MATHEMATICS

Academician L. S. Pontryagin and L. V. Rodygin

APPROXIMATE SOLUTION OF A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS WITH A SMALL PARAMETER AT THE DERIVATIVES*

1. We shall study the behavior, on a finite time interval, of solutions of the system of differential equations

\[ \varepsilon \frac{dx}{dt}=f(x,y),\qquad \frac{dy}{dt}=g(x,y), \tag{1} \]

where \(x=(x_1,\ldots,x_k)\), \(y=(y_1,\ldots,y_l)\) are vectors, and \(\varepsilon>0\) is a small parameter. The functions \(f(x,y)=(f_1(x,y),\ldots,f_k(x,y))\), \(g(x,y)=(g_1,\ldots,g_l)\) are assumed to be twice continuously differentiable in a certain (open) domain \(\Gamma\) of the space of variables \((x,y)\), whose form will be specified below. The projection of the domain \(\Gamma\) onto the space of variables \(y\) is denoted by \(G\).

It is assumed that for any \(y\in G\) the system of fast motions

\[ \frac{d\widetilde{x}}{d\tau}=f(\widetilde{x},y)\qquad (y=\mathrm{const}\text{ — parameter}) \tag{2} \]

has, in the domain under consideration, exactly one rough stable limit cycle \(x^*(\tau,y)\). This means that \(x^*(\tau,y)\) is a periodic solution of system (2), whose period we shall denote by \(T(y)\), and that the multipliers of the system of equations in variations

\[ \frac{d\xi_i}{d\tau}= \sum_{j=1}^{k} \frac{\partial f_i[x^*(\tau,y),y]}{\partial x_j}\,\xi_j \qquad (i=1,\ldots,k) \tag{3} \]

are, in modulus, less than one except for one; these multipliers we denote by \(\lambda_1,\ldots,\lambda_{k-1}\). It follows from this that \(x^*(\tau,y)\) has a domain of attraction \(F(y)\); any solution of system (2) with initial value from \(F(y)\) tends as \(t\to\infty\) to the limit cycle \(x^*\). As is easily seen,

\[ \bigcup_{y\in G} F(y)\times y \]

is a domain in the space \((x,y)\), which we identify with our original domain \(\Gamma\). We shall assume that there exist \(T_1,T_2\) for which \(0<T_1\le T(y)\le T_2\). Generally speaking, as \(y\) approaches the boundary of the domain \(G\), \(T(y)\) could tend to \(0\) or to \(\infty\), but for our further purposes it is always possible to pass from \(G\) to some compact subdomain \(D\subset G\). Under the assumptions made, \(T(y)\) turns out to be a smooth function of \(y\).

Let us introduce into consideration the “averaged system”

\[ \frac{d\bar y}{dt}=\bar g(\bar y)= \frac{1}{T(\bar y)} \int_{0}^{T(\bar y)} g(x^*(\tau,\bar y),\bar y)\,d\tau = \int_{0}^{1} g(X(\varphi,\bar y),\bar y)\,d\varphi, \tag{4} \]

\[ \text{* The principal results of the present work were reported at the All-Union Mathematical Congress } (^{1}). \]

where it is set* that \(X(\varphi,y)=x^*(T(y)\varphi,y)\). We denote by \(\{x(t,\varepsilon),y(t,\varepsilon)\}\) the solution of system (1) with initial value \((x_0,y_0)\in\Gamma,\ x_0\in F(y_0)\), by \(\widetilde{x}(\tau)\) the solution of system (2) with initial value \(\widetilde{x}(0)=x_0\) and with parameter \(y=y_0\), and by \(\overline{y}(t)\) the solution of system (4) with initial value \(\overline{y}(0)=y_0\). It is assumed that \(\overline{y}(t)\in G\) for \(0\leq t\leq L<\infty\). By \(D\) is denoted a compact subdomain of the domain \(G\), containing \(\overline{y}(t)\) \((0\leq t\leq L)\) strictly in its interior. Everywhere below it is assumed that \(y\in D\).

2. We pass in system (1) to the “fast time” \(\tau=t/\varepsilon\):

\[ \frac{dx}{d\tau}=f(x,y),\qquad \frac{dy}{d\tau}=\varepsilon g(x,y) \tag{5} \]

and compare (5) with (2). From the theorem on continuous dependence of solutions on a parameter it follows directly that, on any time interval \(0\leq t\leq \varepsilon t_1\) of length of order \(\varepsilon\), \(|x(t,\varepsilon)-\widetilde{x}(t/\varepsilon)|,\ |y(t,\varepsilon)-y_0|\) can be made arbitrarily small, provided only that \(\varepsilon\) is sufficiently small. Thus, over a time interval of order \(\varepsilon\), the solution of system (1) enters a small neighborhood of the closed curve \(C=C_{y_0}\), where \(C_y\) denotes \(\{x=X(\varphi,y),\ 0\leq\varphi\leq 1\}\times y\).

3. Let us trace the further behavior of \(\{x(t,\varepsilon),y(t,\varepsilon)\}\). It is natural to expect that \(y(t,\varepsilon)\) will be close to the averaged solution \(\overline{y}(t)\), while \(x(t,\varepsilon)\) will remain near the cycles \(C_{\overline{y}(t)}\), performing rapid oscillations along them with period close to \(\varepsilon T(\overline{y}(t))\). The precise formulation is as follows:

Theorem. There exists a function \(\varphi(t,\varepsilon)\) (“phase”), depending smoothly on \(t\), such that if \(\delta>0\), then for \(\delta\leq t\leq L\)

\[ \left|\varepsilon\frac{d\varphi}{dt}-\frac{1}{T(\overline{y}(t))}\right|=O(\varepsilon), \tag{6a} \]

\[ \left|x(t,\varepsilon)-X(\varphi(t,\varepsilon),\overline{y}(t))\right|=O(\varepsilon), \tag{6b} \]

\[ \left|y(t,\varepsilon)-\overline{y}(t)\right|=O(\varepsilon) \tag{6c} \]

uniformly in \(t\).

4. To study system (1) near \(C_y\), we use a nonsingular transformation of the form

\[ \xi=\frac{\partial x^*}{\partial \tau}u_0+A\left(\frac{\tau}{T(y)},y\right)u \]

(\(u_0\) is a scalar, \(u\) is a vector with \(k-1\) components, \(A(\varphi,y)\) is a matrix with \(k\) rows and \(k-1\) columns, having period 1 in \(\varphi\)). With the aid of such a transformation, system (3) can be reduced to the form

\[ \frac{du_0}{d\tau}=0,\qquad \frac{du}{d\tau}=H\left(\frac{\tau}{T(y)},y\right)u, \tag{7} \]

where \(H(\varphi,y)=H(\varphi+1,y)\) is a square matrix of order \(k-1\), and the multipliers of the system \(\dot{u}=Hu\) are our \(\lambda_1,\ldots,\lambda_{k-1}\). Hence there follows the existence of a Lyapunov function \(W\left(\frac{\tau}{T(y)},u,y\right)\), a quadratic function of \(u\) with period 1 in the first argument, whose total derivative with respect to \(\tau\), by virtue of system (7), is

\[ \left[\frac{dW}{d\tau}\right]_{(7)}<-\alpha W\qquad (\alpha>0). \tag{8} \]

* As the initial value \(X(0,y)\), any point of the periodic solution of system (2) may be taken, as long as the dependence on \(y\) is smooth.

Moreover, one can ensure that the coefficients \(W\) are smooth functions of their arguments, so that

\[ \left|\frac{\partial W}{\partial \varphi}\right|,\quad \left|\frac{\partial W}{\partial y_i}\right|<\beta_1 W,\quad \left|\frac{\partial W}{\partial u_i}\right|<\beta_2\sqrt{W}, \]

\[ \beta_3\sqrt{W}<|u|<\beta_4\sqrt{W}\quad (\beta_3>0), \tag{9} \]

and the \(\beta_i\), like \(\alpha\) in (8), can be chosen the same for all \(y\in D\).

By the change of variables \(x=X(\varphi,y)+A(\varphi,y)u\), system (1) is reduced to the form

\[ \varepsilon\frac{d\varphi}{dt}=\frac{1}{T(y)}+O(\varepsilon+|u|), \]

\[ \varepsilon\frac{du}{dt}=H(\varphi,y)u+O(\varepsilon+|u^2|), \tag{10} \]

\[ \varepsilon\frac{dy}{dt}=g[X(\varphi,y),y]+O(\varepsilon+|u|). \]

Computing the total derivative of the function \(W(\varphi,u,y)\) with respect to \(t\), by virtue of system (10), one can show that there exist \(\varepsilon_1,C_1,C_2>0\) such that, for \(\varepsilon<\varepsilon_1\), \(C_1\varepsilon^2\leq W\leq C_2\),

\[ \varepsilon\left[\frac{dW}{dt}\right]_{(10)}<-\alpha_1 W\quad (\alpha_1>0). \]

It follows that the surfaces \(W=C\) \((C_1\varepsilon^2\leq C\leq C_2)\) are contact-free surfaces for system (10), and that if a solution of this system reaches, at time \(t=t_{1\varepsilon}\), the surface \(W=C_2\), then, as long as it lies in the region \(C_1\varepsilon^2\leq W\leq C_2\), the inequality

\[ W(\varphi(t,\varepsilon),u(t,\varepsilon),y(t,\varepsilon))\leq \]

\[ \leq W(\varphi(t_{1\varepsilon},\varepsilon),u(t_{1\varepsilon},\varepsilon),y(t_{1\varepsilon},\varepsilon)) e^{-\frac{1}{\varepsilon}\alpha_1(t-t_{1\varepsilon})} = C_2e^{-\frac{1}{\varepsilon}\alpha(t-t_{1\varepsilon})}, \]

holds; hence after a time \(t_{2\varepsilon}=O(\varepsilon\ln\varepsilon)\), \(W\) becomes \(C_1\varepsilon^2\). The solution thus enters inside the surface \(W=C_1\varepsilon^2\) and can no longer leave the interior of this surface as long as \(y(t,\varepsilon)\in D\). If the solution of system (2) enters, in time \(t_1\), the interior of the surface \(W=\frac12 C_2\), then, for sufficiently small \(\varepsilon\), the solution of system (5) will certainly enter the interior of the surface \(W=C_2\) in time \(t_1\), so that \(t_{1\varepsilon}<\varepsilon t_1\). Using (9), we now obtain that, for \(\varepsilon t_1\leq t\leq t_{2\varepsilon}\), one has

\[ |u|<C_3\exp\{ -\alpha_1(t-\varepsilon t_1)/2\varepsilon\}<C_4e^{-\gamma t/\varepsilon}\quad (\gamma>0), \]

while for \(t>t_{2\varepsilon}\) one has \(|u|<C_5\varepsilon\), as long as \(y(t,\varepsilon)\) does not leave the domain \(D\). The point \(y(0,\varepsilon)\) lies strictly inside \(D\), and the rate of change of \(y(t,\varepsilon)\) is finite, so that \(y(t,\varepsilon)\) can leave \(D\) only after a time \(t_\varepsilon=O(1)\) (it is not excluded, of course, that \(t_\varepsilon=\infty\)). Put \(t_\varepsilon^*=\min(t_\varepsilon,L)\). For \(\varepsilon t_1\leq t<t_\varepsilon^*\) we have, by what has been said,

\[ |u(t,\varepsilon)|\leq C_4e^{-\gamma t/\varepsilon}+C_5\varepsilon. \tag{11} \]

We shall prove later that, for sufficiently small \(\varepsilon\), \(t_\varepsilon^*=L\).

1. We need to estimate \(|y(t,\varepsilon)-\bar y(t)|\) for \(\varepsilon t_1\leq t\leq t_\varepsilon^*\). It is more convenient to estimate on the interval \([\varepsilon t_1,t_\varepsilon^*]\) the quantity

\[ \eta(t,\varepsilon)=y(t,\varepsilon)-\bar y(t)-I(t,\varepsilon), \]

where

\[ I(t,\varepsilon)=\varepsilon T(y(t,\varepsilon)) \int_0^{\varphi(t,\varepsilon)} \{g[X(\theta,\bar y(t)),\bar y(t)]-\bar g[\bar y(t)]\}\,d\theta. \]

We want to prove that for \(\varepsilon t_1 \leqslant t < t_\varepsilon^*\) inequality (6в) holds; for this it is enough to show that, for the indicated values of \(t\), \(|\eta(t,\varepsilon)|=O(\varepsilon)\), \(I(t,\varepsilon)=O(\varepsilon)\) uniformly in \(t\). The second of these relations is obvious, since \(g[X(\theta,\bar y(t)),\bar y(t)]\) is a periodic function of \(\theta\) with period 1 and mean value \(\bar g[\bar y(t)]\), so that the integral entering the expression for \(I(t,\varepsilon)\) is \(O(1)\). Let us now prove that \(|\eta(t,\varepsilon)|=O(\varepsilon)\).

We have

\[ \begin{aligned} \dot\eta(t,\varepsilon) ={}& g[X(\varphi(t,\varepsilon),y(t,\varepsilon)),y(t,\varepsilon)] +O(|u|+\varepsilon)-\bar g[\bar y(t)]\\ &-\varepsilon T(y(t,\varepsilon))\dot\varphi(t,\varepsilon) \{g[X(\varphi(t,\varepsilon),\bar y(t)),\bar y(t)]-\bar g[\bar y(t)]\}\\ &-\varepsilon T\int_0^{\varphi(t,\varepsilon)}\{\dot g-\dot{\bar g}\}\,d\theta -\varepsilon \dot T(y(t,\varepsilon))\int_0^{\varphi(t,\varepsilon)}\{g-\bar g\}\,d\theta . \end{aligned} \]

The last two terms are, obviously, \(O(\varepsilon)\). Further, the product

\[ \varepsilon T(y(t,\varepsilon))\dot\varphi(t,\varepsilon)=1+O(\varepsilon+|u|) \]

and, consequently,

\[ \dot\eta(t,\varepsilon) =g[X(\varphi(t,\varepsilon),y(t,\varepsilon)),y(t,\varepsilon)] -g[X(\varphi(t,\varepsilon),\bar y(t)),\bar y(t)] +O(\varepsilon+|u|). \]

Since \(g[X(\varphi,y),y]\) depends smoothly on \(y\) and, as was proved, \(y(t,\varepsilon)-\bar y(t)=\eta(t,\varepsilon)+O(\varepsilon)\), it follows that for \(\varepsilon t_1 \leqslant t<t_\varepsilon^*\) we have
\[ |\dot\eta(t,\varepsilon)|<B|\eta(t,\varepsilon)|+O(\varepsilon+|u|), \]
or, using (11),

\[ |\dot\eta(t,\varepsilon)|<B|\eta(t,\varepsilon)|+C_6e^{-\gamma t/\varepsilon}+C_7\varepsilon . \tag{12} \]

At the same time, evidently, \(\eta(\varepsilon t_1,\varepsilon)=O(\varepsilon)\).

It is easy to show that from (12) there follows the estimate \(|\eta(t,\varepsilon)|\leqslant \zeta(t,\varepsilon)\) for \(t\in[\varepsilon t_1,t_\varepsilon^*)\), where
\[ \dot\zeta=B\zeta+C_6e^{-\gamma t/\varepsilon}+C_7\varepsilon,\qquad \zeta(\varepsilon t_1,\varepsilon)=\eta(\varepsilon t_1,\varepsilon). \]
Computing \(\zeta\) from this, we obtain that \(\zeta(t,\varepsilon)=O(\varepsilon)\) for \(t\in[\varepsilon t_1,t_\varepsilon^*)\).

1. Inequalities (6а), (6б), obviously, follow for \(t\in[\varepsilon t_1,t_\varepsilon^*)\) from the proved inequalities (6в), (11) and from the system (10). It remains to show that \(t_\varepsilon^*=L\). But this is obvious: if for \(t\in[0,L]\) \(\bar y(t)\) is distant from the boundary of the domain \(D\) by at least \(d\) (\(d>0\)), and if it were the case that \(t_\varepsilon^*=t_\varepsilon<L\), then for sufficiently small \(\varepsilon\) we would have \(|y(t,\varepsilon)-\bar y(t)|<d/2\) for \(t\in[\varepsilon t_1,t_\varepsilon)\), and we would obtain that \(y(t_\varepsilon,\varepsilon)\) lies in \(D\) and is distant from the boundary \(D\) by at least \(d/2\), and hence \(y(t,\varepsilon)\) still does not leave \(D\) over the time \(t_\varepsilon\). Thus the theorem is completely proved.

Received
11 XII 1959

REFERENCES

1. L. S. Pontryagin, Proceedings of the Third All-Union Mathematical Congress, 2, Moscow, 1956, p. 93; 3, Moscow, 1958, p. 570.