\right),\qquad \frac{\partial}{\partial t_0}\left(\int_{t_0}^{t_0+T} (P-A_2)\,dt\right), \]

\[ \frac{\partial}{\partial y_0}\left(\int_{t_0}^{t_0+T} D^{-1}\left(Y_1+\frac{\partial Y_0}{\partial x}u_1-B_1\right)\,dt\right),\qquad \frac{\partial}{\partial t_0}\left(\int_{t_0}^{t_0+T} D^{-1}\left(Y_1+\frac{\partial Y_0}{\partial x}u_1-B_1\right)\,dt\right) \]

(the integrals are taken along (3)) are uniformly bounded for \(0\leq T<\infty\).

15) System (4) possesses “stability”: for every \(K>0\) there exist \(c_1>0\), \(c_2>0\), \(\bar\varepsilon>0\) (\(\bar\varepsilon\leq \varepsilon_0\)) such that if on some interval \([t_0,\bar t(\varepsilon)]\subseteq [t_0,K/\varepsilon]\), for \(0<\varepsilon\leq \bar\varepsilon\), the solutions of (4) and of the equation

\[ z=Y_0(x,z,t)+\varepsilon B_1(x)+\varepsilon\varphi(t,\varepsilon) \]

exist for the initial values \(x_0,y_0,t_0\) and \(z_0,t_0\) (\(\|z_0-y_0\|\leq c_1\), \(x,y\) is the solution of (4) with initial point \(x_0,y_0,t_0\)), and \(\varphi(t,\varepsilon)\) is an arbitrary continuous function such that

\[ \sup_{t_0\leq t\leq \bar t}|\varphi(t,\varepsilon)|\leq c_1, \]

then for \(0<\varepsilon\leq\bar\varepsilon\), \(t\in[t_0,\bar t]\),

\[ |\bar y-z|\leq c_2\left(\varepsilon\sup_{t_0\leq t\leq\bar t}|\varphi(t,\varepsilon)|\cdot |t-t_0|+|y_0-z_0|\right). \]

Define the interval \([t_0,t_1(\varepsilon)]\): \(t_1>t_0\), \(t_1-t_0\leq K/\varepsilon\), on which for \(t\in[t_0,t_1]\) the solution of (4) with initial point \(x_0,y_0,t_0\) does not leave \(G_1(\varepsilon)\).

Theorem 1. Under conditions 1)—15), for arbitrary \(K>0\), \(\delta>0\), there exists \(\varepsilon_1>0\) (\(\varepsilon_1\leq\varepsilon_0\)) such that for \(0<\varepsilon\leq\varepsilon_1\), \(t\in[t_0,t_1(\varepsilon)]\),

\[ \text{1) } |y-\bar y|\leq \delta;\qquad \text{2) } |x-\bar x-\varepsilon u_1(\bar x,\bar y,t)|\leq \varepsilon\delta, \]

where \(x,y\) are solutions of (1) with initial point \(x_0,y_0,t_0\).

From conditions 1)—15) it follows that in \(G\) there exists a continuously differentiable solution of the equation

\[ \frac{\partial u_2}{\partial t}+\frac{\partial u_2}{\partial y}Y_0=P-A_2,\qquad u_2(x,y,t)\big|_{x,y,t\in M}=0, \]

which is easily constructed from the characteristics, as is \(u_1\).

Suppose the following additional conditions are satisfied:

16) In \(G\), \(v_1,u_2\) are bounded and have continuous bounded derivatives with respect to \(x,y,t\).

17) The expressions

\[ \int_{t_0}^{t_0+T} S\,dt,\qquad \int_{t_0}^{t_0+T} (P-A_2)\,dt,\qquad \int_{t_0}^{t_0+T} D^{-1}\left(Y_1+\frac{\partial Y_0}{\partial x}u_1-B_1\right)\,dt \]

(the integrals are taken along (3)) are uniformly bounded for \(0\leq T<\infty\).

If conditions 16), 17) are fulfilled, improved estimates hold:

Theorem 2. Under conditions 1)—17), for arbitrary \(K>0\) there exist \(C>0\), \(\varepsilon_1>0\) (\(\varepsilon_1\leq\varepsilon_0\)) such that for \(0<\varepsilon\leq\varepsilon_1\), \(t\in[t_0,t_1(\varepsilon)]\),

\[ \text{1) } |y-\bar y|\leq C\varepsilon;\qquad \text{2) } |x-\bar x-\varepsilon u_1(\bar x,\bar y,t)|\leq C\varepsilon^2. \]

Remark 1. Restriction 3) can be weakened by allowing, as was done in (³), multiple intersections of (3) with \(M\). Then Theorems 1 and 2 remain valid with some modification of the formulations and conditions.

Remark 2. One may dispense with condition 15). Then, considering the solutions (1) and (4) on an interval \([t_0,t_2(\varepsilon)]\subseteq [t_0,K/\varepsilon]\) such that, for \(t\in[t_0,t_2]\), the solutions (1), (4) lie in \(G_1(\varepsilon)\), one may assert that inequalities 2) of Theorems 1 and 2 (with some modification of the conditions) remain valid; however, the function \(y\) entering the statements of the theorems is then no longer a solution of (4), but is defined as follows:

\[ \bar y\big|_{t=t_0}=y_0,\qquad |\bar y-Y_0(\bar x,\bar y,t)-\varepsilon B_1(\bar x)|\leq \varepsilon\omega, \]

where \(\omega=\alpha(\varepsilon)\) for Theorem 1 and \(O(\varepsilon)\) for Theorem 2, while \(\bar x\) is the solution of (4).

An asymptotic method connected with an averaging scheme different from (³) and having another domain of application was proposed in (⁴).

§ 3. Systems in standard form. In \((^{1})\) systems in standard form \(\dot{x}=\varepsilon X(x,t)\), which are a special case of (1), were studied. From the results of \((^{3})\) follow the approximation equations derived in \((^{1})\); from (4), in particular, follow the equations of the second approximation

\[ \dot{\bar{x}}=\varepsilon X_0(\bar{x})+\varepsilon^2 M_t\left\{\left(\widetilde{X}\frac{\partial}{\partial x}\right)X(\bar{x},t)\right\}, \]

where \(\partial \widetilde{X}/\partial t=X(\bar{x},t)-X_0(\bar{x})\), \(X_0=M_t X\), and \(M_t\) is the averaging operator with respect to the explicitly occurring \(t\).

§ 4. Systems with a rapidly rotating phase. In \((^{4})\) systems with a rapidly rotating phase, which are a special case of (1), were studied; they can be written in the form \(\dot{x}=\varepsilon A(x,\psi,\varepsilon)\), \(\dot{\psi}=\omega(x)+\varepsilon B(x,\psi,\varepsilon)\), where \(A,B\) are periodic in \(\psi\). From the results of \((^{3})\) follow the approximation equations derived in \((^{2})\); from (4), in particular, follow the equations of the second approximation for \(x,\psi\).

Moscow State University
named after M. V. Lomonosov

Received
16 XI 1960

CITED LITERATURE

\(^{1}\) N. N. Bogolyubov, On Certain Statistical Methods in Mathematical Physics, 1945.
\(^{2}\) N. N. Bogolyubov, D. N. Zubarev, Ukr. Mat. Zh., 7, No. 7 (1955).
\(^{3}\) V. M. Volosov, DAN, 137, No. 1 (1961).
\(^{4}\) A. M. Molchanov, DAN, 136, No. 5 (1961).