MATHEMATICS

K. V. ZADIRAKA

INVESTIGATION OF SINGULARLY PERTURBED SYSTEMS IN A NEIGHBORHOOD OF A FAMILY OF CLOSED ORBITS

(Presented by Academician N. N. Bogolyubov, 2 I 1962)

In the present note, N. N. Bogolyubov’s method \((^{1})\) for investigating a system of differential equations

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

in standard form is applied to the study of the properties of solutions of the system

\[ \frac{dx}{dt}=f(x,z,\varepsilon),\qquad \varepsilon\frac{dz}{dt}=F(x,z,\varepsilon), \tag{1} \]

which is singularly perturbed with respect to the system

\[ \frac{d\bar{x}}{dt}=f(\bar{x},\bar{z},0);\qquad F(\bar{x},\bar{z},0)=0, \tag{2} \]

where \(x, f\) and \(z, F\) are vectors of dimensions \(m\) and \(n\), respectively.

It is assumed that the system \(F(x,z,0)=0\) has an isolated solution \(z=\varphi(x)\), while the unperturbed system

\[ \frac{d\bar{x}}{dt}=f(\bar{x},\varphi(\bar{x})),\qquad \bar{z}=\varphi(\bar{x}) \tag{2'} \]

admits a family of solutions

\[ \bar{x}=\bar{x}^{0}(\theta,c),\qquad \bar{z}=\bar{z}^{0}(\theta,c), \tag{3} \]

periodic in \(\theta=\omega(c)t+\varphi_0\) with period \(2\pi\), where \(c\) is a vector with components \(c_1,c_2,\ldots,c_s\), and \(\varphi_0\) is a scalar \((s+1<m)\), with \(\omega\) in the general case being a function of the vector \(c\), continuous and bounded together with its first-order partial derivatives with respect to \(c_1,c_2,\ldots,c_s\) for all \(c\in C\); the parameters \(\varphi_0,c_1,\ldots,c_s\) are essentially distinct.

With respect to the right-hand sides of system (1) and the vector \(\varphi\) we shall assume that in the domain \(x\in U_\rho,\ z\in U_\nu,\ 0<\varepsilon<\varepsilon^*\), where \(U_\rho\) and \(U_\nu\) denote the \(\rho\)- and \(\nu\)-neighborhoods of the family of orbits, \(x^0,\bar{z}^0\), the vector \(f\) and its partial derivatives with respect to \(x\) and \(z\) up to order \(k\), and the vectors \(F\) and \(\varphi\), together with their partial derivatives with respect to \(x\) and \(z\) up to order \((k+1)\) inclusive, are bounded and uniformly continuous.

We shall prove that system (1) also admits a family of periodic solutions of the form

\[ x=x^0(\theta,c,\varepsilon),\qquad z=z^0(\theta,c,\varepsilon) \]

and establish the properties of these solutions.

We transform system (1) by means of the substitution

\[ z=\varphi(x)+y \tag{4} \]

to the form

\[ \frac{dx}{dt}=\Phi(x,y,\varepsilon),\qquad \varepsilon\frac{dy}{dt}=\mathfrak{A}(x)y+Q(x,y,\varepsilon), \tag{5} \]

where the following notation has been introduced:

\[ \Phi(x,y,\varepsilon)=f(x,\varphi(x)+y,\varepsilon),\qquad \mathfrak{A}(x)=\left.\frac{\partial F(x,\varphi(x)+y,\varepsilon)}{\partial z}\right|_{y=0}, \]

\[ Q(x,y,\varepsilon)=F(x,\varphi(x)+y,\varepsilon)-\frac{\partial F(x,\varphi(x))}{\partial z}\,y-\varepsilon \frac{d\varphi}{dx}\,f(x,\varphi(x)+y,\varepsilon). \]

We note that the functions \(\bar{x}=\bar{x}^{0}(\theta,c)\) and \(\bar{y}=0\) are a solution of the system

\[ \frac{d\bar{x}}{dt}=f(\bar{x},\varphi(\bar{x})),\qquad \varepsilon\frac{d\bar{y}}{dt}=\mathfrak{A}(\bar{x})\bar{y}, \tag{6} \]

for which the variational equations corresponding to the family (3) have the form

\[ \frac{d\delta\bar{x}}{dt}=\bar{f}(\bar{x}^{0})\delta\bar{x}, \tag{7} \]

\[ \varepsilon\frac{d\delta\bar{y}}{dt}=\mathfrak{A}(\bar{x}^{0})\delta\bar{y}, \tag{8} \]

where \(\bar{f}=\dfrac{\partial f}{\partial x}+\dfrac{\partial f}{\partial z}\dfrac{d\varphi}{dx}\), and \(\mathfrak{A}\) are periodic functions of \(\theta\) with period \(2\pi\).

We shall assume that all nonzero characteristic exponents \(a_{s+2},\ldots,a_m\) of equation (7) and all \(n\) characteristic exponents of equation (8) have negative real parts. The first \(s+1\) characteristic exponents of equation (7) are equal to zero, since \(\partial \bar{x}^{0}/\partial\theta\) and \(\partial \bar{x}^{0}/\partial c_j\) \((j=1,\ldots,s)\) are solutions of this equation.

According to A. M. Lyapunov’s theorem \((^{2})\) and the remarks made in \((^{3})\), there exist real nonsingular matrices \(P(\theta,c)\) and \(R(\theta,c)\), periodic in \(\theta\) with period \(2\pi\) (if none of the real characteristic numbers of equations (7) and (8) is negative\(*\)), possessing continuous first derivatives with respect to \(\theta\) and \(c\) for all \(\theta\in\Theta,\ c\in C\), as well as such square real matrices \(H(c)\) and \(H_1(c)\), that the transformation

\[ \delta\bar{x}=\frac{\partial\bar{x}^{0}}{\partial\theta}u+\frac{\partial\bar{x}^{0}}{\partial c}v+P(\theta,c)w, \tag{9} \]

\[ \delta\bar{y}=R(\theta,c)\xi \tag{10} \]

reduces systems (7) and (8) to equivalent systems with constant coefficients

\[ \frac{du}{dt}=0,\qquad \frac{dv}{dt}=0,\qquad \frac{dw}{dt}=H(c)w, \tag{11} \]

\[ \varepsilon\frac{d\xi}{dt}=H_1(c)\xi, \tag{12} \]

where the eigenvalues of the matrices \(H(c)\) and \(H_1(c)\) are the nonzero characteristic exponents of equation (7) and the characteristic exponents of equation (8), respectively.

Applying now to system (5) the transformation

\[ x=\bar{x}^{0}(\theta,c)+P(\theta,c)h,\qquad y=R(\theta,c)\xi,\qquad h=(h_{s+2},\ldots,h_m), \tag{13} \]

we obtain the system

\[ \begin{aligned} \frac{d\theta}{dt}&=\omega(c)+A_1(\theta,c,h,\xi,\varepsilon),\\ \frac{dc}{dt}&=A_2(\theta,c,h,\xi,\varepsilon),\\ \frac{dh}{dt}&=H(c)h+A_3(\theta,c,h,\xi,\varepsilon),\\ \varepsilon\frac{d\xi}{dt}&=H_1(c)\xi+A_4(\theta,c,h,\xi,\varepsilon). \end{aligned} \tag{14} \]

\[ \text{* Matrices }P(\theta,c)\text{ and }R(\theta,c)\text{ will have period }4\pi\text{ in }\theta\text{ if among the real characteristic numbers of equations (7) and (8), respectively, there are negative ones.} \]

The functions \(A_i\) \((i=1,\ldots,4)\) are defined in the domain

\[ \theta\in\Theta,\quad c\in C,\quad h\in U_\rho,\quad \xi\in U_\nu,\quad 0<\varepsilon<\varepsilon^{*}, \tag{15} \]

where \(U_\rho\) and \(U_\nu\) denote the \(\rho\)- and \(\nu\)-neighborhoods of the points \(h=0\) and \(\xi=0\), have bounded and uniformly continuous derivatives with respect to \(\theta\) and \(c\), are periodic in \(\theta\), and satisfy the inequalities

\[ \left|A_i(\theta,c,0,0,\varepsilon)\right|\leq M(\varepsilon), \]
\[ \left|A_i(\theta',c',h',\xi',\varepsilon)-A_i(\theta'',c'',h'',\xi'',\varepsilon)\right|\leq \tag{16} \]
\[ \leq \lambda(\varepsilon,\sigma,\mu)\left(|\theta'-\theta''|+|c'-c''|+|h'-h''|+|\xi'-\xi''|\right), \]

where \(\sigma<\rho,\ \mu<\nu\), and moreover \(M(\varepsilon)\to0,\ \lambda(\varepsilon,\sigma,\mu)\to0\) together with \(\varepsilon\).

The following assertions hold:

1. One can specify a positive number \(\varepsilon_1\) such that, for every positive \(\varepsilon<\varepsilon_1\), the system of equations (14) has a unique integral manifold, periodic in \(\theta\) with period \(2\pi\), representable by relations of the form

\[ h=g(\theta,c,\varepsilon),\qquad \xi=g_1(\theta,c,\varepsilon), \]

where the functions \(g\) and \(g_1\) are defined in the domain \(\theta\in\Theta,\ c\in C\) and satisfy the inequalities

\[ |g(\theta,c,\varepsilon)|\leq D(\varepsilon),\qquad |g(\theta',c',\varepsilon)-g(\theta'',c'',\varepsilon)|\leq \Delta(\varepsilon)\bigl(|\theta'-\theta''|+|c'-c''|\bigr), \tag{17} \]
\[ |g_1(\theta,c,\varepsilon)|\leq D(\varepsilon),\qquad |g_1(\theta',c',\varepsilon)-g_1(\theta'',c'',\varepsilon)|\leq \Delta(\varepsilon)\bigl(|\theta'-\theta''|+|c'-c''|\bigr), \]

with \(D(\varepsilon)\to0,\ \Delta(\varepsilon)\to0\) together with \(\varepsilon\). Moreover, \(g\) and \(g_1\) will have bounded and uniformly continuous derivatives up to order \(k\), inclusive.

1. One can specify positive constants \(\varepsilon_0,\gamma_0,C_0,\sigma_0,\gamma_1,C_1,\sigma_1\) \((\sigma_0<\rho,\ \sigma_1<\nu,\ \varepsilon_0<\varepsilon_1)\) such that, for all \(\varepsilon<\varepsilon_0\), any real \(t_0\), and any \(\theta\in\Theta,\ c\in C\), there exist an \([m-(s+1)]\)-dimensional domain \(U_{\sigma_0}\) of points \(\{h\}\) and an \(n\)-dimensional domain \(U_{\sigma_1}\) of points \(\xi\) with the properties

\[ |h_t^{*}-g(\theta_t,c_t,\varepsilon)|\leq C_0e^{-\gamma_0(t-t_0)}|h_0^{*}-g(\theta_0,c_0,\varepsilon)|, \]

\[ |\xi_t^{*}-g_1(\theta_t,c_t,\varepsilon)|\leq C_1e^{-\gamma_1(t-t_0)}|\xi_0^{*}-g_1(\theta_0,c_0,\varepsilon)|, \]

where \(h_t^{*}\) and \(\xi_t^{*}\) are arbitrary solutions of system (14); \(h_0^{*}=h_t^{*}(t_0)\), \(\xi_0^{*}=\xi_t^{*}(t_0)\), \(\theta_0=\theta_t(t_0)\), \(c_0=c_t(t_0)\).

Taking into account that system (14) is equivalent to system (1), we have the following theorem:

Theorem. If the assumptions made above are satisfied, then one can specify a number \(\varepsilon_0>0\) such that, for any positive \(\varepsilon<\varepsilon_0\), the following assertions hold:

1. System (1) has a unique integral manifold \(\mathfrak{M}\).

2. This manifold admits the parametric representation

\[ x=\bar{x}^{0}(\theta,c)+P(\theta,c)g(\theta,c,\varepsilon), \]

\[ z=\varphi(x)+R(\theta,c)g_1(\theta,c,\varepsilon), \]

where the right-hand sides are defined in the domain \(\theta\in\Theta,\ c\in C,\ 0<\varepsilon<\varepsilon_0\), are periodic in the angular variable \(\theta\) with period \(2\pi\), and have bounded and uniformly continuous derivatives with respect to \(\theta\) and \(c\) up to order \(k\), inclusive.

1. One can find a function \(\delta(\varepsilon)\to0\) as \(\varepsilon\to0\) such that

\[ |x(\theta,c,\varepsilon)-\bar{x}^{0}(\theta,c)|<\delta(\varepsilon),\qquad |z(\theta,c,\varepsilon)-\bar{z}^{0}(\theta,c)|<\delta(\varepsilon). \]

1. On the manifold $\mathfrak{M}$, system (1) is equivalent to the system

\[ \frac{d\theta}{dt}=\omega(c)+A_1^*(\theta,c,\varepsilon),\qquad \frac{dc}{dt}=A_2^*(\theta,c,\varepsilon), \]

where

\[ A_1^*=A_1(\theta,c,g(\theta,c,\varepsilon),g_1(\theta,c,\varepsilon),\varepsilon),\quad A_2^*=A_2(\theta,c,g(\theta,c,\varepsilon),g_1(\theta,c,\varepsilon),\varepsilon) \]

are definite functions, periodic in $\theta$ with period $2\pi$ and possessing bounded and uniformly continuous derivatives with respect to $\theta$ and $c$ up to and including order $k-20$.

1. The manifold $\mathfrak{M}$ has the property of attracting solutions of system (1) that are close to it.

I take this opportunity to express my heartfelt gratitude to N. N. Bogolyubov for his attention to this work.

CITED LITERATURE

1. N. N. Bogolyubov, Yu. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, 1955.
2. A. M. Lyapunov, The General Problem of the Stability of Motion, 1935.
3. J. K. Hale, A. P. Stokes, Arch. f. Rat. Mech. and Analysis, 6, No. 2, 133 (1960).