MATHEMATICS

G. E. KUZMAK

ASYMPTOTIC SOLUTIONS OF THE EQUATION OF MOTION OF A DISSIPATIVE SYSTEM WITH ONE DEGREE OF FREEDOM AND SLOWLY VARYING PARAMETERS

(Presented by Academician A. A. Dorodnitsyn, February 22, 1958)

In the present paper we consider the equation

\[ \frac{d^2 y}{dt^2}+f\left(\tau,\frac{dy}{dt}\right)+\varepsilon F(\tau,y)=0; \tag{1} \]

\(\varepsilon\) is a small parameter; \(\tau=\varepsilon t\) is slow time. This equation may be interpreted as the equation of motion of a material point of mass equal to unity under the action of a principal force \(-f(\tau,dy/dt)\), depending on the velocity \(dy/dt\) (a dissipative force), and a small force \(-\varepsilon F(\tau,y)\). The purpose of the work is to compute expressions that uniformly approximate the solution of equation (1) and its derivative with an error of order \(\varepsilon\) on a time interval of order \(1/\varepsilon\).

The problem is considered under the following assumptions:

a) \(f(\tau,0)=0\).

b) \(f_z(\tau,0)\geq \Delta>0\) (the subscript \(z\) denotes derivatives of the function \(f(\tau,dy/dt)\) with respect to \(dy/dt\)).

c) The function \(f(\tau,dy/dt)\) is considered for \(0\leq |dy/dt|\leq w_t,\ 0\leq \tau\leq \tau_0\). In this domain it does not vanish, except for the value \(dy/dt=0\), and is analytic in \(dy/dt\) together with its first and second derivatives with respect to \(\tau\).

d) The function \(F(\tau,y)\) is defined for \(0\leq |y|\leq w\) and \(0\leq \tau\leq \tau_0\), is analytic in this domain with respect to \(y\), and is continuous in \(\tau\).

In items c) and d), \(w_t\) and \(w\) are certain constants.

To solve the problem posed, the method of “model equations” \((^1,^2)\) is used. As the “model” equation we choose the equation

\[ \varphi^2(\tau)\frac{\partial^2 y_0}{\partial \omega^2} + f\left[\tau,\varphi(\tau)\frac{\partial y_0}{\partial \omega}\right]=0. \tag{2} \]

The function \(\varphi(\tau)\) is chosen by means of the equality

\[ \varphi(\tau)=f_z(\tau,0). \tag{3} \]

Then the solution of equation (2) is written in the form

\[ y_0(\tau,\omega)=B_0(\tau)+A[\tau,e^{-\omega-c(\tau)}]. \tag{4} \]

Here the functions \(B_0(\tau)\) and \(c(\tau)\) are arbitrary, while the function \(A[\tau,e^{-\omega-c(\tau)}]\) is determined when solving equation (2) and may be expanded in a series in powers of \(e^{-\omega-c(\tau)}\), convergent for \(0<\Omega\leq \omega<\infty\) and \(0\leq \tau\leq \tau_0\) (\(\Omega\) is a constant independent of \(\varepsilon\)):

\[ A[\tau,e^{-\omega-c(\tau)}] = \sum_{n=1}^{\infty} B_n(\tau)e^{-n[\omega+c(\tau)]}. \tag{5} \]

From the general theorems on the dependence of solutions of differential equations on a parameter and from equation (2) it follows that, if in the expression for \(y_0(\tau,\omega)\) one expresses \(\tau\) and \(\omega\) in terms of \(t\) by means of the relations

\[ \frac{d\tau}{dt}=\varepsilon,\qquad \frac{d\omega}{dt}=\varphi(\tau), \tag{6} \]

then the function of \(t\) thus obtained will differ from the solution of equation (1) by quantities of order \(\varepsilon\) on each time interval of order unity. In order that this function of \(t\) approximate the solution of equation (1) uniformly on a time interval of order \(1/\varepsilon\), it is necessary to determine the functions \(B_0(\tau)\) and \(c(\tau)\) in the appropriate way. To derive the conditions for determining the functions \(B_0(\tau)\) and \(c(\tau)\), substitute the function \(y_0(\tau,\omega)\), where \(\tau\) and \(\omega\) are expressed in terms of \(t\) by means of (6), into equation (1). We obtain:

\[ \varphi^2(\tau)\frac{\partial^2 y_0}{\partial \omega^2} + f\left[\tau,\varphi(\tau)\frac{\partial y_0}{\partial \omega}\right] + \varepsilon \Phi(\tau,\omega) + O(\varepsilon^2)=0. \]

Here

\[ \Phi(\tau,\omega) = 2\varphi(\tau)\frac{\partial^2 y_0}{\partial \omega\,\partial \tau} + \varphi'(\tau)\frac{\partial y_0}{\partial \omega} + f_z\left[\tau,\varphi(\tau)\frac{\partial y_0}{\partial \omega}\right] \frac{\partial y_0}{\partial \tau} + F(\tau,y_0). \tag{7} \]

We obtain the necessary conditions by considering the terms of order \(\varepsilon\). Under the assumptions made above, it follows from equalities (4) and (5) that the function \(\Phi(\tau,\omega)\) can be represented in the form of a series in powers of \(e^{-\omega}\), convergent for \(0<\Omega\leq \omega<\infty\) and \(0\leq \tau\leq \tau_0\):

\[ \Phi(\tau,\omega)=\sum_{n=0}^{\infty}\Phi_n(\tau)e^{-n\omega}. \tag{8} \]

The coefficients \(\Phi_n(\tau)\) are expressed in terms of the functions \(B_0(\tau)\) and \(c(\tau)\), and also in terms of the known functions entering equation (1). From condition b) and equalities (3) and (6) it follows that, as \(t\) increases, each subsequent term of the series (8) decays faster than the preceding one. Accordingly, as conditions for determining the functions \(B_0(\tau)\) and \(c(\tau)\), it is expedient to take the conditions that the coefficients \(\Phi_0(\tau)\) and \(\Phi_1(\tau)\) be equal to zero, since the corresponding terms of the series (8) decay more slowly than the other terms. The resulting relations for the functions \(B_0(\tau)\) and \(c(\tau)\) have the form

\[ \varphi(\tau)B_0'(\tau)+F[\tau,B_0(\tau)]=0, \]

\[ c(\tau) = \int_0^\tau \frac{ \varphi'(\tau)-F_y[\tau,B_0(\tau)]-f_{zz}(\tau,0)F[\tau,B_0(\tau)] }{ \varphi(\tau) }\,d\tau + \ln\left|\frac{B_1(\tau)}{B_1(0)}\right|. \tag{9} \]

Primes here denote derivatives with respect to \(\tau\).

We note that, under condition (9), the inequality

\[ |\Phi(\tau,\omega)|\leq M_\Phi e^{-2\omega}; \tag{10} \]

holds, where \(M_\Phi\) is a constant independent of \(\varepsilon\).

Theorem. If the functions \(f(\tau,dy/dt)\) and \(F(\tau,y)\) satisfy the conditions stated above, and the arbitrary functions \(B_0(\tau)\) and \(c(\tau)\) entering the solution (4) of the “reference” equation are determined by means of equalities (9), then for \(|\varepsilon|\leq \varepsilon_0\) the functions

\[ y_0(t)=y_0(\tau,\omega),\qquad \left(\frac{dy}{dt}\right)_0 = \varphi(\tau)\frac{\partial y_0}{\partial \omega}, \]

where \(\tau=\varepsilon t,\ \omega=\displaystyle\int \varphi(\tau)\,dt\), approximate, respectively, the solution of equation (1) and its derivative with an error of order \(\varepsilon\) on a time interval of order \(1/\varepsilon\).

For the proof, represent the function \(y(t)\) and its derivative* in the form

\[ y(t)=y_0(t)+\varepsilon Y(t),\qquad \frac{dy}{dt}=\left(\frac{dy}{dt}\right)_0+\varepsilon\left[\frac{\partial y_0}{\partial \tau}+\frac{dY}{dt}\right]. \tag{11} \]

Substituting equalities (11) into equation (1), we obtain an equation for the function \(Y(t)\):

\[ \frac{d^2Y}{dt^2}+f_z\left[\tau,\varphi(\tau)\frac{\partial y_0}{\partial \omega}\right]\frac{dY}{dt} =-\Phi(t)+\varepsilon\Psi\left(t,Y,\frac{dY}{dt},\varepsilon\right). \tag{12} \]

Here the function \(\Psi(t,Y,dY/dt,\varepsilon)\) is expressed in terms of the derivatives of the functions \(f(\tau,dy/dt)\) and \(F(\tau,y)\), in which \(y\) and \(dy/dt\) are replaced by means of (11), and also in terms of the function \(y_0(\tau,\omega)\) and its derivatives; the function \(\Phi(t)\) is the function (7), in which \(\tau\) and \(\omega\) are expressed through \(t\) by means of equalities (6).

Next replace equation (12) by an equivalent system of integral equations under the conditions \(Y(0)=0,\ dY/dt|_{t=0}=0\):

\[ Y=\int_0^t Q_0(t,\xi)\left[-\Phi(\xi)+\varepsilon\Psi\left(\xi,Y,\frac{dY}{dt},\varepsilon\right)\right]d\xi, \]

\[ \frac{dY}{dt}=\int_0^t Q_1(t,\xi)\left[-\Phi(\xi)+\varepsilon\Psi\left(\xi,Y,\frac{dY}{dt},\varepsilon\right)\right]d\xi. \tag{13} \]

Here

\[ v(t)=\int_\infty^t \exp\left[-\int_0^t f_z\left[\tau,\varphi(\tau)\frac{\partial y_0}{\partial \omega}\right]dt\right], \]

\[ Q_0(t,\xi)=\frac{v(t)-v(\xi)}{dv(\xi)/d\xi},\qquad Q_1(t,\xi)=\frac{dv(t)/dt}{dv(\xi)/d\xi}. \]

It is not difficult to show that, in the region of variation of the arguments under consideration \(0\leqslant \xi\leqslant t\leqslant \tau_0/\varepsilon\), the kernels \(Q_0(t,\xi)\) and \(Q_1(t,\xi)\) are bounded by some constant \(M_Q\) independent of \(\varepsilon\).

Let \(H\) denote a positive number greater than \(M_\Phi M_Q/2\Delta\). If the values \(|Y|\) and \(|dY/dt|\) do not exceed \(H\), then, under the conditions of the theorem, for \(0\leqslant t\leqslant \tau_0/\varepsilon\) and \(|\varepsilon|\leqslant \varepsilon_0\) the function \(\Psi(t,Y,dY/dt,\varepsilon)\), first, is bounded by a constant \(M_\Psi\), and, second, satisfies the Lipschitz condition in \(Y\) and \(dY/dt\) with constant \(L\). We note that the constants \(H,\ M_\Psi\), and \(L\) can be chosen independent of \(\varepsilon\).

Apply to equations (13) the method of successive approximations:

\[ Y^{(0)}=-\int_0^t Q_0(t,\xi)\Phi(\xi)d\xi,\qquad \frac{dY^{(0)}}{dt}=-\int_0^t Q_1(t,\xi)\Phi(\xi)d\xi, \]

\[ Y^{(n+1)}=Y^{(0)}+\varepsilon\int_0^t Q_0(t,\xi)\Psi\left(\xi,Y^{(n)},\frac{dY^{(n)}}{dt},\varepsilon\right)d\xi, \tag{14} \]

\[ \frac{dY^{(n+1)}}{dt}=\frac{dY^{(0)}}{dt} +\varepsilon\int_0^t Q_1(t,\xi)\Psi\left(\xi,Y^{(n)},\frac{dY^{(n)}}{dt},\varepsilon\right)d\xi \qquad (n=0,1,2,\ldots). \]

Using inequality (10) for \(\Phi(t)\), the boundedness of the kernels, and the above-indicated properties of the function \(\Psi(t,Y,dY/dt,\varepsilon)\), from (14) we obtain the following estimates:

\[ |Y^{(0)}|\leqslant \frac{M_\Phi M_Q}{2\Delta},\qquad |Y^{(n+1)}-Y^{(n)}|\leqslant \frac{M_\Psi}{2L}\frac{(\varepsilon M_Q 2Lt)^{n+1}}{(n+1)!} \qquad (n=0,1,2,\ldots). \tag{15} \]

Exactly the same estimates are obtained for the derivatives. These inequalities are valid under the condition that none of the approximations exceeds \(H\). Below we shall define the number \(\tau_* \leqslant \tau_0\) so that for \(0 \leqslant t \leqslant \tau_*/\varepsilon\) this condition will be satisfied.

In order to estimate \(Y(t)\), represent it in the form

\[ Y(t)=Y^{(0)}+\sum_{n=0}^{\infty}\left(Y^{(n+1)}-Y^{(n)}\right). \tag{16} \]

Using inequalities (15), for \(0 \leqslant t \leqslant \tau_*/\varepsilon\) and \(|\varepsilon| \leqslant \varepsilon_0\), we obtain

\[ |Y(t)| \leqslant \frac{M_{\Phi}M_Q}{2\Delta} +\frac{M_{\Psi}}{2L}\left(e^{2M_Q L\tau}-1\right). \tag{17} \]

We now define the number \(\tau_*\). From (16) we have that \(|Y^{(n+1)}(t)|\), as well as \(|Y(t)|\), satisfies inequality (17). Introduce the number \(\tau_1\) by means of the equality

\[ \frac{M_{\Phi}M_Q}{2\Delta} +\frac{M_{\Psi}}{2L}\left(e^{2M_Q L\tau_1}-1\right)=H. \]

Note that \(\tau_1>0\) because the number \(H\) is greater than \(M_{\Phi}M_Q/2\Delta\). If, as the number \(\tau_*\), we choose the smaller of the numbers \(\tau_0\) and \(\tau_1\), then it is evident that \(|Y^{(n+1)}(t)|\) and \(|dY^{(n+1)}/dt|\) will not exceed \(H\) for \(0 \leqslant t \leqslant \tau_*/\varepsilon\) and \(|\varepsilon| \leqslant \varepsilon_0\). Accordingly, on this same time interval we have

\[ |Y(t)| \leqslant H,\qquad \left|\frac{dY}{dt}\right| \leqslant H. \]

These inequalities, by virtue of (11) and the boundedness of the function \(\partial y_0/\partial \tau\), which holds under the conditions of the theorem, prove the required assertion.

Received
22 II 1958

References

1. A. A. Dorodnitsyn, Uspekhi Mat. Nauk, 7, no. 6 (1952).
2. G. E. Kuzmak, Prikl. Mat. Mekh., 21, no. 2 (1957).