On the Existence of a Family of Periodic Solutions of a System of Differential Equations in the Case of Zero Roots

V. A. Pliss

In his classical work [1], A. M. Lyapunov considered a system of equations

\[ \frac{dx}{dt}=Ax+X(x), \tag{1.1} \]

where \(x\) is an \(n\)-dimensional vector with components \(x_1,\ldots,x_n\), \(A\) is a constant square matrix of order \(n\), and the components \(X_s\) of the vector function \(X\) are series in powers of \(x_s\), beginning with terms of at least second order. With respect to the matrix \(A\), it was assumed that it has a pair of purely imaginary eigenvalues \(\pm \lambda i\) \((\lambda>0)\), and that among the remaining eigenvalues there are no numbers of the form \(m\lambda i\), where \(m\) is an integer, positive, negative, or zero. Lyapunov proved that if the system (1.1) has a holomorphic integral of a special form, then there exists an analytic family of periodic solutions.

In the present paper we shall consider the case where the matrix \(A\) has two zero eigenvalues with a non-simple elementary divisor, and, under some additional assumptions, prove that if the system has an analytic integral of a special form, then there also exists a continuous family of periodic solutions. This family, generally speaking, is not analytic.

1. We shall assume that the matrix \(A\) has two zero eigenvalues with a non-simple elementary divisor, while the remaining eigenvalues have nonzero real parts. Under these assumptions the system (1.1) can be reduced to the form:

\[ \frac{dx}{dt}=y+X(x,y,\xi,\eta), \qquad \frac{dy}{dt}=Y(x,y,\xi,\eta), \]

\[ \frac{d\xi}{dt}=B\xi+P(x,y,\xi,\eta), \qquad \frac{d\eta}{dt}=D\eta+Q(x,y,\xi,\eta), \tag{2.2} \]

where \(x\) and \(y\) are scalar variables, \(\xi\) is a \(k\)-vector, \(\eta\) is an \(l\)-vector with components \(\xi_1,\ldots,\xi_k\) and \(\eta_1,\ldots,\eta_l\), respectively; \(B\) and \(D\) are constant square matrices of orders \(k\) and \(l\), the eigenvalues of the matrix \(B\) have negative real parts, and the eigenvalues of the matrix \(D\) have positive real parts; the scalar functions \(X\) and \(Y\), and the components \(P_s\) and \(Q_s\) of the vector functions \(P\) and \(Q\), are series in powers of \(x,y,\xi_i,\eta_i\), beginning with terms of at least second degree.

Concerning the nonlinearities \(X, Y, P\), and \(Q\), we shall make further assumptions, chosen so that the corresponding “first-approximation” system has a family of periodic solutions. Namely, we shall assume:

1) the function \(X\) vanishes for \(x=0, y=0\) and for \(y=0, \xi=0, \eta=0\), i.e.
\[ X(0,0,\xi,\eta)\equiv 0,\qquad X(x,0,0,0)\equiv 0; \]

2) the function \(Y\) has the form
\[ Y=-x^{2q-1}+g_1x^{2q}+\cdots+y(ax^\alpha+a_1x^{\alpha+1}+\cdots+Y_1(x,y,\xi,\eta)), \]
where \(Y_1(x,0,0,0)=0,\ q\geqslant 2,\ a\alpha>q-1\) or \(\alpha=\infty\), i.e. the function \(Y\), in its expansion into a power series, contains no terms of the form \(x^m y\);

3) the components \(P_s\) and \(Q_s\) of the vector-functions \(P\) and \(Q\), for \(\xi=0,\eta=0\), in the terms independent of \(y\), contain \(x\) only to powers not lower than the \(2q\)-th, and in the terms with the first power of \(y\) contain \(x\) only to powers not lower than the \((\alpha+1)\)-st if \(\alpha<2q-1\), and not lower than the \((2q+1)\)-st if \(\alpha\geqslant 2q-1\).

From the results of A. M. Lyapunov [2] it follows that in many cases, by a suitable change of variables, one can arrange that conditions 1), 2), 3) be satisfied.

Under the assumptions made, system (1.1) has a two-dimensional invariant surface. Indeed, as follows from Theorem (1.1) of [3], there exists an invariant surface
\[ \xi=\bar f(x,y,\eta), \tag{1.2} \]
where the vector-function \(\bar f\) is defined for sufficiently small \(|x|, |y|\), and \(\|\eta\|\) \((\|\eta\|=\sqrt{\eta_1^2+\cdots+\eta_n^2}\)—the Euclidean norm of the vector \(\eta\)), vanishes at the origin
\[ \bar f(0,0,0)=0 \tag{1.3} \]
and satisfies a Lipschitz condition with constant one:
\[ \|\bar f(x^{(1)},y^{(1)},\eta^{(1)})-\bar f(x^{(2)},y^{(2)},\eta^{(2)})\|\leq |x^{(1)}-x^{(2)}|+ \]
\[ +|y^{(1)}-y^{(2)}|+\|\eta^{(1)}-\eta^{(2)}\|. \tag{1.4} \]

Let us study the behavior of the solutions of system (1.1) lying on the invariant surface (1.2). These solutions are described by the following system:
\[ \frac{dx}{dt}=y+X(x,y,\bar f,\eta),\qquad \frac{dy}{dt}=Y(x,y,\bar f,\eta), \tag{1.5} \]
\[ \frac{d\eta}{dt}=D\eta+Q(x,y,\bar f,\eta). \]

Replacing \(t\) by \(-t\) in system (1.5) and using again Theorem 1.1 of [3], we show that system (1.5) has an invariant surface
\[ \eta=g(x,y), \tag{1.6} \]
where the vector-function \(g\) is defined for sufficiently small \(|x|\) and \(|y|\), vanishes at the origin
\[ g(0,0)=0. \tag{1.7} \]

and satisfies the Lipschitz condition:

\[ \left\|g\left(x^{(1)}, y^{(1)}\right)-g\left(x^{(2)}, y^{(2)}\right)\right\| <\left|x^{(1)}-x^{(2)}\right|+\left|y^{(1)}-y^{(2)}\right|. \tag{1.8} \]

Set \(\bar f(x,y,g(x,y))=f(x,y)\); then it is clear that the surface

\[ \xi=f(x,y),\qquad \eta=g(x,y) \tag{1.9} \]

is an invariant surface for the system (1.1).

It can be proved (we shall not dwell on this) that the functions \(f(x,y)\) and \(g(x,y)\) are continuously differentiable \(2q\) times.

Let us study in more detail the location of the invariant surface (1.9). For this purpose, following A. M. Lyapunov [2], we transform the system (1.1). Let \(C(\vartheta)\), \(S(\vartheta)\) be a solution of the system of differential equations

\[ \frac{dC}{d\vartheta}=-S,\qquad \frac{dS}{d\vartheta}=C^{2q-1} \tag{1.10} \]

with initial data \(C(0)=1,\ S(0)=0\). The functions \(C(\vartheta)\) and \(S(\vartheta)\), as is well known (see [2,4]), have a common period \(\omega>0\). We make the change of variables

\[ x=rC(\vartheta),\qquad y=-r^q S(\vartheta), \tag{1.11} \]

then the system (1.1) takes the form:

\[ \frac{dr}{dt}=r^{q+1}R_1(r,\vartheta)+rR_2(r,\vartheta,\xi,\eta), \]

\[ \frac{d\vartheta}{dt}=r^{q-1}+r^q\Theta_1(r,\vartheta)+\Theta_2(r,\vartheta,\xi,\eta), \tag{1.12} \]

\[ \frac{d\xi}{dt}=B\xi+P(r,\vartheta,\xi,\eta),\qquad \frac{d\eta}{dt}=D\eta+Q(r,\vartheta,\xi,\eta), \]

where the functions \(R_1\) and \(\Theta_1\) are series in powers of \(r\) with coefficients \(\omega\)-periodic with respect to \(\vartheta\), converging absolutely and uniformly for sufficiently small \(r\) and all \(\vartheta\); the functions \(R_2\) and \(\Theta_2\) are similar series in powers of \(r,\xi_s,\eta_s\), with \(R_2\) and \(\Theta_2\) vanishing for \(\xi=0,\eta=0\); the components \(P_s\) and \(Q_s\) of the vector-functions \(P\) and \(Q\) are also series in powers of \(r,\xi_s,\eta_s\) with coefficients \(\omega\)-periodic in \(\vartheta\). These series contain no terms below the second dimension in \(r,\xi_s,\eta_s\), and in the terms independent of \(\xi,\eta\), \(r\) enters in powers not lower than \(2q\).

In the variables \(r\) and \(\vartheta\), the invariant surface (1.9) is represented in the form

\[ \xi=f(r,\vartheta),\qquad \eta=g(r,\vartheta). \tag{1.13} \]

Let us prove the following assertion.

Theorem 1. There exists a positive number \(a\) such that, for sufficiently small \(r\), the inequalities

\[ \|f(r,\vartheta)\|<ar^{2q},\qquad \|g(r,\vartheta)\|<ar^{2q}. \tag{1.14} \]

hold.

Proof. Let \(v(\xi)\) be a quadratic form satisfying the equation

\[ (\operatorname{grad}v,B\xi)=-2\|\xi\|^2, \tag{1.15} \]

where the sign \((,)\) denotes, as usual, the scalar product. Since the eigenvalues of the matrix \(B\) are negative, it is well known

as is known, the form \(v\) exists and is positive definite. Analogously, by \(w(\eta)\) we denote the quadratic form satisfying the equation

\[ (\operatorname{grad} w,\, D\eta)=2\|\eta\|^2 . \tag{1.16} \]

The form \(w\) will also be positive definite.

Let us denote by \(h\) and \(H\) such numbers that

\[ h^2\|\xi\|^2 \leq v(\xi)\leq H^2\|\xi\|^2, \tag{1.17} \]

\[ h^2\|\eta\|^2 \leq w(\eta)\leq H^2\|\eta\|^2 . \tag{1.18} \]

Next let \(\beta\) be a number such that, for all \(\xi\) and \(\eta\), the inequalities

\[ \frac{\|\operatorname{grad} v\|}{2\sqrt v}\leq \beta,\qquad \frac{\|\operatorname{grad} w\|}{2\sqrt w}\leq \beta \tag{1.19} \]

hold.

The structure of the vector-functions \(P\) and \(Q\) is such that there exists a number \(\alpha>0\) such that, for sufficiently small \(r\), \(\|\xi\|\), and \(\|\eta\|\), the inequalities hold:

\[ \|P(r,\vartheta,\xi,\eta)\|\leq \alpha\bigl(r^{2q}+\|\xi\|^2+\|\eta\|^2+r\|\xi\|+r\|\eta\|\bigr), \tag{1.20} \]

\[ \|Q(r,\vartheta,\xi,\eta)\|\leq \alpha\bigl(r^{2q}+\|\xi\|^2+\|\eta\|^2+r\|\xi\|+r\|\eta\|\bigr). \tag{1.21} \]

Let us compute the total derivatives with respect to time of the functions \(\sqrt v\) and \(\sqrt w\):

\[ \frac{d}{dt}\sqrt v = \frac{1}{2\sqrt v}\,(\operatorname{grad} v,\, B\xi+P), \tag{1.22} \]

\[ \frac{d}{dt}\sqrt w = \frac{1}{2\sqrt w}\,(\operatorname{grad} w,\, B\eta+Q). \tag{1.23} \]

From these equalities and from the relations (1.15), (1.16), (1.19), (1.20), and (1.21) it follows that, for sufficiently small \(r\), \(\|\xi\|\), and \(\|\eta\|\), the inequalities hold:

\[ \frac{d}{dt}\sqrt v \leq -\frac{\|\xi\|^2}{\sqrt v} +\alpha\beta\bigl(r^{2q}+\|\xi\|^2+\|\eta\|^2+r\|\xi\|+r\|\eta\|\bigr), \]

\[ \frac{d}{dt}\sqrt w \geq \frac{\|\eta\|}{\sqrt w} -\alpha\beta\bigl(r^{2q}+\|\xi\|^2+\|\eta\|^2+r\|\xi\|+r\|\eta\|\bigr). \]

Hence, by virtue of (1.17) and (1.18), we obtain

\[ \frac{d}{dt}\sqrt v \leq -\frac{\sqrt v}{H^2} +\alpha\beta r^{2q} + \alpha\beta\left( \frac{v(\xi)}{h^2} + \frac{w(\eta)}{h^2} + r\frac{\sqrt{v(\xi)}}{h} + r\frac{\sqrt{w(\eta)}}{h} \right), \tag{1.24} \]

\[ \frac{d}{dt}\sqrt w \geq \frac{\sqrt w}{H^2} -\alpha\beta r^{2q} -\alpha\beta\left( \frac{v(\xi)}{h^2} + \frac{w(\eta)}{h^2} + r\frac{\sqrt{v(\xi)}}{h} + r\frac{\sqrt{w(\eta)}}{h} \right). \tag{1.25} \]

Since for sufficiently small \(r,\|\xi\|,\|\eta\|\) we obviously have

\[ \alpha\beta\left(\frac{v(\xi)}{h^2}+r\,\frac{\sqrt{v(\xi)}}{h}\right) < \frac{\sqrt{v(\xi)}}{4H^2}, \tag{1.26} \]

\[ \alpha\beta\left(\frac{w(\eta)}{h^2}+r\,\frac{\sqrt{w(\eta)}}{h}\right) < \frac{\sqrt{w(\eta)}}{4H^2}, \tag{1.27} \]

then from inequalities (1.24) and (1.25) we obtain

\[ \frac{d}{dt}\sqrt{v} \leq -\frac{3\sqrt{v}}{4H^2} +\alpha\beta r^{2q} +\alpha\beta\left(\frac{w(\eta)}{h^2} +r\,\frac{\sqrt{w(\eta)}}{h}\right), \tag{1.28} \]

\[ \frac{d}{dt}\sqrt{w} \geq \frac{3\sqrt{w}}{4H^2} -\alpha\beta r^{2q} -\alpha\beta\left(\frac{v(\xi)}{h^2} +r\,\frac{\sqrt{v(\eta)}}{h}\right). \tag{1.29} \]

Let now \(\rho>0\) be so small a number that on the invariant surface (1.13), for \(0\leq r\leq \rho\), inequalities (1.26) and (1.27) hold and, in addition,

\[ \left|r^{q+1}R_1(r,\vartheta)+rR_2(r,\vartheta,\xi,\eta)\right| < \frac{r}{8qH^2}. \tag{1.30} \]

The existence of such a \(\rho\) follows from the fact that the functions \(f(r,\vartheta)\), \(g(r,\vartheta)\) vanish for \(r=0\) and satisfy a Lipschitz condition.

Choose \(b>6H^2\alpha\beta\) so large that the inequalities

\[ \sqrt{v\bigl(f(\rho,\vartheta)\bigr)}<b\rho^{2q}, \qquad \sqrt{w\bigl(g(\rho,\vartheta)\bigr)}<b\rho^{2q}, \tag{1.31} \]

hold, and let us prove that for all \(r\in(0,\rho]\) the relations

\[ \sqrt{v\bigl(f(r,\vartheta)\bigr)}<br^{2q}, \qquad \sqrt{w\bigl(g(r,\vartheta)\bigr)}<br^{2q} \tag{1.32} \]

hold. Suppose, to the contrary, that there exists a point \(r_0,\vartheta_0\) \((0<r_0<\rho)\) at which one of inequalities (1.32) is violated. Let, for definiteness,

\[ \sqrt{w\bigl(g(r_0,\vartheta_0)\bigr)}\geq br_0^{2q},\qquad r_0>0 \tag{1.33} \]

and

\[ w\bigl(g(r_0,\vartheta_0)\bigr)\geq v\bigl(f(r_0,\vartheta_0)\bigr). \tag{1.34} \]

Denote by \(r(t),\vartheta(t),\xi(t),\eta(t)\) the solution of system (1.12) with initial data \(t=0,\ r=r_0,\ \vartheta=\vartheta_0,\ \xi=f(r_0,\vartheta_0),\ \eta=g(r_0,\vartheta_0)\). This solution lies on the invariant surface (1.13) at least as long as \(r(t)\leq \rho\).

We shall show that along the solution \(r(t),\vartheta(t),\xi(t),\eta(t)\) the inequalities

\[ \sqrt{w(\eta(t))}\geq br^{2q}(t), \tag{1.35} \]

\[ w(\eta(t))\geq v(\xi(t)) \tag{1.36} \]

hold for \(t\geq 0\), as long as \(r(t)\leq \rho\).

Let \(t^*\geq 0\) be a time such that one of inequalities (1.35), (1.36) (or both of them) becomes an equality at \(t=t^*\), and for \(0\leq t\leq t^*\), \(0\leq r(t)\leq \rho\), and both inequalities (1.35) and (1.36) are satisfied. Suppose first that

\[ w\bigl(\eta(t^*)\bigr)=v\bigl(\xi(t^*)\bigr), \tag{1.37} \]

then, by virtue of (1.26) and (1.27), one can write:

\[ \left.\frac{d\sqrt{v}}{dt}\right|_{t=t^*}\leq -\frac{\sqrt{v}}{2H^2}+\alpha\beta r^{2q} \]

and

\[ \left.\frac{d\sqrt{w}}{dt}\right|_{t=t^*}\geq \frac{\sqrt{w}}{2H^2}-\alpha\beta r^{2q}. \]

From these inequalities, the relations (1.35), (1.37), and \(b\geq 6\alpha\beta H^2\) imply that

\[ \left.\frac{d\sqrt{v}}{dt}\right|_{t=t^*}<0,\qquad \left.\frac{d\sqrt{w}}{dt}\right|_{t=t^*}>0. \tag{1.38} \]

It follows that for \(t<t^*\) and sufficiently close to \(t^*\), \(w<v\), while for \(t>t^*\), but sufficiently close to \(t^*\), \(w>v\).

Thus, in the case when equality (1.37) holds, \(t^*=0\), and for sufficiently small positive \(t\) the inequality (1.36) holds in the strict sense. Consequently, inequality (1.36) cannot be violated first.

Let now

\[ \sqrt{w(\eta(t^*))}=br^{2q}(t^*). \tag{1.39} \]

We have:

\[ \frac{d}{dt}br^{2q}=2qbr^{2q-1}\frac{dr}{dt}, \]

whence, by (1.30), we obtain

\[ \left.\frac{d}{dt}br^{2q}\right|_{t=t^*}<\frac{b}{4H^2}\,r^{2q}(t^*). \tag{1.40} \]

Since for \(t=t^*\) inequality (1.36) is satisfied, by virtue of (1.26) and (1.29) we can write:

\[ \left.\frac{d}{dt}\sqrt{w}\right|_{t=t^*} \geq \frac{\sqrt{w}}{2H^2}-\alpha\beta r^{2q}. \]

Hence, from equality (1.39), there follows the inequality:

\[ \left.\frac{d}{dt}\sqrt{w}\right|_{t=t^*} \geq \frac{br^{2q}(t^*)}{2H^2}-\alpha\beta r^{2q}(t^*). \tag{1.41} \]

Since \(b\geq 6\alpha\beta H^2\), it follows from inequalities (1.40) and (1.41) that

\[ \left.\frac{d}{dt}br^{2q}\right|_{t=t^*} < \left.\frac{d}{dt}\sqrt{w}\right|_{t=t^*}. \]

This inequality proves that for \(t<t^*\) and sufficiently close to \(t^*\) inequality (1.35) is violated, while for \(t>t^*\) and sufficiently close to \(t^*\) it is satisfied in the strict sense.

It follows that for \(t>0\) both inequalities (1.36) and (1.35) are satisfied in the strict sense as long as \(r(t)\leq \rho\).

Two possibilities are conceivable: either for all \(t>0\), \(r(t)<\rho\), or not. Suppose first that there exists such a \(t_1>0\) that \(r(t_1)=\rho\) and, for \(0\leq t<t_1\), \(r(t)<\rho\). Then, as was just proved, inequality (1.35) holds for all \(t\in[0,t_1]\), and, in particular, we have

\[ \sqrt{w(\eta(t_1))}\geq b\rho^{2q}. \]

Since the solution chosen by us lies on the invariant surface (1.13), the last inequality contradicts the second of inequalities (1.31).

Suppose now that, for all \(t \geqslant 0\), \(r(t)<\rho\). Then, for all \(t \geqslant 0\), inequalities (1.35) and (1.36) are satisfied, and from (1.29) it follows that, for all \(t \geqslant 0\), the inequality
\[ \frac{d}{dt}\sqrt{w}\geqslant \frac{\sqrt{w}}{2H^2}-\alpha\beta r^{2q}, \]
is satisfied, and, since \(b \geqslant 6\alpha\beta H^2\), we have
\[ \frac{d}{dt}\sqrt{w}\geqslant \frac{\sqrt{w}}{3H^2} \]
for all \(t \geqslant 0\). From this inequality it follows that \(w(\eta(t))\to\infty\) as \(t\to+\infty\), which is impossible, since the solution \(r(t), \vartheta(t), \xi(t), \eta(t)\) lies on the invariant surface (1.13) for all \(t \geqslant 0\).

The contradiction obtained proves inequalities (1.32).

Put \(a=\dfrac{b}{h}\); then from inequalities (1.17), (1.18), and (1.32) we obtain the assertion of the theorem.

1. We shall now assume that system (1.1) has a holomorphic integral of the form
   \[ \frac{1}{2q}x^{2q}+\frac{1}{2}y^2+F(x,y,\xi,\eta)=C, \tag{2.1} \]
   where the function \(F\) is a series in powers of \(x,y,\xi_s,\eta_s\), beginning with terms of not less than the second dimension; the function \(F(x,0,0,0)\), in its expansion as a series in powers of \(x\), contains no terms of dimension less than the \((2q+1)\)-st; the expansion of the function \(F(0,y,0,0)\) in powers of \(y\) contains no terms of degree less than the third; and the series representing the function \(F(x,y,0,0)\), in the terms linear with respect to \(y\), contains no \(x\) in a power lower than \(q+1\).

Under this assumption we shall prove that system (1.1) has a continuous family of periodic solutions. The following assertion is valid.

Theorem 2. If there exists an integral (2.1), then every solution lying on the invariant surface (1.9) is periodic.

Proof. Passing in the integral (2.1) to the variables \(r\) and \(\vartheta\) by formulas (1.11), we obtain
\[ \frac{1}{2}r^{2q}+F_1(r,\vartheta,\xi,\eta)=C. \tag{2.2} \]

The function \(F_1\) expands into a series in powers of \(r,\xi_s,\eta_s\) with \(\omega\)-periodic coefficients in \(\vartheta\); moreover, the expansion of the function \(F_1(r,\vartheta,\xi,\eta)\) begins with terms of not less than the second dimension, while that of the function \(F(r,\vartheta,0,0)\)—with terms of not less than the \((2q+1)\)-st dimension.

The integral (2.2) permits a more detailed study of the behavior of solutions located on the invariant surface (1.13). Substitute in the integral, instead of \(\xi\) and \(\eta\), the vector-functions \(f(r,\vartheta)\) and \(g(r,\vartheta)\); then we obtain the equality
\[ r^{2q}\left(\frac{1}{2}+\frac{F_1(r,\vartheta,f(r,\vartheta),g(r,\vartheta))}{r^{2q}}\right)=C; \tag{2.3} \]

hence we obtain

\[ r\left(\frac{1}{2}+\frac{F_1(r,\vartheta,f(r,\vartheta),g(r,\vartheta))}{r^{2q}}\right)^{\frac{1}{2q}}=c, \tag{2.4} \]

where \(c\) is a new arbitrary constant.

Since the functions \(f(r,\vartheta)\) and \(g(r,\vartheta)\) are continuously differentiable \(2q\) times and, by Theorem 1, are of order \(2q\) with respect to \(r\), equality (2.4) is solvable with respect to \(r\). Let \(r=r(c,\vartheta)\) be the solution of equation (2.4). It is clear that the function \(r(c,\vartheta)\) is continuously differentiable, has period \(\omega\) in \(\vartheta\), and \(r(0,\vartheta)\equiv 0\).

It follows from this that the family of curves

\[ r=r(c,\vartheta),\quad \xi=f(r(c,\vartheta),\vartheta),\quad \eta=g(r(c,\vartheta),\vartheta) \tag{2.5} \]

is a family of closed trajectories of system (1.1), which proves the theorem.

In conclusion, we note that if system (1.1) is canonical, then it always has a family of periodic solutions. Indeed, in this case the Hamiltonian function necessarily has the form of the left-hand side of equality (2.1), and then Theorem 2 is applicable.

References

1. Lyapunov A. M. The General Problem of the Stability of Motion. Moscow—Leningrad, 1950.
2. Lyapunov A. M. Study of one of the special cases of the problem of the stability of motion. Izv. Leningrad State Univ., 1963.
3. Pliss V. A. The reduction principle in the theory of stability of motion. Izv. Academy of Sciences of the USSR, Math. Ser., 28, issue 6, 1964.
4. Lyapunov A. M. Study of one of the special cases of the problem of the stability of motion. Collected Works, vol. II, 1956.

Received by the editors
November 3, 1964

Leningrad State University