Abstract
Full Text
MATHEMATICS
O. B. LYKOVA
ON THE BEHAVIOR OF SOLUTIONS OF A SYSTEM OF DIFFERENTIAL EQUATIONS IN A NEIGHBORHOOD OF AN ISOLATED STATIC SOLUTION
(Presented by Academician N. N. Bogolyubov, 25 II 1957)
Consider the system of differential equations
[
\frac{dx}{dt}=X(x)+\varepsilon X^*(t,x,\varepsilon),
\tag{1}
]
where (\varepsilon) is a small positive parameter.
We shall make the following assumptions:
a) The system of unperturbed equations
[
\frac{dx}{dt}=X(x)
\tag{2}
]
has an isolated static solution corresponding to the equilibrium point
[
x=0,\quad X(0)=0 \quad (X'_x(0)\ne 0).
\tag{3}
]
b) The functions (X(x)+\varepsilon X^*(t,x,\varepsilon)) in the domain
[
-\infty<t<\infty,\quad x\in U_{\sigma_0},\quad 0<\varepsilon<\varepsilon_0,
\tag{4}
]
where (U_{\sigma_0}) is the (\sigma_0)-neighborhood of the point (x=0), are periodic in (t) with period (2\pi) and have bounded and uniformly continuous derivatives with respect to (x,\varepsilon) of arbitrary order.
c) For the variational equations
[
\frac{d\delta x}{dt}=X'_x(0)\delta x,
\tag{5}
]
corresponding to the static solution (3), the characteristic equation
[
|J_n z-A|=0 \quad (A=X'_x(0))
\tag{6}
]
has a pair of purely imaginary roots ((z_1=i\omega,\ z_2=-i\omega)), while the remaining roots ((z_3,\ldots,z_n)) have negative real parts.
Under these assumptions it can be proved that system (1) has a unique local integral manifold, whose parametric representation depends on two arbitrary constants, and to which, as time proceeds, all solutions will tend whose initial values are sufficiently close to the indicated integral manifold.
Let us write system (1) in the form
[
\frac{dx}{dt}=X'_x(0)x+{\overline{X}(x)+\varepsilon X^*(t,x,\varepsilon)},
\tag{7}
]
where (\overline{X}(x)) begins with terms of second order with respect to (x), and, consequently,
respectively, there will also be found (\overline C) and (\delta_0) such that, for (|x|<\delta_0), the inequality
[
|\overline X(x)|\leq \overline C x^2 \qquad (\delta_0<\sigma_0).
\tag{8}
]
will hold.
As is known, the general solution of the system of equations (5) has the form
[
\delta x=C_1Ae^{i\omega t}+C_2Be^{-i\omega t}+D\overline h,
]
where (C_1,C_2) are arbitrary constants ((C_1,C_2) are complex conjugates); (A,B) are constant vectors; (D) is a constant matrix; (\overline h) is a linear combination of products of polynomials by decaying exponentials, containing (n-2) arbitrary constants.
Making in the system (7) the change of variables
[
x=A\xi+B\xi^*+Dh,
\tag{9}
]
we obtain
[
\begin{aligned}
\frac{d\xi}{dt}&=i\omega \xi+P(t,\xi,\xi^,h,\varepsilon),\
\frac{d\xi^}{dt}&=-i\omega \xi^+Q(t,\xi,\xi^,h,\varepsilon),\
\frac{dh}{dt}&=Hh+R(t,\xi,\xi^*,h,\varepsilon),
\end{aligned}
\tag{10}
]
where (H) is a square matrix of order (n-2), for which the equation
[
|J_{n-2}z-H|=0
]
has all roots with negative real parts.
At the same time one can specify such a (\rho_1)
[
\left(\rho_1<\frac{\delta_0}{|A|+|B|+|D|}\right),
]
that for any (|\xi|<\rho_1,\ |\xi^|<\rho_1,\ |h|<\rho_1) we shall have (|x|<\delta_0), and the functions (P(t,\xi,\xi^,h,\varepsilon)), (Q(t,\xi,\xi^,h,\varepsilon)), (R(t,\xi,\xi^,h,\varepsilon)), for (h=0) in the domain
[
-\infty<t<\infty,\qquad |\xi|<\rho,\qquad |\xi^*|<\rho,\qquad |h|<\rho,\qquad 0<\varepsilon<\varepsilon_0,
\tag{11}
]
tend to zero as (\varepsilon\to0,\ \rho^2\to0) ((\rho<\rho_1)).
After this, for the basic equation (1) we can formulate the following theorem.
Theorem. Suppose that, for the system of equations (1), conditions a), b), c) are satisfied. Then one can specify positive numbers (\varepsilon',\delta_1,\rho_1)
[
\left(\delta_1<\rho_1<\frac{\delta_0}{|A|+|B|+|D|}\right),
]
such that, for any positive (\varepsilon<\varepsilon'), (\delta<\delta_1,\rho<\rho_1), the following assertions will be valid:
-
Equation (1) has a unique two-dimensional local(^) integral manifold (\mathfrak M_t), lying in the domain (U^{\varepsilon'}_{\sigma_1}), where (\sigma_1=\rho_1{|A|+|B|+|D|}).*
-
On the local integral manifold, equation (1) is equivalent to the system
[
\begin{aligned}
\frac{d\xi_t}{dt}&=i\omega \xi_t+P_1(t,\xi_t,\xi_t^,\varepsilon),\
\frac{d\xi_t^}{dt}&=-i\omega \xi_t^+Q_1(t,\xi_t,\xi_t^,\varepsilon),
\end{aligned}
\tag{12}
]
(^) We shall call (\mathfrak M_t) a local integral manifold if, from the relation
(x(t_0,\xi_0,\xi_0^,\varepsilon)\in\mathfrak M_t,\ |\xi_0|<\rho_1,\ |\xi_0^|<\rho_1), valid at the moment (t=t_0), it follows that
(x(t,\xi_t,\xi_t^,\varepsilon)\in\mathfrak M_t) for any (t) as long as (|\xi_t|<\rho_1,\ |\xi_t^*|<\rho_1).
where the functions (P_1(t,\xi_t,\xi_t^,\varepsilon)), (Q_1(t,\xi_t,\xi_t^,\varepsilon)) are defined in the domain (11), have bounded and uniformly continuous derivatives, and are periodic in (t) with period (2\pi).
- The local integral manifold (\mathfrak M_t) admits a parametric representation of the form
[
x^(t)=A\xi_t+B\xi_t^+Dh_t^M=F(t,\xi_t,\xi_t^*,\varepsilon),
\tag{13}
]
where (F(t,\xi_t,\xi_t^*,\varepsilon)) is defined in the domain
[
-\infty