Abstract Generated abstract
This paper studies the behavior of solutions of a periodically perturbed system of differential equations near an isolated equilibrium of the unperturbed system when the linearized equations have one pair of purely imaginary characteristic roots and all remaining roots have negative real parts. By transforming the system into variables separating the oscillatory and stable directions, it establishes conditions under which a unique two-dimensional local integral manifold exists and admits a periodic parametric representation with regular dependence on the variables and small parameter. The result shows that nearby solutions are exponentially attracted to this manifold, reducing the local dynamics to a two-dimensional system on it. A linear unperturbed special case is also noted, where an analogous attracting integral manifold exists with an attraction property for arbitrary finite values of the solution.
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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
\[ -\infty<t<\infty,\qquad |\xi_t|<\rho_1,\qquad |\xi_t^*|<\rho_1,\qquad 0<\varepsilon<\varepsilon', \tag{14} \]
has bounded and uniformly continuous derivatives with respect to \(\xi_t,\xi_t^*,\varepsilon\) of any order, and has period \(2\pi\) with respect to \(t\).
- The local integral manifold attracts to itself (as long as \(|\xi_t|<\rho_1,\ |\xi_t^*|<\rho_1\)) any solutions of system (1) \(x(t)\) whose initial values belong to \(U_{\delta_1}\), i.e., the inequalities
\[ |x(t)-x^*(t)|\leq C_1(\varepsilon,\delta^2)e^{-\gamma(t-t_0)}; \tag{15} \]
\[ \left|\frac{d\xi_t}{dt}-i\omega\xi_t-\varepsilon P_1(t,\xi_t,\xi_t^*,\varepsilon)\right| \leq C_2(\varepsilon,\delta^2)e^{-\gamma(t-t_0)}; \]
\[ \left|\frac{d\xi_t^*}{dt}+i\omega\xi_t^*-\varepsilon Q_1(t,\xi_t,\xi_t^*,\varepsilon)\right| \leq C_3(\varepsilon,\delta^2)e^{-\gamma(t-t_0)}. \tag{16} \]
Let us consider the special case of system (1) when \(X(x)=Px\), where \(P=\|p_{ik}\|\) is a constant matrix of order \(n\), i.e., we shall consider the system
\[ \frac{dx}{dt}=Px+\varepsilon X^*(t,x,\varepsilon), \tag{17} \]
where \(\varepsilon\) is a small positive parameter; \(X^*\) is an \(n\)-dimensional vector of Euclidean space. Here \(X^*(t,x,\varepsilon)\) is a periodic function of \(t\) with period \(2\pi\), having bounded and uniformly continuous derivatives of any order in the domain
\[ -\infty<t<\infty,\qquad 0<\varepsilon<\varepsilon_0,\qquad |x|<N,\qquad N=\mathrm{const}. \tag{18} \]
Suppose that the characteristic equation for the system of first approximation \(dx/dt=Ax\) has a pair of purely imaginary roots \(\pm i\omega\), while the remaining roots \(z_3,\ldots,z_n\) have negative real parts.
If these conditions are fulfilled, there is a theorem which, analogously to the preceding theorem, establishes the existence and uniqueness of an integral manifold for system (17), to which, as time passes, any solutions of system (17) will tend. However, in contrast to the preceding theorem, which is local in character, in the present case this attraction property already holds for arbitrary (finite) values of \(x(t)\).
Institute of Mathematics
Academy of Sciences of the Ukrainian SSR
Received
25 II 1957
REFERENCES
- N. N. Bogolyubov, Yu. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, 1955.
- O. B. Lykova, Reports of the Academy of Sciences of the Ukrainian SSR, No. 2 (1957).