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MATHEMATICS
G. S. MAKAEVA
ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH A SMALL PARAMETER, WHOSE SYSTEMS OF “FAST MOTIONS” ARE CLOSE TO HAMILTONIAN ONES
(Presented by Academician P. S. Aleksandrov, 18 IV 1958)
We consider the system of differential equations
\[ \begin{aligned} \varepsilon \frac{dx}{dt} &= \frac{\partial H(x,y,z_1,\ldots,z_l)}{\partial y} +\varepsilon X(x,y,z_1,\ldots,z_l,\varepsilon),\\ \varepsilon \frac{dy}{dt} &= -\frac{\partial H(x,y,z_1,\ldots,z_l)}{\partial x} +\varepsilon Y(x,y,z_1,\ldots,z_l,\varepsilon),\\ \frac{dz_j}{dt} &= Z_j(x,y,z_1,\ldots,z_l,\varepsilon) \qquad (j=1,2,\ldots,l), \end{aligned} \tag{1} \]
where \(\varepsilon\) is a small positive parameter.
The problem of investigating this system and deriving from it the known results of V. M. Volosov \(({}^{1-5})\) was posed by L. S. Pontryagin in his report at V. I. Smirnov’s seminar in Leningrad in April 1957.
In the fast time \(\tau=t/\varepsilon\), system (1) has the form
\[ \begin{aligned} \frac{dx}{d\tau} &= \frac{\partial H(x,y,z_1,\ldots,z_l)}{\partial y} +\varepsilon X(x,y,z_1,\ldots,z_l,\varepsilon),\\ \frac{dy}{d\tau} &= -\frac{\partial H(x,y,z_1,\ldots,z_l)}{\partial x} +\varepsilon Y(x,y,z_1,\ldots,z_l,\varepsilon),\\ \frac{dz_j}{d\tau} &= \varepsilon Z_j(x,y,z_1,\ldots,z_l,\varepsilon) \qquad (j=1,2,\ldots,l). \end{aligned} \tag{2} \]
For \(\varepsilon=0\), system (2) becomes the Hamiltonian system
\[ \begin{aligned} \frac{dx}{d\tau} &= \frac{\partial H(x,y,z_1,\ldots,z_l)}{\partial y},\\ \frac{dy}{d\tau} &= -\frac{\partial H(x,y,z_1,\ldots,z_l)}{\partial x},\\ \frac{dz_j}{d\tau} &=0 \qquad (j=1,\ldots,l). \end{aligned} \tag{3} \]
The first integral
\[ H(x,y,z_1,\ldots,z_l)=h \tag{4} \]
represents the family of all phase trajectories of system (3) on each plane \(z_j=\mathrm{const}\) \((j=1,\ldots,l)\) of the Euclidean space \(E_{2+l}\) of the variables \(x,y,z_1,\ldots,z_l\).
Take some point \((x^0,y^0,z_1^0,\ldots,z_l^0)\) in \(E_{2+l}\) that is not an equilibrium position of system (3). \(h^0=H(x^0,y^0,z_1^0,\ldots,z_l^0)\), \(z_1^0,\ldots,z_l^0\)
determine only one phase trajectory of system (3), \(H(x, y, z_1^0, \ldots, z_l^0)=h^0\), passing through \((x^0, y^0, z_1^0, \ldots, z_l^0)\). Let this phase trajectory be closed. Then in the space \(E_{2+l}\) there exists a certain neighborhood \(G\) of this trajectory in which: 1) the phase trajectories of system (3) are closed; 2) each complex \((h, z_1, \ldots, z_l)\) from the neighborhood of \((h^0, z_1^0, \ldots, z_l^0)\) determines only one phase trajectory of system (3); 3) on each phase trajectory (4) of system (3) one can choose one initial point \(\{\alpha(h, z_1, \ldots, z_l), \beta(h, z_1, \ldots, z_l)\}\), for example as the intersection of the phase trajectories (4) with some smooth curve passing through \(\{x^0, y^0\}\) and not tangent to the phase trajectory \(H(x,y,z_1,\ldots,z_l)=H(x^0,y^0,z_1,\ldots,z_l)\) at the point \((x^0,y^0,z_1,\ldots,z_l)\).
We investigate the solution of system (1) \(x(t,\varepsilon), y(t,\varepsilon), z_1(t,\varepsilon), \ldots, z_l(t,\varepsilon)\), passing through an arbitrary point \((x_0,y_0,z_{10},\ldots,z_{l0})\) of \(G\) at \(t=t_0\).
Theorem. Let the functions \(H(x,y,z_1,\ldots,z_l)\), \(\partial H(x,y,z_1,\ldots,z_l)/\partial x\), \(\partial H(x,y,z_1,\ldots,z_l)/\partial y\) be defined and continuous in \(G\) together with the partial derivatives with respect to all variables up to the second order inclusive, and let the functions \(X(x,y,z_1,\ldots,z_l,\varepsilon)\), \(Y(x,y,z_1,\ldots,z_l,\varepsilon)\), \(Z_j(x,y,z_1,\ldots,z_l,\varepsilon)\) \((j=1,2,\ldots,l)\) be continuous in \(G\) together with partial derivatives up to the first order inclusive and differentiable with respect to \(\varepsilon\) for \(\varepsilon \geqslant 0\).
Then there exists such an \(\varepsilon_0\) that, for any \(0<\varepsilon\leqslant \varepsilon_0\):
1) The functions \(h(t,\varepsilon)=H[x(t,\varepsilon), y(t,\varepsilon), z_1(t,\varepsilon), \ldots, z_l(t,\varepsilon)]\), \(z_1(t,\varepsilon), \ldots, z_l(t,\varepsilon)\) on \([t_0,L]\) coincide, to within quantities of order \(o(\varepsilon)\), with the solution \(\bar h(t), \bar z_1(t), \ldots, \bar z_l(t)\) of the autonomous system of ordinary differential equations, independent of \(\varepsilon\):
\[ \frac{dh}{dt}=A(h,z_1,\ldots,z_l), \]
\[ \frac{dz_j}{dt}=A_j(h,z_1,\ldots,z_l)\qquad (j=1,\ldots,l), \tag{5} \]
passing at \(t=t_0\) through \(h_0=H(x_0,y_0,z_{10},\ldots,z_{l0}), z_{10},\ldots,z_{l0}\) and remaining in \(G\) together with some \(\rho\)-neighborhood on the finite interval \([t_0,L]\). In (5) the functions \(A(h,z_1,\ldots,z_l)\), \(A_j(h,z_1,\ldots,z_l)\) \((j=1,\ldots,l)\) are expressed in terms of the functions of the right-hand sides of system (1) by the formulas
\[ A=\frac{1}{T}\oint_{H=h}\left[ X\frac{\partial H}{\partial x} + Y\frac{\partial H}{\partial y} + \sum_{i=1}^{l} Z_i\frac{\partial H}{\partial z_i} \right] \left[ \left(\frac{\partial H}{\partial x}\right)^2 + \left(\frac{\partial H}{\partial y}\right)^2 \right]^{-1/2} ds, \]
\[ A_j=\frac{1}{T}\oint_{H=h} Z_j \left[ \left(\frac{\partial H}{\partial x}\right)^2 + \left(\frac{\partial H}{\partial y}\right)^2 \right]^{-1/2} ds \qquad (j=1,\ldots,l), \tag{6} \]
where \(X=X(x,y,z_1,\ldots,z_l,0)\); \(Y=Y(x,y,z_1,\ldots,z_l,0)\); \(Z_j=Z_j(x,y,z_1,\ldots,z_l,0)\); \(H=H(x,y,z_1,\ldots,z_l)\); \(T=T(h,z_1,\ldots,z_l)=\)
\[ =\oint_{H=h} \left[ \left(\frac{\partial H}{\partial x}\right)^2 + \left(\frac{\partial H}{\partial y}\right)^2 \right]^{-1/2} ds; \]
\(s\) is the arc length of the phase trajectory (4);
\[ H(x,y,z_1,\ldots,z_l)=h \quad (h,z_1,\ldots,z_l\ \text{fixed}). \]
2) The functions \(x(t,\varepsilon), y(t,\varepsilon)\), to within quantities of order \(o(\varepsilon)\), coincide with the functions
\[ x(t,\varepsilon)= \tag{7} \]
\[ = x^*\left( \varphi(t_0)+\frac{1}{\varepsilon}\int_{t_0}^{t} \frac{1}{T[\bar h(r),\bar z_1(r),\ldots,\bar z_l(r)]}\,dr +\nu(t,\varepsilon), \bar h(t),\bar z_1(t),\ldots,\bar z_l(t) \right), \]
\[ y(t,\varepsilon)= \tag{8} \]
\[ = y^*\left( \varphi(t_0)+\frac{1}{\varepsilon}\int_{t_0}^{t} \frac{1}{T[\bar h(r),\bar z_1(r),\ldots,\bar z_l(r)]}\,dr +\nu(t,\varepsilon), \bar h(t),\bar z_1(t),\ldots,\bar z_l(t) \right), \]
and \(x^*(\varphi,h,z_1,\ldots,z_l)\) and \(y^*(\varphi,h,z_1,\ldots,z_l)\) are periodic in \(\varphi\) with period 1, and
\[ x^*(\varphi,h,z_1,\ldots,z_l)\equiv \widetilde{x}(T\varphi,h,z_1,\ldots,z_l) \equiv \widetilde{x}(\tau,h,z_1,\ldots,z_l), \]
\[ y^*(\varphi,h,z_1,\ldots,z_l)\equiv \widetilde{y}(T\varphi,h,z_1,\ldots,z_l) \equiv \widetilde{y}(\tau,h,z_1,\ldots,z_l); \]
\(\widetilde{x}(\tau,h,z_1,\ldots,z_l)\), \(\widetilde{y}(\tau,h,z_1,\ldots,z_l)\) is a solution of system (3) passing through \(\{\alpha(h,z_1,\ldots,z_l),\beta(h,z_1,\ldots,z_l)\}\) at \(\tau=0\); the period of this solution is \(T(h,z_1,\ldots,z_l)\); \(\varphi_0=\varphi(t_0)\) is found from \(x_0=x^*(\varphi_0,h_0,z_{10},\ldots,z_{l0})\), \(y_0=y^*(\varphi_0,h_0,z_{10},\ldots,z_{l0})\) (the function \(\nu(t,\varepsilon)=\varphi(t,\varepsilon)-\bar{\varphi}(t,\varepsilon)\) will be discussed below).
Proof. The change of variables \(x,y\) to \(\varphi,h\) by the formula
\[ x=x^*(\varphi,h,z_1,\ldots,z_l),\qquad y=y^*(\varphi,h,z_1,\ldots,z_l) \tag{9} \]
transforms system (1) into the system
\[ \frac{dh}{dt}=\mathfrak{A}(\varphi,h,z_1,\ldots,z_l,\varepsilon), \]
\[ \frac{dz_j}{dt}=\mathfrak{A}_j(\varphi,h,z_1,\ldots,z_l,\varepsilon), \tag{10} \]
\[ \frac{d\varphi}{dt}=\frac{1}{\varepsilon T(h,z_1,\ldots,z_l)}+ \]
\[ +\frac{1}{T(h,z_1,\ldots,z_l)} \left[ X\frac{\partial y^*}{\partial h} - Y\frac{\partial x^*}{\partial h} + \sum_{i=1}^{l} Z_i \left( \frac{\partial x^*}{\partial h}\frac{\partial y^*}{\partial z_i} - \frac{\partial y^*}{\partial h}\frac{\partial x^*}{\partial z_i} \right) \right], \]
where
\[ \mathfrak{A}\equiv X\frac{\partial H}{\partial x} + Y\frac{\partial H}{\partial y} + \sum_{i=1}^{l} Z_i\frac{\partial H}{\partial z_i}, \qquad \mathfrak{A}_j\equiv Z_j, \]
\[ X=X[x^*(\varphi,h,z_1,\ldots,z_l),\ y^*(\varphi,h,z_1,\ldots,z_l),\ z_1,\ldots,z_l,\ \varepsilon], \]
\[ Y=Y[x^*(\varphi,h,z_1,\ldots,z_l),\ y^*(\varphi,h,z_1,\ldots,z_l),\ z_1,\ldots,z_l,\ \varepsilon], \]
\[ Z_j=Z_j[x^*(\varphi,h,z_1,\ldots,z_l),\ y^*(\varphi,h,z_1,\ldots,z_l),\ z_1,\ldots,z_l,\ \varepsilon]. \]
But the solution of system (10) \(h(t,\varepsilon),z_1(t,\varepsilon),\ldots,z_l(t,\varepsilon),\varphi(t,\varepsilon)\), passing through \(h_0,z_{10},\ldots,z_{l0},\varphi_0\) at \(t=t_0\), is related to the solution \(\bar{h}(t)\), \(\bar{z}_1(t),\ldots,\bar{z}_l(t),\bar{\varphi}(t,\varepsilon)\) of the averaged system
\[ \frac{dh}{dt}=\int_0^1 \mathfrak{A}(\varphi,h,z_1,\ldots,z_l,0)\,d\varphi \equiv A(h,z_1,\ldots,z_l), \]
\[ \frac{dz_j}{dt}=\int_0^1 \mathfrak{A}_j(\varphi,h,z_1,\ldots,z_l,0)\,d\varphi \equiv A_j(h,z_1,\ldots,z_l) \qquad (j=1,\ldots,l, \]
\[ \frac{d\varphi}{dt}=\frac{1}{\varepsilon T(h,z_1,\ldots,z_l)}, \tag{11} \]
passing through the same initial point \(h_0,z_{10},\ldots,z_{l0},\varphi_0\), as follows:
\[ \begin{aligned} |h(t,\varepsilon)-\bar{h}(t)|&\leq o(\varepsilon),\\ |z_j(t,\varepsilon)-\bar{z}_j(t)|&\leq o(\varepsilon),\\ |\varphi(t,\varepsilon)-\bar{\varphi}(t,\varepsilon)|&\leq o(1)\qquad (o(\varepsilon)>1,\ o(1)>0). \end{aligned} \tag{12} \]
Relations (12) and (9) prove the theorem.
Remark. It is also proved that if the point \((x_0,\ y_0,\ z_{10},\ldots,\ z_{l0})\) of \(E_{2+l}\) is a singular point of system (3) of the type of a nondegenerate center, then there exists some neighborhood \(G_0\) of this point such that, under the hypotheses of the theorem, the solution of system (1) \(x(t,\varepsilon), y(t,\varepsilon), z_1(t,\varepsilon),\ldots,\ z_l(t,\varepsilon)\), passing through \(x_{00}, y_{00}, z_{10},\ldots,\ z_{l0}\) at \(t=t_0\), coincides on the interval \([t_0,L]\), up to an error of order \(o(\varepsilon)\), with the solution of the degenerate system
\[ \frac{\partial H(x,y,z_1,\ldots,z_l)}{\partial y}=0 \qquad \left(|x_0-x_{00}|+|y_0-y_{00}|\leq o(\varepsilon)\right), \]
\[ \frac{\partial H(x,y,z_1,\ldots,z_l)}{\partial x}=0, \tag{13} \]
\[ \frac{dz_j}{dt}=Z_j(x,y,z_1,\ldots,z_l,0) \qquad (j=1,\ldots,l), \]
passing through \(x_0, y_0, z_{10},\ldots,z_{l0}\) at \(t=t_0\) and remaining in \(G_0\) together with some \(\rho_0\)-neighborhood on \([t_0,L]\).
Let us note that the equation
\[ \mu u^{(n)}+Q(t,u,u',\ldots,u^{(n-2)})=0, \tag{14} \]
where \(\mu\) is a small positive parameter; \(Q(t,u,\ldots,u^{(n-2)})=0\) has the root \(u^{(n-2)}=f(t,u,\ldots,u^{(n-s)})\), \(\operatorname{sign} Q=\operatorname{sign}(u^{(n-2)}-f)\); \(m|u^{(n-2)}-f|\leq |Q|\leq M|u^{(n-2)}-f|\); \(m,M\) are certain positive constants, considered in \((1\text{–}4)\), is reduced by the substitution
\[ t=z_1,\qquad u=z_2,\ldots,\qquad u^{(n-3)}=z_{n-1},\qquad u^{(n-2)}=x,\qquad \varepsilon=\sqrt{\mu},\qquad \varepsilon\frac{dx}{dt}=y, \]
\[ H(x,y,z_1,\ldots,z_{n-1}) = \frac{y^2}{2} + \int_0^x Q(z_1,\ldots,z_{n-1},p)\,dp \tag{15} \]
to the system
\[ \varepsilon\frac{dx}{dt} = \frac{\partial H(x,y,z_1,\ldots,z_{n-1})}{\partial y}, \]
\[ \varepsilon\frac{dy}{dt} = - \frac{\partial H(x,y,z_1,\ldots,z_{n-1})}{\partial x}, \]
\[ \frac{dz_1}{dt}=1, \]
\[ \frac{dz_i}{dt}=z_{i+1} \qquad (i=2,\ldots,n-2), \]
\[ \frac{dz_{n-1}}{dt}=x, \tag{16} \]
where the corresponding Hamiltonian system
\[ \varepsilon\frac{dx}{dt}=y, \]
\[ \varepsilon\frac{dy}{dt}=-Q(z_1,\ldots,z_{n-1},x) \]
has the singular point \(x=f(z_1,\ldots,z_{n-1}),\ y=0\) of the type of a nondegenerate center. System (16), however, is a special case of system (1), and the results of papers \((1\text{–}5)\) follow from the theorem obtained in the present article.
In conclusion I express my sincere gratitude to L. S. Pontryagin, under whose supervision this work was carried out.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
17 IV 1958
CITED LITERATURE
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- V. M. Volosov, Uspekhi Mat. Nauk, 5, issue 5 (39), 145 (1950).
- V. M. Volosov, Mat. Sb., 30 (72), 2, 245 (1952).
- V. M. Volosov, Mat. Sb., 31 (73), 3, 645 (1952).
- V. M. Volosov, DAN, 106, No. 1, 7 (1956).