Abstract
Full Text
Mathematics
K. V. Zadiraka
INVESTIGATION OF A SINGULARLY PERTURBED AUTONOMOUS SYSTEM IN A NEIGHBORHOOD OF A TORUS
(Presented by Academician N. N. Bogolyubov, 12 III 1962)
We consider the system of differential equations
[
\frac{dx}{dt}=f(x,u),\qquad
\frac{dy}{dt}=f^(y,v),\qquad
\varepsilon\frac{dv}{dt}=F(x,u),\qquad
\varepsilon\frac{du}{dt}=F^(y,v),
\tag{1}
]
which is singularly perturbed with respect to the system
[
\frac{d\bar x}{dt}=f(\bar x,\varphi(\bar x)),\qquad
\frac{d\bar y}{dt}=f^*(\bar y,\psi(\bar y)),\qquad
\bar u=\varphi(\bar x),\qquad
\bar v=\psi(\bar y),
\tag{2}
]
where (x) and (f), (y) and (f^), (v) and (F), (u) and (F^) are vectors of dimensions (k,l,m), and (n), respectively, and (u=\varphi(x)) and (v=\psi(y)) are isolated solutions of the systems (F(x,u)=0) and (F^*(y,v)=0).
It is assumed that the unperturbed system (2) admits a solution of the form
[
\bar x=\bar x^0(\theta),\qquad
\bar y=\bar y^0(\vartheta),\qquad
\bar u=\varphi(\bar x^0)=\bar u^0(\theta),\qquad
\bar v=\psi(\bar y^0)=\bar v^0(\vartheta),
\tag{3}
]
periodic in (\theta=\omega t) and (\vartheta=\nu t) with period (2\pi).
With respect to the right-hand sides of system (1), and also the vectors (\varphi) and (\psi), we shall assume the following.
In the region (x\in U_{\rho_1}), (y\in U_{\rho_2}), (u\in U_{\rho_3}), (v\in U_{\rho_4}), where (U_{\rho_1}, U_{\rho_2}, U_{\rho_3}, U_{\rho_4}) denote the (\rho_1)-, (\rho_2)-, (\rho_3)- and (\rho_4)-neighborhoods of the periodic solution (\bar x^0,\bar y^0,\bar u^0,\bar v^0), the vectors (f,f^,F,F^,\varphi), and (\psi), together with their partial derivatives with respect to all arguments up to order ((p+1)) inclusive, are bounded and uniformly continuous.
It is established that system (1) admits a unique integral manifold of the form
[
x=x^0(\theta,\vartheta,\varepsilon),\qquad
y=y^0(\theta,\vartheta,\varepsilon),\qquad
u=u^0(\theta,\vartheta,\varepsilon),\qquad
v=v^0(\theta,\vartheta,\varepsilon),
\tag{4}
]
and the properties of this manifold are considered.
By slightly modifying the author’s theorem ((^4)), we can assert that if the real parts of the eigenvalues of the matrices (F_u(x,\varphi(x))) and (F_v^*(y,\psi(y))) are negative, then system (1) has a unique stable integral manifold of the form
[
u=\varphi(x)+\varphi^(x,y,\varepsilon),\qquad
v=\psi(y)+\psi^(x,y,\varepsilon),
\tag{5}
]
where (\varphi^(x,y,\varepsilon)\to0), (\psi^(x,y,\varepsilon)\to0) as (\varepsilon\to0). On this manifold system (1) is rewritten in the form
[
\frac{dx}{dt}=f\bigl(x,\varphi(x)+\varphi^(x,y,\varepsilon)\bigr),\qquad
\frac{dy}{dt}=f^\bigl(y,\psi(y)+\psi^*(x,y,\varepsilon)\bigr),
\tag{6}
]
whose order is (m+n) units less than the order of system (1). For the reduced system (2), the equations in variations corresponding to solution (3) have the form
[
\frac{d\delta\bar x}{dt}=\bar f(\bar x^0)\,\delta\bar x,\qquad
\frac{d\delta\bar y}{dt}=\bar f^*(\bar y^0)\,\delta\bar y,
\tag{7}
]
where the matrices (\bar f=\dfrac{\partial f}{\partial x}+\dfrac{\partial f}{\partial u}\dfrac{d\varphi}{dx}) and (\bar f^{}=\dfrac{\partial f^{}}{\partial y}+\dfrac{\partial f^{*}}{\partial v}\dfrac{d\psi}{dy}) are periodic in (\theta) and (\vartheta) with period (2\pi).
We shall assume that (k-1) characteristic exponents of the first of equations (7) and (l-1) characteristic exponents of the second of equations (7) have negative real parts. The first and the second of equations (7) each have one characteristic exponent equal to zero, since (\partial \bar x^{0}/\partial\theta) and (\partial \bar y^{0}/\partial\vartheta) are solutions of equations (7).
According to A. M. Lyapunov’s theorem ((^{1})) and the remarks made in ((^{3})), there exist real nonsingular matrices (P(\theta)) and (P^{}(\vartheta)), periodic in (\theta) and (\vartheta) with period (\,2\pi), having continuous first derivatives with respect to (\theta) and (\vartheta), and also square real matrices (H) and (H^{*}), such that the transformation
[
\delta \bar x=\frac{\partial \bar x^{0}}{\partial\theta}\rho+P(\theta)q,\qquad
\delta \bar y=\frac{\partial \bar y^{0}}{\partial\vartheta}r+P^{*}(\vartheta)s
\tag{8}
]
reduces equations (7) to the equivalent systems with constant coefficients
[
\frac{d\rho}{dt}=0,\qquad \frac{dq}{dt}=Hq;
]
[
\frac{dr}{dt}=0,\qquad \frac{ds}{dt}=H^{*}s,
\tag{9}
]
where the eigenvalues of the matrices (H) and (H^{*}) are equal to the nonzero characteristic exponents of equations (7).
We now transform system (6) by means of the substitution
[
x=\bar x^{0}(\theta)+P(\theta)h,\qquad
y=\bar y^{0}(\vartheta)+P^{}(\vartheta)h^{}
\tag{10}
]
to the equivalent system
[
\frac{d\theta}{dt}=\omega+A_{1}(\theta,\vartheta,h,h^{*},\varepsilon),
]
[
\frac{d\vartheta}{dt}=\nu+A_{2}(\theta,\vartheta,h,h^{*},\varepsilon),
]
[
\frac{dh}{dt}=Hh+A_{3}(\theta,\vartheta,h,h^{*},\varepsilon),
]
[
\frac{dh^{}}{dt}=H^{}h^{}+A_{4}(\theta,\vartheta,h,h^{},\varepsilon)
\tag{11}
]
of (k+l) equations with (k+l) unknowns (\theta,\vartheta,h=(h_{2},\ldots,h_{k}), h^{}=(h_{2}^{},\ldots,h_{l}^{*})).
The functions (A_i) ((i=1,2,\ldots,4)) are defined in the domain (\theta\in\Theta,\ \vartheta\in\Omega,\ h\in U_{\rho_1},\ h^{}\in U_{\rho_2},\ 0\le \varepsilon<\varepsilon^{}), where (U_{\rho_1}) and (U_{\rho_2}) denote the (\rho_1)- and (\rho_2)-neighborhoods of the points (h=0) and (h^{*}=0), possess bounded and uniformly continuous derivatives with respect to (\theta) and (\vartheta), and are periodic in (\theta) and (\vartheta) with period (2\pi).
Applying the method of proof set forth in the monograph ((^{2})), we verify the validity of the following assertions:
- There exists a positive number (\varepsilon_1<\varepsilon^{*}) such that, for any positive number (\varepsilon<\varepsilon_1), the system of equations (11) has a unique integral manifold, periodic in (\theta) and (\vartheta) with period (2\pi), representable by relations of the form
[
h=g(\theta,\vartheta,\varepsilon),\qquad
h^{}=g^{}(\theta,\vartheta,\varepsilon),
\tag{12}
]
* The matrices (P(\theta)) and (P^{*}(\vartheta)) will have period (4\pi) in (\theta) and (\vartheta) if there are negative ones among the real characteristic numbers of equations (7).
where the functions (g) and (g^*) are defined in the domain (\theta \in \Theta,\ \vartheta \in \Omega) and satisfy the inequalities
[
\begin{gathered}
|g(\theta,\vartheta,\varepsilon)| \leq D(\varepsilon),\
|g(\theta',\vartheta',\varepsilon)-g(\theta'',\vartheta'',\varepsilon)|
\leq \Delta(\varepsilon)\bigl(|\theta'-\theta''|+|\vartheta'-\vartheta''|\bigr),\
|g^(\theta,\vartheta,\varepsilon)| \leq D^(\varepsilon),\
|g^(\theta',\vartheta',\varepsilon)-g^(\theta'',\vartheta'',\varepsilon)|
\leq \Delta^*(\varepsilon)\bigl(|\theta'-\theta''|+|\vartheta'-\vartheta''|\bigr),
\end{gathered}
\tag{13}
]
where (D(\varepsilon)\to 0,\ D^(\varepsilon)\to 0,\ \Delta(\varepsilon)\to 0,\ \Delta^(\varepsilon)\to 0) together with (\varepsilon); (g) and (g^*) have bounded and uniformly continuous derivatives with respect to (\theta) and (\vartheta) up to order (p), inclusive.
- There exist positive constants (\varepsilon_0,\gamma_0,C_0,\sigma_0,\gamma_1,C_1,\sigma_1) ((\sigma_0<\rho_1,\ \sigma_1<\rho_2,\ \varepsilon_0<\varepsilon_1)) such that, for all (\varepsilon<\varepsilon_0), any real (t_0), and any (\theta\in\Theta,\ \vartheta\in\Omega), there exists a ((k-1))-dimensional domain (U_{\sigma_0}) of points ({h}) and an ((l-1))-dimensional domain of points ({h^*}) with the properties
[
|h_t-g(\theta_t,\vartheta_t,\varepsilon)|
\leq C_0 e^{-\gamma_0(t-t_0)}
|h_0-g(\theta_0,\vartheta_0,\varepsilon)|,
]
[
|h_t^-g^(\theta_t,\vartheta_t,\varepsilon)|
\leq C_1 e^{-\gamma_1(t-t_0)}
|h_0^-g^(\theta_0,\vartheta_0,\varepsilon)|,
]
where (h_t) and (h_t^) are arbitrary solutions of system (11); (h_0=h_t(t_0),\ h_0^=h_t^*(t_0),\ \theta_0=\theta_t(t_0),\ \vartheta_0=\vartheta_t(t_0)).
Transferring these assertions to system (1), which is equivalent to system (11), we obtain the following theorem:
Theorem. Under the assumptions indicated above, there exists a number (\varepsilon_0>0) such that, for any positive (\varepsilon<\varepsilon_0), the following assertions hold:
-
System (1) has a unique integral manifold (\mathfrak{M}), which is a torus in ((k+l+m+n))-dimensional space.
-
This manifold admits the parametric representation
[
\begin{gathered}
x=\bar{x}^{\,0}(\theta)+P(\theta)g(\theta,\vartheta,\varepsilon),
\qquad
y=\bar{y}^{\,0}(\vartheta)+P^(\vartheta)g^(\theta,\vartheta,\varepsilon),\
u=\varphi(x)+\varphi^(x,y,\varepsilon),
\qquad
v=\psi(y)+\psi^(x,y,\varepsilon),
\end{gathered}
]
where the right-hand sides are defined in the domain (\theta\in\Theta,\ \vartheta\in\Omega,\ 0<\varepsilon<\varepsilon_0), are periodic in the angular variables (\theta) and (\vartheta) with period (2\pi), and have bounded and uniformly continuous derivatives with respect to (\theta) and (\vartheta) up to order (p), inclusive.
- On the manifold (\mathfrak{M}), system (1) is equivalent to the system
[
\frac{d\theta}{dt}
=
\omega+A_1(\theta,\vartheta,g(\theta,\vartheta,\varepsilon),g^*(\theta,\vartheta,\varepsilon),\varepsilon),
]
[
\frac{d\vartheta}{dt}
=
\nu+A_2(\theta,\vartheta,g(\theta,\vartheta,\varepsilon),g^*(\theta,\vartheta,\varepsilon),\varepsilon),
]
where the right-hand sides are definite functions, periodic in (\theta) and (\vartheta) with period (2\pi), and possessing bounded and uniformly continuous derivatives with respect to (\theta) and (\vartheta) up to order (p), inclusive.
- 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 gratitude to Academician N. N. Bogolyubov for his attention to this work.
Received
1 III 1962
REFERENCES
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- N. N. Bogolyubov, Yu. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, 1955.
- J. K. Hale, A. P. Stokes, Arch. for Ration. Mech. and Analysis, 6, No. 2, 133 (1960).
- K. V. Zadiraka, DAN, 115, No. 4, 646 (1957).