UDC 517.919

ON THE THEORY OF INVARIANT SURFACES

V. A. PLISS

Many works have been devoted to the theory of invariant surfaces of systems of ordinary differential equations. The fundamental results in this direction are due to N. M. Krylov and N. N. Bogolyubov [1]. Subsequently many authors investigated questions of the existence of invariant surfaces. A detailed survey of works devoted to these questions was made by N. N. Bogolyubov and Yu. A. Mitropolsky [2].

In the works published up to the present time, questions of perturbation of systems possessing invariant surfaces are considered. The known results have the following character. One considers the system

\[ \frac{dx}{dt}=X(x,t)+R(x,t), \tag{1} \]

where it is known that the “generating” system

\[ \frac{dx}{dt}=X(x,t) \tag{2} \]

has an invariant surface, and the perturbation \(R(x,t)\) is small in some sense. In various works conditions are given under which system (1) has an invariant surface tending to the invariant surface of system (2) as the perturbation \(R(x,t)\) decreases.

In the present paper we shall give conditions for the existence of invariant surfaces of system (2) which do not fit into the usual framework of perturbation theory.

1. Consider the system

\[ \frac{dx}{dt}=X(x,y,t),\qquad \frac{dy}{dt}=Y(x,y,t), \tag{1.1} \]

where \(x\) and \(X\) are \(n\)-dimensional vectors; \(y\) and \(Y\) are \(m\)-dimensional vectors.

With respect to the right-hand sides of system (1.1) we make the following assumptions.

The vector functions \(X\) and \(Y\) are continuous and have continuous and bounded partial derivatives with respect to \(x\) and \(y\) for all \(x,t\) and for \(\|y\|\le K\), where \(\|y\|\) is the Euclidean norm of the vector \(y\), and \(K\) is some positive number. The vector functions \(X\) and \(Y\) have period \(\omega\) in \(t\):

\[ X(x,y,t+\omega)=X(x,y,t),\qquad Y(x,y,t+\omega)=Y(x,y,t). \]

We form the Jacobi matrices of \(X\) with respect to \(x\) and of \(Y\) with respect to \(y\), i.e. the matrices \(\partial X/\partial x\) and \(\partial Y/\partial y\), and consider, along with them, the symmetrized Jacobi matrices

\[ V(x,y,t)=\frac{1}{2}\left[\frac{\partial X}{\partial x}+\left(\frac{\partial X}{\partial x}\right)^*\right] \tag{1.2} \]

and

\[ W(x,y,t)=\frac{1}{2}\left[\frac{\partial Y}{\partial y}+\left(\frac{\partial Y}{\partial y}\right)^*\right], \tag{1.3} \]

where, as usual, \(A^*\) is the transpose of the matrix \(A\).

Let \(\lambda_k(x,y,t)\) \((k=1,2,\ldots,n)\) be the eigenvalues of the matrix \(V(x,y,t)\), and let \(\mu_k(x,y,t)\) \((k=1,2,\ldots,m)\) be the eigenvalues of the matrix \(W(x,y,t)\). We shall assume that there exist numbers \(\lambda\) and \(\mu\) such that \(\mu<0\), \(\lambda>\mu\), and for all \(x,t\) and \(\|y\|\le K\) the inequalities

\[ \lambda_k(x,y,t)\ge \lambda \quad (k=1,2,\ldots,n), \tag{1.4} \]

\[ \mu_k(x,y,t)\le \mu \quad (k=1,2,\ldots,n) \tag{1.5} \]

are satisfied.

Form the Jacobi matrices of \(X\) with respect to \(y\) and of \(Y\) with respect to \(x\), i.e., the matrices \(\partial X/\partial y\) and \(\partial Y/\partial x\). We shall assume that the norms of these matrices satisfy the inequalities

\[ \left\|\frac{\partial X}{\partial y}\right\|\le \alpha,\qquad \left\|\frac{\partial Y}{\partial x}\right\|\le \beta, \tag{1.6} \]

where \(\|A\|\) is the Euclidean norm of the matrix \(A\), and the numbers \(\alpha\) and \(\beta\) satisfy the inequality

\[ 4\alpha\beta<(\lambda-\mu)^2. \tag{1.7} \]

Moreover, in what follows we shall assume that

\[ \frac{d}{dt}\|y\|<0 \tag{1.8} \]

for all \(x,t\) and for \(\|y\|=K\).

Under the conditions formulated, we shall prove the existence of an invariant surface.

1. Consider in the space \(x,y\) a surface of dimension \(n\), defined by the equation

\[ y=g(x), \tag{2.1} \]

where \(g(x)\) is an \(m\)-dimensional vector function of the \(n\)-dimensional vector argument \(x\), defined for all \(x\) and possessing the properties

\[ \|g(x)\|\le K, \tag{2.2} \]

and \(g(x)\) satisfies the Lipschitz condition for all \(x\),

\[ \|g(x_1)-g(x_2)\|\le L\|x_1-x_2\|. \tag{2.3} \]

Here

\[ L=\frac{\lambda-\mu}{2\alpha}. \tag{2.4} \]

Through the surface (2.1) at \(t=t_0\), pass all possible solutions of the system (1.1); then, for each fixed \(t\), we shall obtain a surface of dimension \(n\). This surface is defined by the equations

\[ x=x(t,t_0,\xi,g(\xi)),\qquad y=y(t,t_0,\xi,g(\xi)), \tag{2.5} \]

where \(x(t,t_0,x_0,y_0)\), \(y(t,t_0,x_0,y_0)\) is the solution of system (1.1) with initial data \(t=t_0\), \(x=x_0\), \(y=y_0\).

The following assertion is valid.

Lemma 1. For \(t>t_0\), the surface (2.5) is represented in the form

\[ y=g(x,t), \tag{2.6} \]

where the vector function \(g(x,t)\) is continuous for all \(x\) and \(t>t_0\), is bounded,

\[ \|g(x,t)\|\leq K \tag{2.7} \]

and satisfies the Lipschitz condition

\[ \|g(x_1,t)-g(x_2,t)\|\leq L\|x_1-x_2\|. \tag{2.8} \]

Proof. From the general theorems of the theory of differential equations it follows that there exists a \(t^*>0\) (not depending on \(t_0\) or on the particular choice of the function \(g(x)\) with the listed properties) such that, for
\(t_0\leq t\leq t_0+t^*\), the surface (2.5) is represented in the form (2.6); moreover, the function \(g(x,t)\) is continuous for all \(x\) and \(t_0\leq t\leq t_0+t^*\) and satisfies a Lipschitz condition in \(x\)

\[ \|g(x_1,t)-g(x_2,t)\|\leq L^*\|x_1-x_2\|. \tag{2.9} \]

From inequality (1.8) it follows that on all solutions of system (1.1) with initial data \(t=t_0\), \(x=\xi\), \(y=g(\xi)\), the inequality \(\|y\|\leq K\) is satisfied for \(t>t_0\). Hence it follows that, for \(t_0\leq t\leq t_0+t^*\), \(g(x,t)\) satisfies the inequality

\[ \|g(x,t)\|\leq K. \]

We shall show that in condition (2.9) the constant \(L^*\) may be taken equal to \(L\), i.e., that condition (2.8) is satisfied.

Take any two points \((\xi_1,\eta_1)\) and \((\xi_2,\eta_2)\) of the space \(x,y\), subject to the condition \(\|\eta_1\|\leq K\), \(\|\eta_2\|\leq K\), and consider the solutions of system (1.1) passing through these points at \(t=t_0\):

\[ x_1=x(t,t_0,\xi_1,\eta_1),\qquad y_1=y(t,t_0,\xi_1,\eta_1), \]

\[ x_2=x(t,t_0,\xi_2,\eta_2),\qquad y_2=y(t,t_0,\xi_2,\eta_2). \]

Form the differences

\[ \Delta x=x_1-x_2,\qquad \Delta y=y_1-y_2. \tag{2.10} \]

These differences satisfy the following system of equations:

\[ \frac{d\Delta x}{dt}=X(x_1,y_1,t)-X(x_2,y_2,t), \]

\[ \frac{d\Delta y}{dt}=Y(x_1,y_1,t)-Y(x_2,y_2,t), \tag{2.11} \]

and the norms of these differences satisfy the equations

\[ \frac{d\|\Delta x\|}{dt} = \frac{([X(x_1,y_1,t)-X(x_2,y_2,t)],\Delta x)}{\|\Delta x\|}, \]
\[ \frac{d\|\Delta y\|}{dt} = \frac{([Y(x_1,y_1,t)-Y(x_2,y_2,t)],\Delta y)}{\|\Delta y\|}, \tag{2.12} \]

where \((\, ,\,)\) is the scalar product of vectors.

Let us estimate the scalar products appearing on the right-hand sides of equations (2.12). We have

\[ X(x_1,y_1,t)-X(x_2,y_2,t) = X(x_1,y_1,t)- \]
\[ -\,X(x_1,y_2,t)+X(x_1,y_2,t)-X(x_2,y_2,t). \tag{2.13} \]

It is not difficult to see that

\[ X(x_1,y_1,t)-X(x_2,y_2,t) = \int_0^1 \frac{\partial X(x_1,y_2+u(y_1-y_2),t)}{\partial y} \,\Delta y\,du. \]

Hence the estimate follows:

\[ \|X(x_1,y_1,t)-X(x_1,y_2,t)\|\leq \]
\[ \leq \int_0^1 \left\| \frac{\partial X(x_1,y_2+u(y_1-y_2),t)}{\partial y} \right\| \|\Delta y\|\,du. \tag{2.14} \]

From inequalities (1.6) and (2.14) we obtain

\[ \|X(x_1,y_1,t)-X(x_1,y_2,t)\|\leq a\|\Delta y\|. \tag{2.15} \]

Let us estimate the scalar product \(([X(x_1,y_2,t)-X(x_2,y_2,t)],\Delta x)\). We shall show that, whatever the vectors \(x\) and \(x'\) and the vector \(y\) satisfying the inequality \(\|y\|\leq K\), the inequality

\[ (V(x',y,t)x,x)\geq \lambda\|x\|^2 \tag{2.16} \]

holds.

Considering the function \((V(x',y,t)x,x)\) as a quadratic form in the components of the vector \(x\), we reduce it to canonical form by means of an orthogonal transformation \(x=Pz\):

\[ (V(x',y,t)x,x)=\sum_{k=1}^{n}\lambda_k(x',y,t)z_k^2, \tag{2.17} \]

where \(z_k\) \((k=1,2,\ldots,n)\) are the components of the vector \(z\). From equality (2.17) and inequality (1.4) it follows that

\[ (V(x',y,t)x,x)\geq \lambda\|z\|^2. \]

But the transformation \(x=Pz\) is orthogonal; consequently, \(\|z\|=\|x\|\), whence inequality (2.16) follows.

Next we have

\[ X(x_1,y_2,t)-X(x_2,y_2,t)= \]
\[ = \int_0^1 \frac{\partial X(x_2+u(x_1-x_2),y_2,t)}{\partial x} \,\Delta x\,du. \tag{2.18} \]

It is not difficult to verify the validity of the equality

\[ \left(\int_0^1 \frac{\partial X(x_2+u(x_1-x_2),\,y_2,\,t)}{\partial x}\,\Delta x\,du,\ \Delta x\right) = \]

\[ =\left(\int_0^1 V(x_2+u(x_1-x_2),\,y_2,\,t)\Delta x\,du,\ \Delta x\right). \]

Hence, and from (2.18), we obtain

\[ ([X(x_1,y_2,t)-X(x_2,y_2,t)],\Delta x) = \]

\[ =\int_0^1 (V(x_2+u(x_1-x_2),\,y_2,\,t)\Delta x,\Delta x)\,du. \]

The last equality, together with (2.16), gives

\[ ([X(x_1,y_2,t)-X(x_2,y_2,t)],\Delta x)\geq \lambda\|\Delta x\|^2 . \tag{2.19} \]

From inequality (2.15) there follows the estimate

\[ ([X(x_1,y_1,t)-X(x_1,y_2,t)],\Delta x)\leq a\|\Delta y\|\,\|\Delta x\|. \tag{2.20} \]

From the first equation of system (2.12), equality (2.13), and the estimates (2.19) and (2.20), we obtain

\[ \frac{d\|\Delta x\|}{dt}\geq \lambda\|\Delta x\|-a\|\Delta y\|. \tag{2.21} \]

Similarly, from the second equation of system (2.12) and conditions (1.5) and (1.6), we obtain the estimate

\[ \frac{d\|\Delta y\|}{dt}\leq \mu\|\Delta y\|+\beta\|\Delta x\|. \tag{2.22} \]

Let now the initial points \((\xi_1,\eta_1)\) and \((\xi_2,\eta_2)\) of the solutions \((x_1,y_1)\) and \((x_2,y_2)\) lie on the surface (2.1), i.e.

\[ \eta_1=g(\xi_1),\qquad \eta_2=g(\xi_2). \]

From condition (2.3) it follows that then the inequality

\[ \|\eta_1-\eta_2\|\leq L\|\xi_1-\xi_2\| \]

holds, i.e. for \(t=t_0\) the inequality

\[ \|\Delta y\|\leq L\|\Delta x\| \tag{2.23} \]

holds.

We shall show that this inequality holds for all \(t\geq t_0\). Indeed, suppose that at some time \(t=t'>t_0\) inequality (2.23) becomes an equality,

\[ \|\Delta y\|=L\|\Delta x\|. \tag{2.24} \]

Then from the estimates (2.21) and (2.22) it follows...

\[ \frac{d\|\Delta x\|}{dt} \geqslant \lambda \|\Delta x\|-\alpha L\|\Delta x\|, \tag{2.25} \]

\[ \frac{d\|\Delta y\|}{dt} \leqslant \mu L\|\Delta x\|+\beta\|\Delta x\|. \tag{2.26} \]

From equality (2.4) and inequality (1.7) it follows that

\[ \mu L+\beta<\lambda L-\alpha L^{2}, \]

and hence, from (2.25), (2.26), it follows that on the solutions \(x_1, y_1\) and \(x_2, y_2\), for \(t=t'\), the inequality

\[ \frac{d\|\Delta y\|}{dt}<L\,\frac{d\|\Delta x\|}{dt} \]

holds.

This inequality proves that (2.23) is satisfied for all \(t\geqslant t_0\).

Thus, for any two points of the surface (2.6), inequality (2.23) holds. This proves that on the time interval \(t_0\leqslant t\leqslant t_0+t^*\) the function \(g(x,t)\) satisfies the Lipschitz condition (2.8).

Thus the assertion of the lemma has been established for \(t\in [t_0,t_0+t^*]\). But the surface \(y=g(x,t_0+t^*)\) is no different from the surface \(y=g(x)\), and the instant \(t_0+t^*\) is no different from the instant \(t_0\). Continuing our reasoning further for \(t\geqslant t_0+t^*\), we also establish the validity of the assertion of the lemma for all \(t\geqslant t_0\).

1. Let us introduce the space \(G\) of \(m\)-dimensional vector functions \(y(x)\) of the \(n\)-dimensional vector argument \(x\), defined for all \(x\) and possessing properties (2.2) and (2.3). We introduce in this space the metric

\[ \rho(g_1,g_2)=\sup \|g_1(x)-g_2(x)\|. \tag{3.1} \]

It is not difficult to see that the space \(G\) is complete. Denote by \(F_{t_0,t}\) (\(t_0\) is an arbitrary number, \(t\geqslant t_0\)) the operator which assigns to the function \(g(x)\) the function \(g(x,t)\) defined by equality (2.6).

It follows from Lemma 1 that the operator \(F_{t_0,t}\), for any \(t_0\) and \(t\geqslant t_0\), maps the space \(G\) into itself.

Since the partial derivatives of the right-hand sides of the system (1.1) are bounded by the hypothesis, the operator \(F_{t_0,t}\) is continuous in \(t\) and \(g\) for \(t\geqslant t_0\) and \(g\in G\).

The following assertion is true.

Lemma 2. The operator \(F_{t_0,t}\), for any \(t_0\) and \(t>t_0\), is a contraction operator.

Proof. Let \(g_1(x)\in G\), \(g_2(x)\in G\). Consider all possible solutions of the system (1.1) with initial points on the surfaces \(y=g_1(x)\) and \(y=g_2(x)\) at \(t=t_0\); then, for each \(t\geqslant t_0\), we obtain the surfaces:

\[ y=F_{t_0,t}g_1,\qquad y=F_{t_0,t}g_2. \]

Choose any two of these solutions with common initial data in \(x\):

\[ x_1=x(t,t_0,\xi,g_1(\xi)),\qquad y_1=y(t,t_0,\xi,g_1(\xi)), \]

\[ x_2=x(t,t_0,\xi,g_2(\xi)),\qquad y_2=y(t,t_0,\xi,g_2(\xi)). \]

Form the differences

\[ \Delta x=x_1-x_2,\qquad \Delta y=y_1-y_2. \]

These differences satisfy the system of equations (2.11), and their norms satisfy the system (2.12). For \(t=t_0\), \(\Delta x=0\), \(\Delta y=g_1(\xi)-g_2(\xi)=\Delta y_0\). We choose the point \(\xi\) so that \(\Delta y_0\ne 0\), i.e., \(g_1(\xi)\ne g_2(\xi)\); then it is clear that for any \(\varepsilon>0\) one can indicate such a \(t^*>0\) that, for \(t_0\leq t\leq t_0+t^*\),

\[ \|\Delta x\|<\varepsilon\|\Delta y\|. \tag{3.2} \]

Since, by assumption, the partial derivatives of the right-hand sides are bounded, the quantity \(t^*\) may be regarded as independent of the particular choice of the vector functions \(g_1(x)\) and \(g_2(x)\), of the point \(\xi\), and of the value \(t_0\).

In the case under consideration, the estimate (2.22) is evidently valid. From this estimate and inequality (3.2) it follows that, for \(t_0\leq t\leq t_0+t^*\), the inequality

\[ \frac{d\|\Delta y\|}{dt}<(\mu+\beta\varepsilon)\|\Delta y\| \tag{3.3} \]

holds. Hence it follows that, for \(t_0\leq t\leq t_0+t^*\), the relation

\[ \|\Delta y\|\leq \|\Delta y_0\| e^{(\mu+\beta\varepsilon)(t-t_0)} \tag{3.4} \]

is satisfied.

Consider the difference \(X(x_1,y_2,t)-X(x_2,y_2,t)\). Since, by assumption, the partial derivatives of \(X\) with respect to \(x\) and \(y\) are bounded, there exists a \(\nu>0\) such that the inequality

\[ \|X(x_1,y_2,t)-X(x_2,y_2,t)\|\leq \nu\|\Delta x\|. \tag{3.5} \]

holds. From the first equation of the system (2.12), equality (2.13), and inequalities (2.14) and (3.5), there follows the estimate

\[ \frac{d\|\Delta x\|}{dt}\leq \nu\|\Delta x\|+\alpha\|\Delta y\|. \tag{3.6} \]

Thus, in view of (3.4), for sufficiently small \(\varepsilon\), \(\|\Delta y\|\) decreases as \(t\) increases; therefore from (3.6) we obtain the inequality

\[ \frac{d\|\Delta x\|}{dt}\leq \nu\|\Delta x\|+\alpha\|\Delta y_0\|, \tag{3.7} \]

valid for \(t_0\leq t\leq t_0+t^*\). Integrating this inequality, we obtain

\[ \|\Delta x\|\leq \frac{\alpha}{\nu}\,\|\Delta y_0\|\left(e^{\nu(t-t_0)}-1\right). \tag{3.8} \]

On the surface \(y=F_{t_0,t}g_1(x)\), consider the point \(R_1\) with coordinates

\[ x=x(t,t_0,\xi,g_1(\xi)),\qquad y=y(t,t_0,\xi,g_1(\xi)), \]

i.e., the point

\[ x=x(t,t_0,\xi,g_1(\xi)),\qquad y=F_{t_0,t}g_1\big|_{x=x(t,t_0,\xi,g_1(\xi))}, \]

and on the surface \(y=F_{t_0,t}g_2\)—the point \(R_2\) with coordinates

\[ x=x(t,t_0,\xi,g_1(\xi)),\qquad y=F_{t_0,t}g_2\big|_{x=x(t,t_0,\xi,g_1(\xi))}. \]

Thus, the points \(R_1\) and \(R_2\) lie on the surfaces \(y=F_{t_0,t}g_1\) and \(y=F_{t_0,t}g_2\), respectively, and have one and the same value of the vector coordinate \(x\).

By Lemma 1, the function \(F_{t_0,t}g\) satisfies the Lipschitz condition with constant \(L\); therefore

\[ \left\|y(R_2)-y(t,t_0,\xi,g_2(\xi))\right\|\leq L\left\|x(R_2)- x(t,t_0,\xi,g_2(\xi))\right\|=L\|\Delta x\|, \tag{3.9} \]

where \(x(R_2)\) and \(y(R_2)\) are the vector coordinates of the point \(R_2\). From the last inequality (3.8) we obtain the estimate:

\[ \left\|y(R_2)-y(t,t_0,\xi,g_2(\xi))\right\|\leq \frac{L\alpha}{\nu}\,\|\Delta y_0\|\left(e^{\nu(t-t_0)}-1\right), \tag{3.10} \]

but

\[ \left\|y(t,t_0,\xi,g_1(\xi))-y(t,t_0,\xi,g_2(\xi))\right\|=\|\Delta y\|, \]

and \(y(t,t_0,\xi,g_1(\xi))=y(R_1)\). Consequently, by virtue of inequality (3.4), we obtain

\[ \left\|y(R_1)-y(R_2)\right\|\leq \|\Delta y_0\| \left[ e^{(\mu+\beta\varepsilon)(t-t_0)} + \frac{L\alpha}{\nu}\left(e^{\nu(t-t_0)}-1\right) \right]. \tag{3.11} \]

The last inequality is valid for \(t_0\leq t\leq t_0+t^*\). Put

\[ s(t)=e^{(\mu+\beta\varepsilon)(t-t_0)} + \frac{L\alpha}{\nu}\left(e^{\nu(t-t_0)}-1\right). \]

It is clear that \(s(t_0)=1\), and
\[ s'(t_0)=\mu+\beta\varepsilon+L\alpha=\mu/2+\lambda/2+\beta\varepsilon. \]

Since, by hypothesis, \(\mu+\lambda<0\), by choosing \(\varepsilon\) sufficiently small one can achieve the inequality \(s'(t_0)<0\). Then, for \(t_0<t\leq t_0+T\) and sufficiently small \(T\in(0,t^*]\), the inequality

\[ s(t)<1. \tag{3.12} \]

will hold.

Thus, taking an arbitrary pair of points \(R_1\) and \(R_2\) on the surfaces \(y=F_{t_0,t}g_1\) and \(y=F_{t_0,t}g_2\), respectively, with a common vector coordinate \(x\), we have obtained for them inequality (3.11). From this inequality there follows the relation

\[ \rho(F_{t_0,t}g_1,\ F_{t_0,t}g_2)\leq s(t)\rho(g_1,g_2), \tag{3.13} \]

and this, in conjunction with (3.12), proves that for \(t_0<t\leq t_0+T\) the operator \(F_{t_0,t}\) is a contraction operator.

We shall show that also for \(t>t_0+T\) the operator \(F_{t_0,t}\) is a contraction operator. Let \(t>t_0+T\); then \(t\) can be represented in the form \(t=t_0+kT+\tau\), where \(k\) is a natural number and \(\tau\in[0,T)\). From the very definition of the operator \(F_{t_0,t}\) it follows that it can be represented in the following way:

\[ F_{t_0,t}=F_{t_0+kT,\ t_0+kT+\tau}F_{t_0+(k-1)T,\ t_0+kT}\cdots F_{t_0,\ t_0+T}. \]

Hence, from (3.13), it follows that

\[ \rho(F_{t_0,t}g_1,\ F_{t_0,t}g_2)\leq s(\tau)s^k(T)\rho(g_1,g_2). \]

This proves that for all \(t>t_0\) the operator \(F_{t_0,t}\) is a contraction operator.

The lemma is proved.

1. Lemmas 1 and 2 make it possible to prove the following assertion.

Theorem 1. Suppose that the conditions listed in Sec. 1 are satisfied; then system (1.1) has an invariant surface of the form

\[ y=g(x,t), \tag{4.1} \]

where the \(m\)-dimensional vector-function \(g\) has the following properties: \(g(x,t)\) is defined and continuous for all \(x,t\), is bounded,

\[ \|g(x,t)\|\leq K, \tag{4.2} \]

satisfies the Lipschitz condition

\[ \|g(x_1,t)-g(x_2,t)\|\leq L\|x_1-x_2\| \tag{4.3} \]

and has period \(\omega\) in \(t\).

Proof. Apply the operator \(F_{0,\omega}\) to the space \(G\). From Lemmas 1 and 2 it follows that this operator has a fixed point \(g^*\). Through the surface \(y=g^*(x)\) at \(t=0\), let us pass all possible solutions of system (1.1); then for each fixed \(t\) we shall obtain the surface

\[ y=g(x,t), \tag{4.4} \]

where, by Lemma 1, the function \(g(x,t)\) has the properties (4.2) and (4.3). Since \(g^*(x)\) is a fixed point of the operator \(F_{0,\omega}\), by the definition of this operator we shall have

\[ g(x,0)=g^*(x)=g(x,\omega), \tag{4.5} \]

and hence, from the \(\omega\)-periodicity of the right-hand sides of system (1.1), it follows that for every \(t\)

\[ g(x,t+\omega)=g(x,t). \tag{4.6} \]

The theorem is proved.

Let us note that, if one uses the arguments of N. N. Krasovskii [3], then Theorem 1 can be generalized. Conditions (1.4), (1.5) may be imposed not on the symmetrized Jacobi matrices (1.2), (1.3), but on the symmetrized matrices \(A\,\partial X/\partial x\) and \(B\,\partial Y/\partial y\), where \(A\) and \(B\) are arbitrary constant, square, symmetric positive definite matrices of orders \(n\) and \(m\), respectively.

1. In the preceding sections, conditions were obtained for the existence of invariant surfaces representable in the form (4.1). However, for a number of questions in the qualitative theory of differential equations, invariant surfaces with a more complicated topological structure are of interest. In particular, in the theory of dissipative systems, an important role is played by invariant surfaces homeomorphic to the topological product of an \((n-1)\)-dimensional sphere and a line (see [4], § 2). The preceding arguments extend without difficulty to this important case.

Consider the system of equations

\[ \frac{dx}{dt}=X(x,t), \tag{5.1} \]

where \(x\) is an \(n\)-dimensional vector with components \(x_1,x_2,\ldots,x_n\); the vector-function \(X(x,t)\) is defined, continuous, and continuously differentiable with respect to \(x\) for

for all \(t\) and \(\rho_1 \leqslant \|x\| \leqslant \rho_2\); \(\rho_1\) and \(\rho_2\) are positive numbers; \(X(x,t)\) is \(\omega\)-periodic in \(t\): \(X(x,t+\omega)=X(x,t)\). We shall further assume that for \(\|x\|=\rho_1\)

\[ \frac{d\|x\|}{dt}>0, \tag{5.2} \]

and for \(\|x\|=\rho_2\)

\[ \frac{d\|x\|}{dt}<0. \tag{5.3} \]

Take an arbitrary point \(x_0\) from the region \(\rho_1 \leqslant \|x\| \leqslant \rho_2\). In the space \(x\), perform a rotation of the coordinate axes \(y=P(x_0)x\), where \(P(x_0)\) is the rotation matrix, and \(y\) is an \(n\)-dimensional vector with components \(y_1,y_2,\ldots,y_n\). We shall carry out this rotation in such a way that the point \(x_0\) lies on the axis \(Oy_n\), i.e. so that \(y_{10}=0,\ y_{20}=0,\ \ldots,\ y_{n-1,0}=0,\ y_{n0}>0\), where \(y_{i0}\) are the components of the vector \(y_0=P(x_0)x_0\). In the coordinates \(y\), system (5.1) has the form

\[ \frac{dy}{dt}=Y(y,t). \tag{5.4} \]

Let us pass in system (5.4) to spherical coordinates by the formulas

\[ \begin{aligned} y_1&=r\sin\varphi_1,\\ y_2&=r\cos\varphi_1\sin\varphi_2,\\ &\cdots\cdots\cdots\cdots\\ y_{n-1}&=r\cos\varphi_1\cos\varphi_2\cdots\cos\varphi_{n-2}\sin\varphi_{n-1},\\ y_n&=r\cos\varphi_1\cos\varphi_2\cdots\cos\varphi_{n-2}\cos\varphi_{n-1}. \end{aligned} \tag{5.5} \]

The point \(y_0\) will correspond to the point \(y_{n0}=r_0,\ \varphi_1=\cdots=\varphi_{n-1}=0\). The Jacobian of the transformation (5.5) for \(\varphi_1=\varphi_2=\cdots=\varphi_{n-2}=0\) is equal to \(r^{\,n-1}\). System (5.4) in the spherical coordinates (5.5) has the form

\[ \frac{dr}{dt}=R(r,\varphi,t), \]

\[ \frac{d\varphi}{dt}=\Phi(r,\varphi,t), \tag{5.6} \]

where \(\varphi\) is the vector with components \(\varphi_1,\varphi_2,\ldots,\varphi_{n-1}\); \(\Phi\) is an \((n-1)\)-dimensional vector function; \(r\) and \(R\) are scalar quantities. The functions \(R\) and \(\Phi\), for sufficiently small \(\|\varphi\|\) and for \(\rho_1 \leqslant r \leqslant \rho_2\), are defined, continuous, and continuously differentiable with respect to \(r\) and \(\varphi\). Moreover, they have period \(\omega\) in \(t\).

We shall make the following assumptions concerning system (5.1). We shall assume that, for any choice of the point \(x_0\) (and hence also of the coordinates \(y\) and \(r,\varphi\)), the inequality

\[ \frac{\partial R(r,0,t)}{\partial r}\leqslant \mu \tag{5.7} \]

is satisfied for \(\rho_1 \leqslant r \leqslant \rho_2,\ -\infty<t<+\infty\).

Form the Jacobi matrix \(\partial\Phi/\partial\varphi\) and set

\[ V(r,\varphi,t)=\frac{1}{2}\left[\frac{\partial\Phi}{\partial\varphi}+\left(\frac{\partial\Phi}{\partial\varphi}\right)^{*}\right]. \]

Let \(\lambda_k(r,\varphi,t)\) \((k=1,2,\ldots,n-1)\) be the eigenvalues of the matrix \(V(r,\varphi,t)\). Suppose that for any choice of the point \(x_0\) the inequality

\[ \lambda_k(r,0,t)\geq \lambda \tag{5.8} \]

holds for \(\rho_1\leq r\leq \rho_2,\; -\infty<t<+\infty\).

In addition, we shall assume that for any \(x_0\) the inequalities

\[ \left\|\frac{\partial \Phi(r,0,t)}{\partial r}\right\|\leq \alpha,\qquad \left\|\frac{\partial R(r,0,t)}{\partial \varphi}\right\|\leq \beta \tag{5.9} \]

hold.

The numbers \(\lambda,\mu,\alpha,\beta\) satisfy the inequalities \(\mu<0,\; \lambda>\mu,\; 4\alpha\beta<(\lambda-\mu)^2\).

Theorem 2. Under the assumptions made, system (5.1) has in the space \((x,t)\) an \(n\)-dimensional invariant surface \(\mathrm{M}\) with the following properties:

1) the surface \(\mathrm{M}\) is \(\omega\)-periodic, i.e. its intersections with the \(n\)-dimensional hyperplanes \(t=\tau\) and \(t=\tau+\omega\) coincide for every \(\tau\);

2) for each \(t\), the surface \(\mathrm{M}\) has one and only one point of intersection with each ray issuing from the origin of coordinates;

3) for any choice of the point \(x_0\), an element of the surface \(\mathrm{M}\) is represented in the form

\[ r=g(\varphi,t), \tag{5.10} \]

where the scalar function \(g(\varphi,t)\) has the following properties: it is defined and continuous for all \(-\infty<t<+\infty\) and for sufficiently small \(\|\varphi\|\); for such values of the arguments \(g\) is bounded, \(\rho_1<g<\rho_2\); it satisfies the Lipschitz condition in \(\varphi\)

\[ \left|g(\bar{\varphi},t)-g(\bar{\bar{\varphi}},t)\right|\leq L\|\bar{\varphi}-\bar{\bar{\varphi}}\|, \tag{5.11} \]

where

\[ L=\frac{\lambda-\mu}{2\alpha}; \]

\(g(\varphi,t)\) has period \(\omega\) in \(t\).

The proof of this theorem differs from the proof of Theorem 1 only in insignificant details.

The last theorem also makes it possible to establish the following assertion.

Corollary. Let the conditions of Theorem 1 be satisfied, and let the order \(n\) of the system be odd; then system (5.1) has at least one periodic solution in the domain \(\rho_1\leq \|x\|\leq \rho_2\).

Indeed, introduce the Poincaré transformation \(T\), which assigns to a point \(x_0\) of the domain \(\rho_1\leq \|x\|\leq \rho_2\) the point \(x(\omega,x_0)\), where \(x(t,x_0)\) is the solution of system (5.1) with initial data \(x=x_0\) at \(t=0\).

Let \(\mathrm{M}_0\) be the intersection of the surface \(\mathrm{M}\) with the hyperplane \(t=0\). According to the preceding theorem, the surface \(\mathrm{M}_0\) is homeomorphic to a sphere and is invariant with respect to the transformation \(T\). Since the surface \(\mathrm{M}_0\) is a topological sphere of even dimension \(n-1\), and the transformation \(T\), as is not hard to see, is homotopic to the identity, it follows that on the sphere \(\mathrm{M}_0\) the transformation \(T\) has a fixed point. This proves the corollary.

References

1. Krylov N. M., Bogolyubov N. N. Application of the methods of nonlinear mechanics to the theory of stationary oscillations. Kiev, 1934.

2. Bogoliubov N. N., Mitropolsky Ju. A. Contrib. to Differ. Equations, v. II, 1963.

3. Krasovskii N. N. PMM, 21, 63, 1957.

4. Pliss V. A. Nonlocal problems in the theory of oscillations. Moscow—Leningrad, 1964.

Received by the editors
April 26, 1966

Leningrad State University
named after A. A. Zhdanov