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MATHEMATICS
K. V. ZADIRAKA
ON AN INTEGRAL MANIFOLD OF A SYSTEM OF DIFFERENTIAL EQUATIONS CONTAINING A SMALL PARAMETER
(Presented by Academician N. N. Bogolyubov on 11 III 1957)
Consider the system of differential equations
\[ \frac{dx}{dt}=f(t,x,z,t/\mu), \qquad \mu \frac{dz}{dt}=F(t,x,z) \tag{1} \]
with initial conditions \(x^0, y^0\) for \(t=t_0\), where \(x\) and \(f\) are \(n\)-dimensional, while \(z\) and \(F\) are \(m\)-dimensional vectors, and the so-called degenerate \((\mu=0)\) system and the system averaged with respect to the argument \(t/\mu\),
\[ \frac{d\overline{x}}{dt}=f_0(t,\overline{x},\overline{z}), \qquad \overline{z}=\varphi(t,\overline{x}), \qquad \overline{x}(t_0)=x^0, \tag{2} \]
where \(z=\varphi(t,x)\) is a root of the system \(F(t,x,z)=0\);
\[ f_0(t,x,z)=\lim_{T\to\infty}\frac{1}{T}\int_0^T f(t,x,z,\nu)\,d\nu . \]
In the absence of the argument \(t/\mu\), system (1) reduces to the system considered by A. N. Tikhonov \((^2)\) and I. S. Gradshtein \((^3)\) on a finite interval of \(t\). We shall consider the question of the existence of an integral manifold of system (1) close to the integral manifold \(z=\varphi(t,x)\).
We shall assume that in the domain
\[ -\infty<t<\infty,\qquad x\in G,\qquad |z-\varphi(t,x)|\le \rho,\qquad 0<\mu<\mu^* \]
the following conditions are satisfied:
a) the vector \(f\) is continuous, bounded, and satisfies a Lipschitz condition in \(x\) and \(z\);
b) the vectors \(F\) and \(\varphi\) and their derivatives with respect to all arguments up to and including the second order are continuous and bounded;
c) the second mixed derivatives of the vector \(F\) with respect to \(x\) and \(z\) satisfy a Lipschitz condition in \(z\);
d) uniformly with respect to \(x\in G\), there exists the limit
\[ f_0(t,x,z)=\lim_{T\to\infty}\frac{1}{T}\int_0^T f(t,x,z,\nu)\,d\nu; \]
e) the roots of the characteristic equation \(\det \|pE-\mathcal U\|=0\), where the matrix \(\mathcal U(t,x)=F_z\big|_{z=0}\), \(E\) is the identity matrix, satisfy the condition \(\operatorname{Re}\{p_i(t,x)\}\le -\alpha<0\).
Theorem 1. Under these conditions one can indicate such a positive number \(\mu_0\) that, for every positive number \(\mu<\mu_0\), the system
of equations (1) we have a unique integral manifold*, representable by a relation of the form \(z(t,x,\mu)=\varphi(t,x)+\psi(t,x,\mu)\), in which \(\psi(t,x,\mu)\), as a function of \((t,x)\), is defined in the domain \(-\infty<t<\infty,\ x\in G\), and satisfies the inequalities
\[ |\psi(t,x,\mu)|\leq D(\mu)<\rho,\qquad |\psi(t,x',\mu)-\psi(t,x'',\mu)|\leq \Delta(\mu)|x'-x''|, \]
where \(D(\mu)\to 0,\ \Delta(\mu)\to 0\) as \(\mu\to 0\).
Proof. Passing in system (1) to the new argument by means of the substitution \(t=\mu\tau\), making the change \(z=\varphi(\mu\tau,x)+\xi\), and separating out the linear part in the second of them, we bring system (1) to the form
\[ \frac{dx}{d\tau}=\mu\Phi(\tau,x,\xi,\mu),\qquad \frac{d\xi}{d\tau}=\mathcal U(\mu\tau,x)\xi+Q(\tau,x,\xi,\mu), \tag{3} \]
where
\[ \mathcal U(\mu\tau,x)=F_\xi\big|_{\xi=0},\qquad Q(\tau,x,\xi,\mu)=Z(\mu\tau,x,\xi)-\mu\left(\frac{\partial\varphi}{\partial x}f+\frac{\partial\varphi}{\partial t}\right), \]
with
\[ |Z(\mu\tau,x,\xi)|\leq M|\xi|^2,\qquad Z(\mu\tau,x,0)=0. \]
Under the conditions imposed on the vectors \(f,F\) and \(\varphi\), the inequalities hold
\[ \mu|\Phi(\tau,x,\xi,\mu)|\leq \mu L,\qquad |Q(\tau,x,\xi,\mu)|\leq M|\xi|^2+\mu N, \tag{4} \]
\[ \mu|\Phi(\tau,x',\xi',\mu)-\Phi(\tau,x'',\xi'',\mu)| \leq \mu A\{|x'-x''|+|\xi'-\xi''|\}, \]
\[ |Q(\tau,x',\xi',\mu)-Q(\tau,x'',\xi'',\mu)| \leq (B\sigma+\mu C)\{|x'-x''|+|\xi'-\xi''|\}, \tag{5} \]
where \(L,M,N,A,B,C\) are constant numbers; \(B\sigma=\max\{B_1|\xi|^2,\ B_2|\xi|\}\), with \(\sigma\to 0\) as \(|\xi|\to 0\).
Consider the matrix equation
\[ \frac{dU(\tau,\tau_1,x)}{d\tau}=\mathcal U(\mu\tau,x)U(\tau,\tau_1,x),\qquad U\big|_{\tau=\tau_1}=E. \tag{6} \]
Since the matrix \(\mathcal U(\mu\tau,x)\) is bounded and satisfies the Lipschitz condition
\[ |\mathcal U(\mu\tau',x)-\mathcal U(\mu\tau'',x)| \leq \mu H|\tau'-\tau''|\qquad (H=\mathrm{const}), \tag{7} \]
then, according to a theorem of N. Ya. Lyashenko \((^4)\), for the matrix \(U\) the estimate holds
\[ |U(\tau,\tau_1,x)|\leq K e^{-\frac{\alpha}{4}(\tau-\tau_1)} \qquad (\tau>\tau_1). \tag{8} \]
Instead of system (3), we shall consider the system
\[ \frac{dx}{d\tau}=\mu\Phi_1(\tau,x,\xi,\mu), \]
\[ \frac{d\xi}{d\tau}=\mathcal U_1(\mu\tau,x)\xi+Q_1(\tau,x,\xi,\mu), \tag{3'} \]
where \(\Phi_1,\ Q_1,\ \mathcal U_1\) coincide with \(\Phi,\ Q,\ \mathcal U\) in the domain \(-\infty<t<\infty,\ x\in G,\ |\xi|\leq \rho,\ 0<\mu<\mu^*\), and outside this domain satisfy conditions (4), (5), (7), (8) with constants \(L_1,\ M_1,\ N_1,\ A_1,\ B_1,\ C_1,\ H_1,\ K_1\).
* In speaking of an integral manifold, we mean that from the relation \(z_t=z(t,x_t,\mu)\), valid at some instant of time \(t_0\), there follows the validity of this relation for some \(t\), as long as \(x_t\) remains in the domain \(G\).
Considering the recurrence relation
\[ \psi_{n+1}(\tau,x)=\int_{-\infty}^{\tau} U_1(\tau,\tau-z,x)Q_1\{\tau+z;\,x_n^\tau(x);\,\psi_n(\tau+z;\,x_n^\tau(x));\,\mu\}\,dz, \tag{9} \]
where \(x_n^\tau\) and \(\dot{x}_n^\tau\) are solutions of the equation
\[ \frac{dx}{d\tau}=\mu\Phi_1(\tau,x,\psi_n(\tau,x),\mu), \tag{10} \]
with initial values, respectively, \(x^0\) and \(x^*\) at \(\tau=\tau_0\), and using the method of N. N. Bogolyubov \((^1)\), we establish the inequalities
\[ |\psi_n(\tau,x)|\leq D(\mu),\qquad |\psi_{n+1}(\tau,x)-\dot{\psi}_n(\tau,x)|\leq \]
\[ \leq \Delta(\mu)|x^0-x^*|+\frac12|\psi_n-\dot{\psi}_{n-1}| \tag{11} \]
and, in particular,
\[ |\psi_n(\tau,x)-\dot{\psi}(\tau,x)|\leq \Delta(\mu)|x^0-x^*|, \tag{12} \]
\[ \|\psi_{n+1}(\tau,x)-\psi_n(\tau,x)\|\leq \frac12\|\psi_n-\psi_{n-1}\|, \tag{13} \]
where for \(\mu<\mu_0\), \(D(\mu)\to 0\) and \(\Delta(\mu)\to 0\) as \(\mu\to0\).
These inequalities guarantee that the functions \(\psi_n\) belong to the domain of definition \(\xi\) and that there exists a unique solution \(\psi(\tau,x,\mu)\) of the equation
\[ \psi(\tau,x,\mu)=\int_{-\infty}^{\tau} U(\tau,\tau-z,x)Q\{\tau+z;\,x^\tau(x);\,\psi(\tau+z;\,x^\tau(x));\,\mu\}\,dz, \]
which, obviously, satisfies the requirements of Theorem 1 and, as is easy to show, determines an integral manifold for the system of differential equations (3) under consideration.
Theorem 2. If, in addition to the hypotheses of Theorem 1, the integral manifold of system (2) has bounded and uniformly continuous derivatives with respect to \(x\) up to order \((m+1)\) inclusive in the domain \(-\infty<t<\infty\), \(x\in G\), \(|\xi|\leq \rho\), \(0<\mu<\mu^*\), the vector \(f\) has bounded and uniformly continuous derivatives with respect to \(x\) and \(z\) up to order \(m\), and the vector \(F\) up to order \((m+2)\) inclusive, then the integral manifold \(z(t,x,\mu)\equiv\varphi(t,x)+\psi(t,x,\mu)\) of system (1) will have bounded and uniformly continuous derivatives with respect to \(x\) up to order \(m\) inclusive.
Proof. Differentiating the recurrence relation (17) with respect to \(x\), we have
\[ \frac{\partial\psi_{n+1}(\tau,x)}{\partial x} = \int_{-\infty}^{\tau} \left[ U'(\tau,\tau-z,x)\frac{dQ}{dx_n^\tau}\frac{dx_n^\tau}{dx} + \frac{\partial U(\tau,\tau-z,x)}{\partial x}Q \right]dz. \tag{14} \]
To estimate \(dx_n^\tau/dx\), consider the matrix equation
\[ \frac{d}{d\tau}\frac{dx_n^\tau}{dx} = \mu\frac{d\Phi}{dx_n^\tau}\frac{dx_n^\tau}{dx}, \qquad \left.\frac{dx_n^\tau}{dx}\right|_{\tau=\tau_0}=E, \tag{15} \]
obtained by differentiating equation (10) with respect to \(x\).
Since the matrix of coefficients of equation (15) satisfies the hypotheses of the theorem cited earlier \((^4)\), we have the estimate
\[ \left|\frac{dx_n^\tau}{dx}\right|\leq K_2 e^{-\frac{\alpha}{4}(\tau-\tau_0)} \qquad (\tau>\tau_0). \tag{16} \]
To estimate \(U_x\), we differentiate the matrix equation (6) with respect to \(x\)
\[ \frac{dU_x(\tau,\tau_1,x)}{d\tau} = \mathcal U(\mu\tau,x)U_x + \frac{\partial\mathcal U}{\partial x_n^\tau}\frac{dx_n^\tau}{dx}\,U, \qquad \left. U_x \right|_{\tau=\tau_1}=0. \tag{17} \]
Taking into account that for the matrix equation
\[ \frac{dW}{d\tau}=\mathcal U(\mu\tau,x)W,\qquad \left. W \right|_{\tau=\tau_1}=E, \]
we have the estimate
\[ \left|W(\tau,\tau_1,x)\right| \le K e^{-\frac{\alpha}{4}(\tau-\tau_1)} \qquad (W\equiv U), \]
we obtain for \(U_x\) from (17)
\[ U_x(\tau,\tau_1,x) = \int_0^\tau W(\tau,\theta,x) \frac{\partial\mathcal U}{\partial x_n^\tau} \frac{dx_n^\tau}{dx} U(\theta,\tau_1,x)\,d\theta, \]
\[ \left|U_x(\tau,\tau_1,x)\right| \le \frac{8K^2K_2c_0}{\alpha} e^{-\frac{\alpha}{4}(\tau-\tau_1)} \left( \left|\frac{\partial\mathcal U}{\partial x_n^\tau}\right| \le c_0 \right). \]
Using the conditions of Theorem 2 and the estimates (16), (18), from relation (14) we have
\[ \left| \frac{\partial\psi_{n+1}(\tau,x)}{\partial x} \right| \le \int_{-\infty}^{\tau} \left[ K e^{-\frac{\alpha}{4}|z|} \left| \frac{dQ}{dx_n^\tau} \right| K_2e^{-\frac{\alpha}{4}|z|} + \frac{8K^2K_2c_0}{\alpha} e^{-\frac{\alpha}{4}|z|} |Q| \right]dz \le \]
\[ \le \frac{4K}{\alpha} \left( a_1+\frac{16K^2K_2c_0a_0}{\alpha} \right) =c_1 \qquad \left( |Q|\le a_0,\quad \left|\frac{dQ}{dx_n^\tau}\right|\le a_1 \right). \]
Similarly we obtain the estimates
\[ \left| \frac{d^l\psi_n}{dx^l} \right| \le c_l \qquad (l=1,2,\ldots,k). \]
Applying now to the recurrence relation (14) the method of proof of Theorem 1, we obtain for \(\partial^l\psi_n/\partial x^l\) inequalities analogous to inequalities (11) and (13), ensuring uniform convergence of \(\partial^l\psi_n/\partial x^l\) to \(\partial^l\psi/\partial x^l\) \((l=1,2,\ldots,k)\).
In conclusion, the author takes this opportunity to express deep gratitude to N. N. Bogolyubov for valuable advice in carrying out the present work.
Institute of Mathematics
Academy of Sciences of the Ukrainian SSR
Received
27 II 1957
CITED LITERATURE
- N. N. Bogolyubov, Yu. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, 1955, p. 397.
- A. N. Tikhonov, Mat. sbornik, 22 (64), No. 2, 193 (1948).
- I. S. Gradshtein, DAN, 65, No. 6, 789 (1949).
- N. Ya. Lyashchenko, DAN, 96, 237 (1954).