Families of Periodic Solutions of Systems of Second-Order Differential Equations Without Dissipation

V. A. Pliss

In this paper we study questions of the existence and construction of families of periodic solutions of systems of ordinary differential equations in a neighborhood of an equilibrium state. A. M. Lyapunov in [1] considered, from this same point of view, the system of equations

\[ \frac{dx}{dt}=Ax+X(x), \tag{0.1} \]

where \(x\) is an \(n\)-dimensional vector with components \(x_1, x_2,\ldots,x_n\), \(A\) is a constant square matrix of order \(n\), and the components \(X_s\) of the vector-function \(X\) are series in powers of \(x_1,x_2,\ldots,x_n\) beginning with terms no lower than the second order of smallness. It was assumed with respect to the matrix \(A\) 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 a positive, negative, or zero integer, i.e. only the nonresonant case was considered. Under this assumption Lyapunov proved that if the system (0.1) has a formal family of periodic solutions, then it also has a true family, i.e. he proved the convergence of the series representing the formal family. In [2] an analogous assertion is proved for the resonant case.

We shall consider systems of second-order equations that do not contain first derivatives. For such systems M. A. Krasnosel’skii [3, 4] gave conditions for the existence of continual families of periodic solutions.

In the present paper conditions will be given for the existence of families of periodic solutions that differ from the conditions formulated in the works cited above. Under these conditions methods for constructing families of periodic solutions will be proved; in passing, some properties of such families will also be investigated.

Consider a system of second-order equations of the following form:

\[ \frac{d^2x_i}{dt^2}+\lambda_i^2 x_i = g_i(x_1,\ldots,x_n) \quad (i=1,\ldots,n), \tag{0.2} \]

where \(\lambda_i\) are positive numbers, and \(g_i(x_1,\ldots,x_n)\) are series in powers of \(x_1,\ldots,x_n\) without constant and linear terms, convergent for sufficiently small \(|x_j|\).

We note that every solution of the system (0.2) satisfying boundary conditions of the form

\[ \dot{x}_i(0)=\dot{x}_i(\omega)=0 \quad (i=1,\ldots,n), \tag{0.3} \]

has period \(2\omega\). Indeed, let \(x_i=\varphi_i(t)\) be a solution of system (0.2) with property (0.3). Define the functions \(\psi_i(t)\) on the interval \([-\omega,0]\) by the equalities

\[ \psi_i(t)=\varphi_i(-t)\quad (i=1,\ldots,n). \tag{0.4} \]

It is not difficult to see that the functions \(\psi_i(t)\) \((i=1,\ldots,n)\) constitute a solution of system (0.2) on the interval \([-\omega,0]\). Moreover, it is clear that \(\psi_i(0)=\varphi_i(0)\) and \(\dot\psi_i(0)=\dot\varphi_i(0)=0\). Thus the functions \(\psi_i(t)\) constitute a continuation of the solution \(\varphi_i(t)\) \((i=1,\ldots,n)\) to the interval \([-\omega,0]\). Since \(\psi_i(-\omega)=\varphi_i(\omega)\) and \(\dot\psi_i(-\omega)=-\dot\varphi_i(\omega)=0=\dot\varphi_i(\omega)\), it follows that \(\varphi_i(t)\) \((i=1,\ldots,n)\) is a \(2\omega\)-periodic solution of system (0.2). In this case all the functions \(\varphi_i(t)\) are even.

Definition 0.1. A periodic solution of system (0.2) with property (0.3) will be called symmetric.

In what follows we shall be interested only in symmetric periodic solutions.

§ 1. THE NONRESONANT CASE

Let us first consider the case where one of the numbers \(\lambda_i\) is such that all the others are not its integral multiples. Without loss of generality we may assume that the number \(\lambda_1\) has this property, i.e., that for all integers \(m\) the inequalities

\[ \lambda_i\ne m\lambda_1\quad (i=2,\ldots,n) \tag{1.1} \]

hold.

1. We shall show that in this case there exists an analytic family of symmetric periodic solutions whose periods are close to

\[ \frac{2\pi}{\lambda_1}. \]

Make in system (0.2) the change of variables \(\dot x_i=-\lambda_i y_i\) \((i=1,\ldots,n)\); then we obtain the system

\[ \dot x_i=-\lambda_i y_i,\quad \dot y_i=\lambda_i x_i+G_i(x_1,\ldots,x_n)\quad (i=1,\ldots,n), \tag{1.2} \]

where \(G_i(x_1,\ldots,x_n)=-\dfrac{1}{\lambda_i}g_i(x_1,\ldots,x_n)\). Passing in the first two equations of system (1.2) to polar coordinates by the formulas

\[ x_1=r\cos\varphi,\quad y_1=r\sin\varphi, \tag{1.3} \]

we obtain

\[ \dot r=G_1(r\cos\varphi,x_2,\ldots,x_n)\sin\varphi,\quad \dot\varphi=\lambda_1 r+G_1(r\cos\varphi,x_2,\ldots,x_n)\cos\varphi, \]

\[ \dot x_i=-\lambda_i y_i,\quad \dot y_i=\lambda_i x_i+G_i(r\cos\varphi,x_2,\ldots,x_n)\quad (i=2,\ldots,n). \tag{1.4} \]

As the parameter of the family we choose the initial datum of the quantity \(r\). Denote this parameter by \(a\). It is convenient to introduce the parameter \(a\) into the system of differential equations itself; for this purpose we make the change of variables:

\[ r=a(1+\rho),\quad x_i=a\xi_i,\quad y_i=a\eta_i\quad (i=2,\ldots,n). \tag{1.5} \]

Since the functions \(G_i\) are series in powers of their arguments without constant and linear terms, as a result of the substitution (1.5) we obtain the system

\[ \dot\rho=a\overline{R}(r,\varphi,\xi_2,\ldots,\xi_n,a), \tag{1.6} \]

\[ \begin{gathered} \dot{\varphi}=\lambda_1+a\Phi(r,\varphi,\xi_2,\ldots,\xi_n,a),\\ \dot{\xi}_i=-\lambda_i\eta_i,\quad \dot{\eta}_i=\lambda_i\xi_i+a\overline{Q}_i(r,\varphi,\xi_2,\ldots,\xi_n,a)\quad (i=2,\ldots,n), \end{gathered} \tag{1.6} \]

where \(\overline{R}\), \(\Phi\), and \(\overline{Q}_i\) are series in powers of \(r,\xi_2,\ldots,\xi_n,a\) with coefficients \(2\pi\)-periodic in \(\varphi\); these series converge for sufficiently small \(r,|\xi_2|,\ldots,|\xi_n|,a\).

Eliminating \(t\) from system (1.6), we obtain

\[ \frac{d\rho}{d\varphi}=aR(\rho,\varphi,\xi_2,\ldots,\xi_n,a), \]

\[ \frac{d\xi_i}{dt}=-\mu_i\eta_i+aP_i(\rho,\varphi,\xi_2,\ldots,\xi_n,a), \tag{1.7} \]

\[ \frac{d\eta_i}{dt}=\mu_i\xi_i+aQ_i(\rho,\varphi,\xi_2,\ldots,\xi_n,a)\quad (i=2,\ldots,n), \]

where \(\mu_i=\dfrac{\lambda_i}{\lambda_1}\) \((i=2,\ldots,n)\), and \(R,P_i,Q_i\) \((i=2,\ldots,n)\) are functions of the same character as \(\overline{R}\), \(\Phi\), and \(\overline{Q}_i\). In view of condition (1.1), all \(\mu_i\) are not integers.

Let \(\rho(\varphi,a,a_2,\ldots,a_n)\), \(\xi_i(\varphi,a,a_2,\ldots,a_n)\), \(\eta_i(\varphi,a,a_2,\ldots,a_n)\) \((i=2,\ldots,n)\) be a solution of system (1.7) with initial data \(\varphi=0\), \(\rho=0\), \(\xi_i=a_i\), \(\eta_i=0\) \((i=2,\ldots,n)\). The functions \(\xi_i(\varphi,a,a_2,\ldots,a_n)\), \(\eta_i(\varphi,a,a_2,\ldots,a_n)\) are analytic in their arguments for sufficiently small \(a,|a_2|,\ldots,|a_n|\); therefore they can be represented in the form

\[ \xi_i=\xi_{i0}(\varphi,a_2,\ldots,a_n)+aX_i(\varphi,a,a_2,\ldots,a_n), \]

\[ \eta_i=\eta_{i0}(\varphi,a_2,\ldots,a_n)+aY_i(\varphi,a,a_2,\ldots,a_n)\quad (i=2,\ldots,n), \tag{1.8} \]

where \(\xi_{i0},\eta_{i0},X_i,Y_i\) are analytic functions of their arguments for sufficiently small \(a,|a_2|,\ldots,|a_n|\), \(\xi_{i0}(0,a_2,\ldots,a_n)=a_i\), \(\eta_{i0}(0,a_2,\ldots,a_n)=0\), \(X_i(0,a,a_2,\ldots,a_n)=0\), \(Y_i(0,a,a_2,\ldots,a_n)=0\) \((i=2,\ldots,n)\). Substituting expression (1.8) into system (1.7), we obtain, for \(a=0\),

\[ \frac{d\xi_{i0}}{d\varphi}=-\mu_i\eta_{i0},\quad \frac{d\eta_{i0}}{d\varphi}=\mu_i\xi_{i0}. \tag{1.9} \]

Hence it follows that

\[ \xi_{i0}=a_i\cos\mu_i\varphi,\quad \eta_{i0}=a_i\sin\mu_i\varphi. \tag{1.10} \]

The right-hand side of the first equation of system (1.7) contains the factor \(a\), and therefore it is clear that the function \(\rho(\varphi,a,a_2,\ldots,a_n)\) also has the factor \(a\), i.e., it is representable in the form

\[ \rho=a\overline{\rho}(\varphi,a,a_2,\ldots,a_n). \tag{1.11} \]

Since \(y_1=r\sin\varphi\), and \(r=a(1+\rho)\), it follows from (1.11) that, for sufficiently small positive \(a\), \(y_1\) vanishes for the first time after \(\varphi=0\) at \(\varphi=\pi\). We shall try to choose the initial data \(a_i\) so that at \(\varphi=\pi\) all the functions \(y_i\) \((i=2,\ldots,n)\) vanish. For this it is necessary to satisfy the system of equations

\[ \eta_i(\pi,a,a_2,\ldots,a_n)=0\quad (i=2,\ldots,n). \tag{1.12} \]

This system, in view of (1.8) and (1.10), can be rewritten in the form

\[ a_i \sin \mu_i \pi + aY_i(\pi, a, a_2,\ldots,a_n)=0 \quad (i=2,\ldots,n). \tag{1.13} \]

Since, by condition (1.1), the \(\mu_i\) are not integers, it is clear that the system of equations (1.13) has a unique analytic solution \(a_i=a_i(a)\) \((i=2,\ldots,n)\) for sufficiently small \(a\).

Thus, on the family of solutions of system (1.7) with initial data \(\varphi=0\), \(\rho=0\), \(\xi_i=a_i(a)\), \(\eta_i=0\) \((i=2,\ldots,n)\), the equalities (1.12) are satisfied. From the second equation of system (1.6) it follows that on this family, for \(\varphi=\pi\), \(t\) is an analytic function of \(a\):

\[ t=\omega(a) \tag{1.14} \]

and

\[ \lim_{a\to0}\omega(a)=\frac{\pi}{\lambda_1}. \tag{1.15} \]

It follows from this that on the solutions of system (1.4) with initial data \(x_i=aa_i(a)\), \(y_i=0\) \((i=2,\ldots,n)\), the equalities

\[ \varphi(\omega(a))=\pi,\quad y_i(\omega(a))=0 \quad (i=2,\ldots,n) \tag{1.16} \]

are satisfied. Consequently, the family of solutions

\[ x_i=f_i(t,a)\quad (i=1,\ldots,n) \tag{1.17} \]

of system (0.2) with initial data \(x_1=a\), \(\dot x_1=0\), \(x_i=aa_i(a)\), \(\dot x_i=0\) \((i=2,\ldots,n)\) is a family of symmetric periodic solutions with period \(2\omega(a)\)*.

1. Let us note one property of the family (1.17). Suppose that the functions \(g_i(x_1,0,\ldots,0)\) \((i=2,\ldots,n)\) can be represented in the form

\[ g_i(x_1,0,\ldots,0)=A_{i0}x_1^{m_i}+A_{i1}x_1^{m_i+1}+\cdots \quad (i=2,\ldots,n), \tag{1.18} \]

where, among the numbers \(m_i\), there may, of course, also be improper ones (in this case \(g_i(x_1,0,\ldots,0)\equiv0\)). Put \(m=\min m_i\) \((i=2,\ldots,n)\). Then it is clear that

\[ Q_i(\rho,\varphi,0,\ldots,0,a)=B_{i0}(\rho,\varphi)a^{k_i}+\cdots \quad (i=2,\ldots,n), \tag{1.19} \]

where \(k_i\ge m-2\), and the omitted terms have order higher than \(k_i\) in \(a\). It follows from this that

\[ Y_i(\varphi,a,0,\ldots,0)=D_{i0}(\varphi)a^{l_i}+\cdots \quad (i=2,\ldots,n) \tag{1.20} \]

and \(l_i\ge m-2\). Consequently, the solution of the system of equations (1.13) has the form

\[ a_i(a)=a^{m-1}\bar a_i(a)\quad (i=2,\ldots,n), \tag{1.21} \]

where \(\bar a_i(a)\) are analytic functions for sufficiently small \(a\).

Thus, the family of periodic solutions (1.17) can be represented in the form

\[ x_1=\sum_{k=1}^{\infty} f_1^{(k)}(t)a^k,\quad x_i=\sum_{k=m}^{\infty} f_i^{(k)}(t)a^k \quad (i=2,\ldots,n), \tag{1.22} \]

i.e. in the family (1.17) the functions \(x_i\) \((i=2,\ldots,n)\) have order not lower than the \(m\)-th in \(a\).

* The fact of existence of a family of periodic solutions of system (0.1) in the nonresonant case can, of course, also be proved otherwise, relying on the results of A. M. Lyapunov [1].

In conclusion, we note that the family (1.17) can be constructed in the form (1.22) by the method of undetermined coefficients.

§ 2. RESONANT CASE. SYSTEMS OF TWO EQUATIONS

In this section we shall consider systems of two equations of the form (0.1), i.e. the case where \(n=2\); here we shall assume that the numbers \(\lambda_1\) and \(\lambda_2\) are equal to each other. In this case, by changing the scale of the time \(t\), one can arrange that \(\lambda_1=\lambda_2=1\). We shall consider the system

\[ \ddot{x}_1+x_1=g_1(x_1,x_2), \]

\[ \ddot{x}_2+x_2=g_2(x_1,x_2). \tag{2.1} \]

As before, we assume that the functions \(g_1(x_1,x_2)\) and \(g_2(x_1,x_2)\) are represented by series in powers of \(x_1\) and \(x_2\), beginning with terms of no lower than second order and converging for sufficiently small \(|x_1|, |x_2|\).

We shall be interested in questions of the existence of families of symmetric periodic solutions with periods close to \(2\pi\).

1. Let \(M\) be a set of points of the plane with coordinates \((a,b)\), and let the point \((0,0)\) be a limit point for \(M\). The set of solutions \(x_1=x_1(t,a,b)\), \(x_2=x_2(t,a,b)\) with initial data \(x_1(0,a,b)=a\), \(x_2(0,a,b)=b\), \(\dot{x}_1(0,a,b)=0\), \(\dot{x}_2(0,a,b)=0\), possessing the property

\[ \dot{x}_1(\omega(a,b),a,b)=\dot{x}_2(\omega(a,b),a,b)=0 \tag{2.2} \]

for \((a,b)\in M\), will be called a family of symmetric periodic solutions.

As was shown above, all solutions of such a family have period \(2\omega(a,b)\).

In what follows we shall be interested only in those families of symmetric periodic solutions whose periods are close to \(2\pi\), i.e. for which the relation

\[ \omega(a,b)\to\pi \quad \text{as } (a,b)\in M,\ (a,b)\to(0,0). \tag{2.3} \]

holds.

For such families the following assertion is valid.

Theorem 2.1. Suppose that the function \(g_2(x_1,0)\) is represented in the form

\[ g_2(x_1,0)=\alpha x_1^k+\cdots, \tag{2.4} \]

where \(\alpha\ne0\), \(k\) is an odd number, and the unwritten terms have order higher than \(k\). Suppose that \(M\) is the set of initial data of a family of symmetric periodic solutions with periods close to \(2\pi\). Then there exists a \(\chi>0\) such that, for sufficiently small \(|a|\), \(|b|\) and \((a,b)\in M\), the inequality

\[ |b|>\chi |a|^{k-2} \tag{2.5} \]

holds.

Proof. In system (2.1) make the change of variables \(\dot{x}_1=-y_1\), \(\dot{x}_2=-y_2\); then we obtain the system

\[ \dot{x}_1=-y_1,\quad \dot{y}_1=x_1-g_1(x_1,x_2), \]

\[ \dot{x}_2=-y_2,\quad \dot{y}_2=x_2-g_2(x_1,x_2). \tag{2.6} \]

In the first two equations of this system let us pass to polar coordinates by the formulas

\[ x_1=r\cos\varphi,\quad y_1=r\sin\varphi, \]

then we shall have

\[ \dot r=-g_1(r\cos\varphi,x_2)\sin\varphi,\quad r\dot\varphi=r-g_1(r\cos\varphi,x_2)\cos\varphi . \tag{2.7} \]

In what follows we shall assume that \(a>0\) and \(b>0\) (generality is not lost by this assumption, since, if necessary, the corresponding \(x_i\) can be replaced by \(-x_i\)).

In proving inequality (2.5) one may restrict oneself to the case when for \((a,b)\in M\), \(b<a\) (in the opposite case it would suffice to pass to the corresponding submanifold).

Make the change of variables

\[ r=a(1+\rho),\quad x_2=ax,\quad y_2=ay . \]

In the new variables system (2.6) takes the following form:

\[ \frac{d\rho}{dt}=aP(\rho,\varphi,x,a),\quad \frac{d\varphi}{dt}=1+\Phi(\rho,\varphi,x,a), \]

\[ \frac{dx}{dt}=-y,\quad \frac{dy}{dt}=x+Y(\rho,\varphi,x,a), \tag{2.8} \]

where

\[ aP(\rho,\varphi,x,a)=- \frac{g_1\bigl(a(1+\rho)\cos\varphi,ax\bigr)\sin\varphi}{a}, \]

\[ \Phi(\rho,\varphi,x,a)=- \frac{g_1\bigl(a(1+\rho)\cos\varphi,ax\bigr)\cos\varphi}{a(1+\rho)}, \]

\[ Y(\rho,\varphi,x,a)=- \frac{g_2\bigl(a(1+\rho)\cos\varphi,ax\bigr)}{a}. \]

The functions \(P,\Phi,Y\) are analytic in their arguments for sufficiently small \(a,\rho,x\) and are \(2\pi\)-periodic in \(\varphi\).

The initial data for system (2.8) are as follows:

\[ t=0,\ \varphi=0,\ \rho=0,\ y=0,\ x=\frac{b}{a}=c. \tag{2.9} \]

From the form of the function \(Y\) and from condition (2.4) of the theorem it follows that the first terms in the expansion of the function \(Y\) in a series have the form

\[ Y=-\alpha a^{k-1}\cos^k\varphi+\beta ax\cos\varphi+\gamma ax^2+\cdots . \tag{2.10} \]

The terms not written out in this formula either have order not less than the \(k\)-th with respect to \(a\) and \(\rho\), or contain the factor \(x\) and have order not less than the second with respect to \(a\) and \(\rho\).

The function \(\Phi\) has the following representation:

\[ \Phi=\alpha_1 a\cos^3\varphi+\beta_1 ax\cos^2\varphi+\gamma_1 ax^2\cos\varphi+\cdots, \tag{2.11} \]

where the terms not written out have order not less than the second with respect to \(a\) and \(\rho\).

Eliminating \(t\) in system (2.8), we obtain

\[ \frac{d\rho}{d\varphi}=aP(1+\Phi)^{-1}, \]

\[ \frac{dx}{d\varphi}=-y(1+\Phi)^{-1},\quad \frac{dy}{d\varphi}=(x+Y)(1+\Phi)^{-1}. \tag{2.12} \]

From (2.10) and (2.11) it follows that the last two equations of system (2.12) have the form

\[ \frac{dx}{d\varphi}=-y+ay\left(\alpha_1\cos^3\varphi+\beta_1 x\cos^2\varphi+\gamma_1\cos\varphi+\ldots\right), \]

\[ \begin{aligned} \frac{dy}{d\varphi}={}&x-\alpha a^{k-1}\cos^k\varphi+\beta ax\cos\varphi+\gamma ax^2+\\ &+ax\left(\alpha_1\cos^3\varphi+\beta_1x\cos^2\varphi+\gamma_1x^2\cos\varphi\right)+\ldots . \end{aligned} \tag{2.13} \]

From the first equation of system (2.8) and the third of the initial conditions (2.9), it follows that the quantity \(\rho\) is of order \(a\) for bounded \(\varphi\); therefore the unwritten terms in the right-hand sides of system (2.13) either have order higher than \(k-1\) in \(a\), or contain \(x\) or \(y\) as a factor and have order lower than the second in \(a\).

The solution of system (2.12) is represented by analytic functions of \(a\) and \(c\) for sufficiently small values of these variables and for all \(\varphi\in[0,\pi]\). Let us find the first terms of the expansion of these functions in powers of \(a\) and \(c\):

\[ \begin{aligned} x(\varphi,a,c)={}&x_{01}(\varphi)c+x_{k-10}(\varphi)a^{k-1}+\\ &+x_{11}(\varphi)ac+x_{12}(\varphi)ac^2+x_{13}(\varphi)ac^3+\ldots, \end{aligned} \]

\[ \begin{aligned} y(\varphi,a,c)={}&y_{01}(\varphi)c+y_{k-10}(\varphi)a^{k-1}+y_{11}(\varphi)ac+\\ &+y_{12}(\varphi)ac^2+y_{13}(\varphi)ac^3+\ldots, \end{aligned} \tag{2.14} \]

where the unwritten terms either have order in \(a\) not lower than the \(k\)-th, or contain \(c\) and have order in \(a\) not lower than the second.

Substituting (2.14) into (2.13), we find

\[ \frac{dx_{01}}{d\varphi}=-y_{01},\qquad \frac{dy_{01}}{d\varphi}=x_{01}, \]

whence, by virtue of (2.9), we obtain

\[ x_{01}=\cos\varphi,\qquad y_{01}=\sin\varphi. \tag{2.15} \]

Similarly,

\[ \frac{dx_{k-10}}{d\varphi}=-y_{k-10},\qquad \frac{dy_{k-10}}{d\varphi}=x_{k-10}-\alpha\cos^k\varphi, \tag{2.16} \]

whence it follows that

\[ x_{k-10}=\bar{\alpha}\varphi\sin\varphi+\bar{\psi}(\varphi),\qquad y_{k-10}=-\bar{\alpha}\varphi\cos\varphi+\bar{\psi}(\varphi), \tag{2.17} \]

where \(\bar{\alpha}\ne0\), and the function \(\bar{\psi}\) has the property

\[ \bar{\psi}(0)=\bar{\psi}(\pi)=0. \tag{2.18} \]

For the functions \(x_{11}, y_{11}\) we obtain the system

\[ \frac{dx_{11}}{d\varphi}=-y_{11}+\alpha_1\cos^3\varphi\sin\varphi, \]

\[ \frac{dy_{11}}{d\varphi}=x_{11}+\beta\cos^2\varphi+\alpha_1\cos^4\varphi. \]

whence it follows that

\[ y_{11}(0)=y_{11}(\pi)=0. \tag{2.19} \]

Similarly we obtain the equalities

\[ y_{12}(0)=y_{13}(0)=y_{12}(\pi)=y_{13}(\pi)=0. \tag{2.20} \]

From the equalities (2.14), (2.15), (2.17), (2.18), (2.19), and (2.20) it follows that \(y\) for \(\varphi=\pi\) has the form

\[ y(\pi,a,c)=\bar{\alpha}\pi a^{k-1}+\bar{\beta}a^2c+\ldots, \tag{2.21} \]

where the terms not written out either have order not lower than the \(k\)-th in \(a\), or contain the factor \(c\) and have order not lower than the third in \(a\), or contain the factor \(a^2c^2\).

Since \(y_t=r\sin\varphi\), and \(r=a(1+\rho)\), it follows from the first equation of system (2.8) that the function \(y_1\) vanishes for the first time after \(t=0\) at \(\varphi=\pi\). Therefore, in order that the set \(M\) be the set of initial data of a family of symmetric periodic solutions with periods close to \(2\pi\), it is necessary and sufficient that for \((a,b)\in M\) the equality \(y(\pi)=0\) hold, or

\[ \bar{\alpha}\pi a^{k-1}+\bar{\beta}a^2c+\ldots=0. \tag{2.22} \]

Hence it follows that, under the hypotheses of the theorem, there exists a \(\chi>0\) such that, for sufficiently small \(a\) and \(c\), the inequality \(c>\chi a^{k-3}\) is satisfied; multiplying this inequality by \(a\) and taking into account that \(c=b/a\), we obtain (2.5). The theorem is proved.

In a number of cases it turns out that the presence in the expansion of the function \(g_2(x_1,0)\) of a series of even powers of \(x_1\) does not affect the conclusion of Theorem 2.1. In particular, the following is true.

Theorem 2.2. Suppose that the expansions of the functions \(g_1(x_1,x_2)\) and \(g_2(x_1,x_2)\) begin with terms of degree not lower than the third, and the function \(g_2(x_1,0)\) can be represented in the form

\[ g_2(x_1,0)=\delta x_1^{k-1}+\alpha x_1^k+\ldots, \tag{2.23} \]

where \(\alpha\ne0\), \(k\) is an odd number, \(k>3\), and the terms not written out have order higher than \(k\). Suppose that \(M\) is the set of initial data of a family of symmetric periodic solutions with periods close to \(2\pi\). Then there exists a \(\chi>0\) such that, for sufficiently small \(|a|\), \(|b|\) and \((a,b)\in M\), the inequality

\[ |b|>\chi |a|^{k-2} \tag{2.24} \]

is satisfied.

Proof. Just as in the proof of the preceding theorem, we pass from system (2.1) to system (2.6), and from it to system (2.8). In the case under consideration, the functions \(aP\), \(\Phi\), and \(Y\) will contain the factor \(a^2\). The initial data for system (2.8) in the case under consideration will be the same, (2.9).

From the form of the function \(Y\) and from condition (2.23) of the theorem being proved it follows that, in the case under consideration, the function \(Y\) can be represented in the form

\[ Y=-\delta a^{k-2}\cos^{k-1}\varphi-\alpha a^{k-1}\cos^k\varphi+\ldots . \tag{2.25} \]

The terms not written out in this formula either have order not lower than the \(k\)-th in \(a\), or contain a factor of the form \(a^{k-2}\rho\), or contain the factor \(x\) and have order not lower than the second in \(a\).

Eliminating \(t\) in system (2.8), we arrive, just as before, at system (2.12). From (2.25) and the fact that the functions \(Y\) and \(\Phi\) contain

by the factor \(a^2\), it follows that the last two equations of system (2.12) can be written in the form

\[ \frac{dx}{d\varphi}=-y+\ldots, \]

\[ \frac{dy}{d\varphi}=x-\delta a^{k-2}\cos^{k-1}\varphi-\alpha a^{k-1}\cos^k\varphi+\ldots . \tag{2.26} \]

From the first equation of system (2.8), in view of the fact that \(aP\) contains the factor \(a^2\), and from the third condition (2.9), it follows that for bounded \(\varphi\) the quantity \(\rho\) has order no lower than the second in \(a\); therefore the terms not written out in the right-hand sides of system (2.26) either have order no lower than the \(k\)-th in \(a\), or contain \(x\) or \(y\) and have order no lower than the second in \(a\).

The solution of system (2.26) is represented by analytic functions of \(a\) and \(c\) for sufficiently small values of these variables and for all \(\varphi\in[0,\pi]\). Let us find the first terms in the expansion of these functions in a series in \(a\) and \(c\):

\[ x(\varphi,a,c)=x_{01}(\varphi)c+x_{k-20}(\varphi)a^{k-2}+x_{k-10}(\varphi)a^{k-1}+\ldots, \]

\[ y(\varphi,a,c)=y_{01}(\varphi)c+y_{k-20}(\varphi)a^{k-2}+y_{k-10}(\varphi)a^{k-1}+\ldots, \tag{2.27} \]

where the terms not written out either have, with respect to \(a\), order no lower than the \(k\)-th, or contain \(c\) and have order with respect to \(a\) no lower than the second.

Just as in the proof of the preceding theorem, we establish that the functions \(x_{01}\) and \(y_{01}\) have the form (2.15).

The equations for \(x_{k-20}\) and \(y_{k-20}\) in our case have the form

\[ \frac{dx_{k-20}}{d\varphi}=-y_{k-20},\qquad \frac{dy_{k-20}}{d\varphi}=x_{k-20}-\delta\cos^{k-1}\varphi, \]

whence, by virtue of the oddness of \(k\), it follows that the function \(y_{k-20}(\varphi)\) has the property

\[ y_{k-20}(0)=y_{k-20}(\pi)=0. \tag{2.28} \]

The equations for \(x_{k-10}\), \(y_{k-10}\) in the case under consideration have the form (2.16); consequently, these functions themselves are represented in the form (2.17)—(2.18).

From the equalities (2.27), (2.15), (2.28), and (2.17) it follows that the function \(y\) for \(\varphi=\pi\) takes the form

\[ y(\pi,a,c)=\bar{\alpha}\pi a^{k-1}+\bar{\beta}a^2c+\ldots, \tag{2.29} \]

where the terms not written out have the same meaning as in equality (2.21).

From (2.29), just as before, we obtain equation (2.22), and from it, assertion (2.24) of our theorem. The theorem is proved.

1. For what follows it is convenient in system (2.6) to pass to polar coordinates for \(x_1,y_1\) and \(x_2,y_2\). Put

\[ x_1=r_1\cos\varphi,\quad y_1=r_1\sin\varphi;\quad x_2=\rho_1\cos\vartheta,\quad y_2=\rho_1\sin\vartheta, \tag{2.30} \]

then we obtain the system

\[ \dot r_1=-g_1(r_1\cos\varphi,\rho_1\cos\vartheta)\sin\varphi,\quad r_1\dot\varphi=r_1-g_1(r_1\cos\varphi,\rho_1\cos\vartheta)\cos\varphi, \]

\[ \dot\rho_1=-g_2(r_1\cos\varphi,\rho_1\cos\vartheta)\sin\vartheta,\quad \rho_1\dot\vartheta=\rho_1-g_2(r_1\cos\varphi,\rho_1\cos\vartheta)\cos\vartheta. \tag{2.31} \]

Definition 2.1. Let \(x_1(t,a,b)\), \(x_2(t,a,b)\), \((a,b)\in M\), be a family of symmetric periodic solutions of system (2.1) with periods close to \(2\pi\). We shall call this family circular if there exist functions \(\chi_1(b)\) and \(\chi_2(a)\) such that

\[ \lim_{b\to 0}\chi_1(b)=\infty,\qquad \lim_{a\to 0}\chi_2(a)=\infty, \tag{2.32} \]

and, for \((a,b)\in M\), the inequalities

\[ |a|>\chi_1(b)|g_1(0,b)|,\qquad |b|>\chi_2(a)|g_2(a,0)|. \tag{2.33} \]

It follows from the very form of the equations for \(r_1\) and \(\rho_1\) that in circular families the time derivatives of the radius vectors (i.e., the quantities \(\dot r_1\) and \(\dot \rho_1\)) have an order of smallness higher than the initial data of the radius vectors (i.e., the quantities \(a\) and \(b\)), and, consequently, as the initial data in a circular family decrease, the quantities \(r_1\) and \(\rho_1\) tend to constants.

When the conditions of Theorems 2.1 and 2.2 (and, respectively, the conditions symmetric to them) are fulfilled, all families of symmetric periodic solutions with periods close to \(2\pi\) are circular.

We shall study families of circular periodic solutions. In doing so, as before, we shall assume that \(a>0\), \(b>0\) (if this were not so, it would suffice to replace the corresponding \(x_i\) by \(-x_i\)). In system (2.31) make the substitution

\[ r_1=a(1+r),\qquad \rho_1=b(1+\rho), \tag{2.34} \]

then we obtain

\[ a\dot r=-g_1\bigl(a(1+r)\cos\varphi,\ b(1+\rho)\cos\vartheta\bigr)\sin\varphi, \]

\[ a(1+r)\dot\varphi=a(1+r)-g_1\bigl(a(1+r)\cos\varphi,\ b(1+\rho)\cos\vartheta\bigr)\cos\varphi, \]

\[ b\dot\rho=-g_2\bigl(a(1+r)\cos\varphi,\ b(1+\rho)\cos\vartheta\bigr)\sin\vartheta, \tag{2.35} \]

\[ b(1+\rho)\dot\vartheta=b(1+\rho)-g_2\bigl(a(1+r)\cos\varphi,\ b(1+\rho)\cos\vartheta\bigr)\cos\vartheta. \]

The initial data for this system are as follows: \(\varphi=0\), \(\vartheta=0\), \(r=0\), \(\rho=0\) for \(t=0\).

Suppose that the lowest exponent in the expansion of the function \(g_1(0,x_2)\) in a series in \(x_2\) is equal to \(m\), and the lowest exponent in the expansion of the function \(g_2(x_1,0)\) is equal to \(k\). Here, of course, the numbers \(m\) and \(k\) may also be improper, i.e., the functions \(g_1(0,x_2)\) and \(g_2(x_1,0)\) may be identically zero. Thus, the functions \(g_1(x_1,x_2)\) and \(g_2(x_1,x_2)\) can be represented in the form

\[ g_1(x_1,x_2)=x_2^m\bar g_1(x_2)+x_1G_1(x_1,x_2), \]

\[ g_2(x_1,x_2)=x_1^k\bar g_2(x_1)+x_2G_2(x_1,x_2), \tag{2.36} \]

where \(\bar g_1(x_2)\) and \(\bar g_2(x_1)\) are analytic functions and \(\bar g_1(0)\ne 0\), \(\bar g_2(0)\ne 0\) (in the case where \(m=\infty\) or \(k=\infty\), the first term in the corresponding equality is simply absent), the functions \(G_1\) and \(G_2\) are analytic, and \(G_1(0,0)=0\), \(G_2(0,0)=0\).

Put

\[ R_1(r,\rho,\varphi,\vartheta,a,b,\lambda)=-\lambda(1+\rho)^m\cos^m\vartheta\,\bar g_1\bigl(b(1+\rho)\cos\vartheta\bigr)\sin\varphi- \]

\[ {}-(1+r)G_1\bigl(a(1+r)\cos\varphi,\ b(1+\rho)\cos\vartheta\bigr)\sin\varphi\cos\varphi, \]

\[ \Phi(r,\rho,\varphi,\vartheta,a,b,\lambda) = -\lambda\frac{(1+\rho)^m}{1+r}\cos^m\vartheta\,\bar g_1(b(1+\rho)\cos\vartheta)\cos\varphi \]
\[ {}-G_1(a(1+r)\cos\varphi,\ b(1+\rho)\cos\vartheta)\cos^2\varphi, \]

\[ P_1(r,\rho,\varphi,\vartheta,a,b,\mu) = -\mu(1+r)^k\cos^k\varphi\,\bar g_2(a(1+r)\cos\varphi)\sin\vartheta \]
\[ {}-(1+\rho)G_2(a(1+r)\cos\varphi,\ b(1+\rho)\cos\vartheta)\sin\vartheta\cos\vartheta, \]

\[ \theta_1(r,\rho,\varphi,\vartheta,a,b,\mu) = -\mu\frac{(1+r)^k}{1+\rho}\cos^k\varphi\,\bar g_2(a(1+r)\cos\varphi)\cos\vartheta \]
\[ {}-G_2(a(1+r)\cos\varphi,\ b(1+\rho)\cos\vartheta)\cos^2\vartheta. \]

The functions \(R_1,\Phi,P_1\) and \(\theta_1\) are expanded in series in powers of \(r,\rho,a,b,\lambda,\mu\); these series have coefficients that are \(2\pi\)-periodic in \(\varphi\) and \(\vartheta\), and converge for sufficiently small \(r,\rho,a,b,\lambda,\mu\).

Consider the system of equations

\[ \frac{dr}{dt}=R_1,\qquad \frac{d\varphi}{dt}=1+\Phi,\qquad \frac{d\rho}{dt}=P_1,\qquad \frac{d\vartheta}{dt}=1+\theta_1 \tag{2.37} \]

with initial data \(r=\varphi=\rho=\vartheta=0\) at \(t=0\). It is not difficult to see that system (2.37) passes into (2.35) when \(\lambda=b^m/a,\ \mu=a^k/b\).

Eliminating \(t\) in system (2.37), we then obtain

\[ \frac{dr}{d\varphi}=R,\qquad \frac{d\rho}{d\varphi}=P,\qquad \frac{d\vartheta}{d\varphi}=1+\theta, \tag{2.38} \]

where

\[ R=\frac{R_1}{1+\Phi},\qquad P=\frac{P_1}{1+\Phi},\qquad \theta=\frac{1+\theta_1}{1+\Phi}-1. \]

The functions \(R,P\) and \(\theta\) are functions of the same nature as \(R_1,P_1,\Phi\) and \(\theta_1\). Moreover, it is clear that \(R=P=\theta=0\) for \(a=b=\lambda=\mu=0\); therefore, for sufficiently small \(a,b,\lambda\) and \(\mu\), the solution of system (2.38) with initial data \(r=\rho=\vartheta=0\) at \(\varphi=0\) is analytic in the parameters \(a,b,\lambda\) and \(\mu\).

From the definition of a symmetric periodic solution (Definition 0.1) it follows that if \(a\) and \(b\) are such that, on the solution of system (2.38) with initial data \(r=\rho=\vartheta=\varphi=0\), the relation

\[ \vartheta(\pi,a,b,\lambda,\mu)=\pi \quad\text{for}\quad \lambda=b^m/a,\ \mu=a^k/b, \tag{2.39} \]

holds, and the quantities \(b^m/a,\ a^k/b\) are sufficiently small, then the solution of system (2.1) with initial data \(x_1=a,\ \dot x_1=0,\ x_2=b,\ \dot x_2=0\) is a symmetric periodic solution.

On the other hand, if a set \(M\) of points \((a,b)\) is a set of initial data of a circular family of periodic solutions with periods close to \(2\pi\), then it is clear that, for sufficiently small \(a\) and \(b\) such that \((a,b)\in M\), the quantities \(\lambda=b^m/a,\ \mu=a^k/b\) are arbitrarily small and relation (2.39) is satisfied. Hence the assertion of the following theorem follows.

Theorem 2.3. In order that a set \(M\) of points \((a,b)\), having the point \((0,0)\) as its limit point, be a set of initial data of a circular family of symmetric periodic solutions with periods,

close to \(2\pi\), it is necessary and sufficient that inequality (2.33) and equality (2.39) be satisfied.

1. We now turn directly to the question of the existence and construction of families of symmetric periodic solutions with periods close to \(2\pi\). The following is true.

Theorem 2.4. Suppose that the functions \(g_1(x_1,x_2)\) and \(g_2(x_1,x_2)\) are represented in the form

\[ \begin{aligned} g_1(x_1,x_2)&=W_1(x_1,x_2)+\ldots,\\ g_2(x_1,x_2)&=W_2(x_1,x_2)+\ldots, \end{aligned} \tag{2.40} \]

where \(W_1(x_1,x_2)\) and \(W_2(x_1,x_2)\) are forms of degree \(k\), \(k\) is an odd number, and the unwritten terms have order higher than \(k\). If the form of degree \((k+1)\)

\[ W(x_1,x_2)=x_2W_1(x_1,x_2)-x_1W_2(x_1,x_2) \tag{2.41} \]

vanishes only for \(x_1=x_2=0\), then system (2.1) has no family of symmetric periodic solutions with periods close to \(2\pi\).

Proof. We shall assume, for definiteness, that the form (2.41) is positive definite. This is possible only under the condition that

\[ W(0,x_2)=\alpha x_2^k\;(\alpha>0),\quad W_2(x_1,0)=\beta x_1^k\;(\beta<0). \tag{2.42} \]

Hence, by Theorem 2.1, it follows that any family of symmetric periodic solutions with periods close to \(2\pi\) is circular; therefore it suffices to prove that system (2.1) has no circular families. In doing so, one may restrict oneself only to the case \(a>0,\ b>0\), since all the remaining cases are exhausted by replacing \(x_i\) by \(-x_i\) for the corresponding \(i\).

Set

\[ \begin{aligned} x_1 w_1(x_1,x_2)&=W_1(x_1,x_2)-W_1(0,x_2),\\ x_2 w_2(x_1,x_2)&=W_2(x_1,x_2)-W_2(x_1,0); \end{aligned} \tag{2.43} \]

thus \(w_1\) and \(w_2\) are forms of degree \((k-1)\).

Now consider the set \(M\) of points \((a,b)\) (\(a\) and \(b\) positive and sufficiently small) having the point \((0,0)\) as its limit point and possessing property (2.32)—(2.33). We shall show that equality (2.39) cannot be realized on this set.

Expand the solution of system (2.38) in a series in powers of \(a,b,\lambda,\mu\) and substitute this expansion into the third equation of this system; then, by virtue of (2.42), (2.43), we obtain

\[ \frac{d\vartheta}{d\varphi} = 1-\beta\mu\cos^{k+1}\varphi -w_2(a\cos\varphi,b\cos\varphi)\cos^2\varphi + \]

\[ +\alpha\lambda\cos^{k+1}\varphi +w_1(a\cos\varphi,b\cos\varphi)\cos^2\varphi+\ldots, \tag{2.44} \]

where the unwritten part of the series standing on the right-hand side, in the terms not depending on \(\lambda\) and \(\mu\), has order not lower than the \(k\)-th in \(a\) and \(b\), and in the terms linear with respect to \(\lambda\) and \(\mu\) contains as a factor either \(a\) or \(b\).

Integrating equality (2.44) with respect to \(\varphi\) from \(0\) to \(\pi\) and multiplying by \(ab\), we obtain:

\[ [\vartheta(\pi,a,b,\lambda,\mu)-\pi]ab= \]

\[ = ab\left\{[\alpha\lambda+w_1(a,b)-\beta\mu-w_2(a,b)] \int_0^\pi \cos^{k+1}\varphi\,d\varphi+\ldots\right\}. \tag{2.45} \]

In this equality, the unwritten part of the series in the terms not depending on \(\lambda\) and \(\mu\) has order not lower than the \(k\)-th in \(a\) and \(b\), while the terms of this part of the series containing \(\lambda\) or \(\mu\) have one of the quantities \(a\), \(b\), \(\lambda\), or \(\mu\) as a factor.

Noting this, we substitute in (2.45) \(b^k/a\) for \(\lambda\) and \(a^k/b\) for \(\mu\), and regard the points \((a,b)\) as lying on the set \(M\). Since, by our assumption, on the set \(M\) the relations (2.32)—(2.33) are satisfied, it is clear that

\[ \frac{b^k}{a}\to 0,\qquad \frac{a^k}{b}\to 0 \quad \text{as } (a,b)\in M,\ (a,b)\to (0,0). \tag{2.46} \]

But

\[ ab\left[\alpha\frac{b^k}{a}+w_1(a,b)-\beta\frac{a^k}{b}-w_2(a,b)\right]=W(a,b), \]

whence, from the positive definiteness of \(W\), the oddness of the number \(k+1\), and the relations (2.45), (2.46), it follows that

\[ \vartheta(\pi,a,b,\lambda,\mu)-\pi>0 \]

for

\[ (a,b)\in M,\qquad \lambda=\frac{b^k}{a},\qquad \mu=\frac{a^k}{b}. \]

Thus, for \((a,b)\in M\), relation (2.39) is not satisfied; consequently, by Theorem 2.3, the set \(M\) cannot be the set of initial data of a circular family of periodic solutions. The theorem is proved.

Theorem 2.5. Suppose, as before, that the functions \(g_1(x_1,x_2)\) and \(g_2(x_1,x_2)\) are represented in the form (2.40) with odd \(k\), and that the form \(W(x_1,x_2)\) has the form (2.41).

If the equation

\[ W(1,z)=0 \tag{2.47} \]

has a nonzero root \(z=h\) of odd multiplicity \(l\), then system (2.1) has a circular family of symmetric periodic solutions with periods close to \(2\pi\).

Proof. We shall assume, for definiteness, that \(h>0\) (this can be achieved, if necessary, by replacing \(x_1\) by \(-x_1\)). Just as in the proof of the preceding theorem, we introduce forms of degree \(k-1\), \(w_1(x_1,x_2)\) and \(w_2(x_1,x_2)\), by means of the equalities (2.43), and set \(W_1(0,x_2)=\alpha x_2^k,\ W_2(x_1,0)=\beta x_1^k\) (in the case under consideration \(\alpha\) and \(\beta\) may be zero).

We expand the solution of system (2.38) in a series in powers of \(a,b,\lambda,\mu\) and substitute this expansion into the third equation of system (2.38); then we obtain equation (2.44). Integrating this equation with respect to \(\varphi\) from \(0\) to \(\pi\) and multiplying by \(ab\), we obtain (2.45). In equality (2.45) we make the change of variables \(b=(h+c)a\), where the variable \(c\) is subject to the inequality \(|c|<h\); then we obtain

\[ [\vartheta(\pi,a,b,\lambda,\mu)-\pi]ab=(h+c)a^2\left\{[\alpha\lambda+ \tag{2.48} \]

\[ +w_1(a,a(h+c))-\beta\mu-w_2(a,a(h+c))]\int_0^\pi \cos^{k+1}\varphi\,d\varphi+\cdots\right\}. \]

Substituting here, instead of \(\lambda\),

\[ \frac{b^k}{a}=(h+c)^k a^{k-1}, \]

and, instead of \(\mu\),

\[ \frac{a^k}{b}=\frac{a^{k-1}}{h+c}, \]

we obtain

\[ [\vartheta(\pi,a,b,\lambda,\mu)-\pi]ab = a^{k+1} W(1,h+c)\int_0^\pi \cos^{k+1}\varphi\,d\varphi+\cdots, \tag{2.49} \]

since

\[ \alpha(h+c)^{k+1}+(h+c)w_1(1,h+c)-\beta-(h+c)w_2(1,h+c)=W(1,h+c). \]

The terms not written out on the right-hand side of equality (2.49) have order higher than \(k+1\) in \(a\). Since, by assumption, \(h\) is a root of equation (2.47) of odd multiplicity \(l\), it follows that \(c=0\) is a root of multiplicity \(l\) of the polynomial \(W(1,h+c)\). Hence, by the well-known Weierstrass theorem, it follows that equality (2.49) can be represented in the form

\[ [\vartheta(\pi,a,b,\lambda,\mu)-\pi]ab = a^{k+1}E(a,c)[c^l+ \]

\[ +\gamma_i(a)c^{l-1}+\cdots+\gamma_l(a)], \tag{2.50} \]

where the function \(E(a,c)\) is analytic in a neighborhood of the point \(a=c=0\), \(E(0,0)\ne0\), and \(\gamma_i(a)\) \((i=1,\ldots,l)\) are analytic functions, with \(\gamma_i(0)=0\) \((i=1,\ldots,l)\) in a neighborhood of the point \(a=0\).

Equating the right-hand side of (2.50) to zero, we obtain the equation

\[ c^l+\gamma_1(a)c^{l-1}+\cdots+\gamma_l(a)=0. \tag{2.51} \]

Since \(l\) is odd by assumption, this equation has at least one real solution \(c=c(a)\), where \(c(a)\) is a function analytic in \(a^{1/\nu}\), and \(\nu\) is an integer subject to the inequalities \(1\le \nu\le l\). Hence it follows that, for \(b=(h+c(a))a\), equality (2.39) is satisfied.

This, by Theorem 2.3, proves our assertion. The theorem is proved.

Remark 2.1. From the method of proof of Theorem 2.5 it follows that the function \(c(a)\) can be constructed by the method of undetermined coefficients, as a series in powers of \(a^{1/\nu}\), i.e., under the conditions of Theorem 2.5 the family of periodic solutions can be found with arbitrary accuracy.

1. In conclusion of this section we consider the case where the expansions of the functions \(g_1(x_1,x_2)\) and \(g_2(x_1,x_2)\) in powers of \(x_1\) and \(x_2\) begin with terms of even order, and show that this circumstance does not always have an essential influence on the question of the existence of families of symmetric periodic solutions.

In particular, the following holds.

Theorem 2.6. Suppose that the functions \(g_1(x_1,x_2)\) and \(g_2(x_1,x_2)\) have the form

\[ \begin{aligned} g_1(x_1,x_2)&=V_1(x_1,x_2)+W_1(x_1,x_2)+\cdots,\\ g_2(x_1,x_2)&=V_2(x_1,x_2)+W_2(x_1,x_2)+\cdots, \end{aligned} \tag{2.52} \]

where \(V_1(x_1,x_2)\) and \(V_2(x_1,x_2)\) are forms of degree \(k-1\), \(W_1(x_1,x_2)\) and \(W_2(x_1,x_2)\) are forms of degree \(k\), \(k\) is an odd number and \(k\ge5\), while the terms not written out on the right-hand side of (2.52) have order higher than \(k\) in \(x_1\) and \(x_2\).

If the form of degree \(k+1\)

\[ W(x_1,x_2)=x_2W_1(x_1,x_2)-x_1W_2(x_1,x_2) \tag{2.53} \]

vanishes only when \(x_1=x_2=0\), then system (2.1) has no family of symmetric periodic solutions with periods close to \(2\pi\).

Proof. As in the proof of Theorem 2.4, set \(W_1(0,x_2)=\alpha x_2^k\), \(W_2(x_1,0)=\beta x_1^k\). For definiteness

we shall assume that the form \(W(x_1,x_2)\) is positive definite; then it is clear that \(\alpha>0,\ \beta<0\).

By virtue of Theorem 2.2, in the case under consideration it suffices to prove that system (2.1) has no circular families of symmetric periodic solutions with periods close to \(2\pi\). In doing so, one may restrict oneself to the case where \(a>0,\ b>0\).

Introduce the notation (2.42) and, in addition, put

\[ x_1 v_1(x_1,x_2)=V_1(x_1,x_2)-V_1(0,x_2),\quad V_1(0,x_2)=\alpha_1 x_2^{k-1}, \]
\[ x_2 v_2(x_1,x_2)=V_2(x_1,x_2)-V_2(x_1,0),\quad V_2(x_1,0)=\beta_1 x_1^{k-1}. \tag{2.54} \]

Consider the set \(M\) of points \((a,b)\) (\(a\) and \(b\) are positive and sufficiently small), having the point \((0,0)\) as its limit point, and suppose that this set is the set of initial data of a family of symmetric periodic solutions with periods close to \(2\pi\). Since in our case \(k-1\ge 4,\ \alpha>0\) and \(\beta<0\), by Theorem 2.2 there exists a number \(\chi>0\) such that for points of the set \(M\) the inequalities

\[ a>\chi b^{k-2},\quad b>\chi a^{k-2}. \tag{2.55} \]

hold.

Replace in system (2.35) the quantities \(b^{k-1}/a\) and \(a^{k-1}/b\) by \(\lambda\) and \(\mu\), respectively, and eliminate \(t\) from the resulting system; then, exactly as above, we obtain system (2.38).

Expand the solution of system (2.38) in a series in powers of \(a,b,\lambda,\mu\) and substitute this expansion into the third equation of system (2.38); then, taking into account that \(k\ge 5\), we obtain

\[ \frac{d\vartheta}{d\varphi} =1-\beta_1\mu\cos^k\varphi -v_2(a\cos\varphi,b\cos\varphi)\cos^2\varphi +\alpha_1\lambda\cos^k\varphi+ \]
\[ +v_1(a\cos\varphi,b\cos\varphi)\cos^2\varphi -\beta\mu a\cos^{k+1}\varphi -w_2(a\cos\varphi,b\cos\varphi)\cos^2\varphi+ \]
\[ +a\lambda b\cos^{k+1}\varphi +w_1(a\cos\varphi,b\cos\varphi)\cos^2\varphi+\cdots , \tag{2.56} \]

where the unwritten part of the series in the terms not depending on \(\lambda\) and \(\mu\) has order no lower than the \((k+1)\)-st in \(a\) and \(b\), while in the terms linear in \(\lambda\) and \(\mu\) it has order no lower than the second in \(a\) and \(b\).

Integrating equality (2.56) with respect to \(\varphi\) from \(0\) to \(\pi\) and multiplying by \(a^2b^2\), we obtain, because \(k\) is odd,

\[ [\vartheta(\pi,a,b,\lambda,\mu)-\pi]a^2b^2 =a^2b^2\{[\alpha\lambda b+w_1(a,b)- \]
\[ -\beta\mu a-w_2(a,b)]\int_0^\pi \cos^{k+1}\varphi\,d\varphi+\cdots\}; \tag{2.57} \]

the unwritten part of the series has the same meaning as in the preceding equality.

Substituting \(b^{k-1}/a\) for \(\lambda\) and \(a^{k-1}/b\) for \(\mu\) in the right-hand side of the last equality and taking (2.55) into account, we obtain

\[ [\vartheta(\pi,a,b,\lambda,\mu)-\pi]a^2b^2= \]
\[ =abW(a,b)\int_0^\pi \cos^{k+1}\varphi\,d\varphi +o\left((a^2+b^2)^{\frac{k+3}{2}}\right), \tag{2.58} \]

where, as usual, \(\dfrac{o(\varepsilon)}{\varepsilon}\to 0\) as \(\varepsilon\to 0\). Since, by assumption, the form \(W\) of degree \((k+1)\) is positive definite, it follows from (2.58) that equality (2.39) cannot hold on the set \(M\). The theorem is thus proved.

Using the arguments carried out in the proof of the last two theorems, it is not difficult to prove the following assertion.

Theorem 2.7. Let the functions \(g_1(x_1,x_2)\) and \(g_2(x_1,x_2)\) have the form (2.52), let \(k\) be an odd number, \(k\ge 5\), and let the equation

\[ W(1,z)=0, \]

where the form \(W(x_1,x_2)\) is given by equality (2.53), have a nonzero root of odd multiplicity. Then system (2.1) has a circular family of periodic solutions.

§ 3. THE RESONANT CASE. SYSTEMS OF HIGHER ORDERS

In this section we consider the case when in system (0.2) some of the numbers \(\lambda_i\) are equal to one another, while the remaining ones are not integral multiples of them; that is, we consider the system

\[ \ddot{x}_i+\lambda^2 x_i=g_i(x_1,\ldots,x_n) \quad (i=1,\ldots,s), \]

\[ \ddot{x}_i+\lambda_i^2 x_i=g_i(x_1,\ldots,x_n) \quad (i=s+1,\ldots,n), \]

where \(\lambda_i\ne m\lambda\) \((i=s+1,\ldots,n,\; m=0,1,\ldots)\). By a change of the variable \(t\) one can arrange that \(\lambda=1\); therefore we shall study the system

\[ \begin{aligned} \ddot{x}_i+x_i&=g_i(x_1,\ldots,x_n) \quad (i=1,\ldots,s),\\ \ddot{x}_i+\lambda_i^2 x_i&=g_i(x_1,\ldots,x_n) \quad (i=s+1,\ldots,n), \end{aligned} \tag{3.1} \]

where \(\lambda_i\ne m\) \((i=s+1,\ldots,n;\; m=0,1,\ldots)\).

In this section we shall indicate conditions for the existence of an analytic family of symmetric periodic solutions with periods close to \(2\pi\).

We pass from system (3.1) to a system of first-order equations. With the substitution \(\dot{x}_i=-y_i\) \((i=1,\ldots,s)\), \(\dot{x}_i=-\lambda_i y_i\) \((i=s+1,\ldots,n)\), we obtain

\[ \begin{aligned} \dot{x}_i&=-y_i,\quad \dot{y}_i=x_i-g_i(x_1,\ldots,x_n) \quad (i=1,\ldots,s),\\ \dot{x}_i&=-\lambda_i y_i,\quad \dot{y}_i=\lambda_i x_i+G_i(x_1,\ldots,x_n) \quad (i=s+1,\ldots,n), \end{aligned} \tag{3.2} \]

where \(G_i(x_1,\ldots,x_n)=-\dfrac{1}{\lambda_i}g_i(x_1,\ldots,x_n)\) \((i=s+1,\ldots,n)\).

In the first \(2s\) equations of system (3.2) we pass to polar coordinates by the formulas

\[ x_i=r_i\cos\varphi_i,\quad y_i=r_i\sin\varphi_i \quad (i=1,\ldots,s), \tag{3.3} \]

then we obtain

\[ \dot{r}_i=-g_i(r_1\cos\varphi_1,\ldots,r_s\cos\varphi_s,x_{s+1},\ldots,x_n)\sin\varphi_i, \]

\[ r_i\dot{\varphi}_i=r_i-g_i(r_1\cos\varphi_1,\ldots,r_s\cos\varphi_s,x_{s+1},\ldots,x_n)\cos\varphi_i \tag{3.4} \]

\[ (i=1,\ldots,s), \]

\[ \dot{x}_i=-\lambda_i y_i,\quad \dot{y}_i=\lambda_i x_i+G_i(r_1\cos\varphi_1,\ldots,r_s\cos\varphi_1,x_{s+1},\ldots,x_n) \]

\[ (i=s+1,\ldots,n). \]

Since we are interested in families of symmetric periodic solutions, we shall study the solution of system (3.4) with initial data \(t=0\), \(r_i=a_i\), \(\varphi_i=0\) \((i=1,\ldots,s)\); \(x_i=a_i\), \(y_i=0\) \((i=s+1,\ldots,n)\). We introduce the first \(s\) initial data \(a_i\) into system (3.4) itself by the substitution

\[ r_i=a_i(1+\rho_i)\quad (i=1,\ldots,s), \tag{3.5} \]

then we obtain

\[ a_i\dot{\rho}_i=-g_i\sin\varphi_i, \]

\[ \dot{\varphi}_i=1-\frac{g_i\cos\varphi_i}{a_i(1+\rho_i)} \quad (i=1,\ldots,s), \tag{3.6} \]

\[ \dot{x}_i=-\lambda_i y_i,\quad \dot{y}_i=\lambda_i x_i+G_i\quad (i=s+1,\ldots,n). \]

1. Let us first consider the case where the expansions of the functions \(g_i(x_1,\ldots,x_s,0,\ldots,0)\) \((i=1,\ldots,s)\) in powers of \(x_1,\ldots,x_s\) begin with forms of odd order. The following is valid.

Theorem 3.1. Suppose that the functions \(g_i(x_1,\ldots,x_n,0,\ldots,0)\) are representable in the form

\[ g_i(x_1,\ldots,x_s,0,\ldots,0)=W_i(x_1,\ldots,x_s)+\ldots \quad (i=1,\ldots,s), \tag{3.7} \]

where \(W_i(x_1,\ldots,x_s)\) \((i=1,\ldots,s)\) are forms of odd degree \(k\), and the omitted terms have order higher than \(k\). Suppose that the system of equations

\[ U_i(1,z_2,\ldots,z_s)=0\quad (i=2,\ldots,s), \tag{3.8} \]

where the functions \(U_i\) are given by the formulas

\[ U_i(x_1,\ldots,x_s)=x_i W_1(x_1,\ldots,x_s)-x_1 W_i(x_1,\ldots,x_s) \tag{3.9} \]

\[ (i=2,\ldots,s), \]

has a solution \(z_i=h_i,\ h_i\ne0\) \((i=2,\ldots,s)\), and

\[ \left. \frac{D(U_2,\ldots,U_s)}{D(z_2,\ldots,z_s)} \right|_{z_i=h_i} \ne 0. \tag{3.10} \]

Suppose, in addition, that the expansions of the functions \(g_i(x_1,\ldots,x_s,0,\ldots,0)\) \((i=s+1,\ldots,n)\) in powers of \(x_1,\ldots,x_n\) begin with terms of dimension not lower than \(k\). Then system (3.1) has an analytic family of symmetric periodic solutions with periods close to \(2\pi\).

Proof. Since \(h_i\ne0\), we may assume that \(h_i>0\) \((i=2,\ldots,s)\) (generality is not restricted by this assumption, since positivity of \(h_i\) can be achieved by replacing the corresponding \(x_i\) by \(-x_i\)). Put

\[ a_i=(h_i+c_i)a_1\quad (i=2,\ldots,s) \tag{3.11} \]

and

\[ x_i=a_1\xi_i,\quad y_i=a_1\eta_i\quad (i=s+1,\ldots,n), \tag{3.12} \]

then, instead of system (3.6), we obtain

\[ \dot{\rho}_i=P_i(\rho_1,\ldots,\rho_s,\varphi_1,\ldots,\varphi_s,\xi_{s+1},\ldots,\xi_n,a_1,c_2,\ldots,c_s), \]

\[ \dot\varphi_i=1+\Phi_i(\rho_1,\ldots,\rho_s,\varphi_1,\ldots,\varphi_s,\xi_{s+1},\ldots,\xi_n, \tag{3.13} \]
\[ a_1,c_2,\ldots,c_s)\quad (i=1,\ldots,s); \]
\[ \dot\xi_i=-\lambda_i\eta_i, \]
\[ \dot\eta_i=\lambda_i\xi_i+Q_i(\rho_1,\ldots,\rho_s,\varphi_1,\ldots,\varphi_s,\xi_{s+1},\ldots,\xi_n, \]
\[ a_1,c_2,\ldots,c_s)\quad (i=s+1,\ldots,n). \]

The functions \(\Phi_i\) and \(Q_i\) have the following form:
\[ \Phi_i=-\frac{g_i\cos\varphi_i}{a_1(h_i+c_i)(1+\rho_i)} \quad (i=1,2,\ldots,s), \tag{3.14} \]
where \(h_1+c_1=1\) has been set, and
\[ Q_i=\frac{1}{a_1}G_i\quad (i=s+1,\ldots,n). \tag{3.15} \]

Since \(h_i>0\) \((i=2,\ldots,s)\), it is clear that the functions \(P_i,\Phi_i\) \((i=1,\ldots,s)\) and \(Q_i\) \((i=s+1,\ldots,n)\) are analytic in the variables \(\rho_1,\ldots,\rho_s,\xi_{s+1},\ldots,\xi_n,a_1,c_2,\ldots,c_n\) for sufficiently small values of these variables; moreover, \(P_i\to0\), \(Q_i\to0\), and \(\Phi_i\to0\) as \(a_1\to0\).

Eliminating \(t\) from system (3.13), we then obtain
\[ \frac{d\rho_i}{d\varphi_1}=\frac{P_i}{1+\Phi_1} \quad (i=1,\ldots,s),\qquad \frac{d\varphi_i}{d\varphi_1}=\frac{1+\Phi_i}{1+\Phi_1} \]
\[ (i=2,\ldots,s), \tag{3.16} \]
\[ \frac{d\xi_i}{d\varphi_1}=-\frac{\lambda_i\eta_i}{1+\Phi_1},\qquad \frac{d\eta_i}{d\varphi_1}=\frac{\lambda_i\xi_i+Q_i}{1+\Phi_1} \]
\[ (i=s+1,\ldots,n). \]

Let \(\rho_i(\varphi_1,a_1,c_2,\ldots,c_n)\) \((i=1,\ldots,s)\), \(\varphi_i(\varphi_1,a_1,c_1,\ldots,c_n)\) \((i=2,\ldots,s)\), \(\xi_i(\varphi_1,a_1,c_2,\ldots,c_n)\), \(\eta_i(\varphi_1,a_1,c_2,\ldots,c_n)\) \((i=s+1,\ldots,n)\) be the solution of system (3.16) with initial data \(\rho_i=0\), \(\varphi_i=0\) \((i=1,\ldots,s)\), \(\xi_i=c_i\), \(\eta_i=0\) \((i=s+1,\ldots,n)\). Since the right-hand sides of system (3.16) are analytic, the functions \(\rho_i,\varphi_i,\xi_i,\eta_i\) are also analytic in the variables \(a_1,c_2,\ldots,c_n\) for sufficiently small values of these variables. From formulas (3.14) and (3.15) it follows that the functions \(\xi_i\) and \(\eta_i\) can be represented in the form
\[ \xi_i=\xi_{i0}(\varphi_1,c_{s+1},\ldots,c_n)+a_1\xi_{i1}(\varphi_1,a_1,c_2,\ldots,c_s)+ \]
\[ {}+a_1\xi_{i2}(\varphi_1,a_1,c_2,\ldots,c_n), \]
\[ \eta_i=\eta_{i0}(\varphi_1,c_{s+1},\ldots,c_n)+a_1\eta_{i1}(\varphi_1,a_1,c_2,\ldots,c_s)+ \tag{3.17} \]
\[ {}+a_1\eta_{i2}(\varphi_1,a_1,c_2,\ldots,c_n)\quad (i=s+1,\ldots,n). \]

Here
\[ \xi_{i0}(0,c_{s+1},\ldots,c_n)=c_i,\quad \eta_{i0}(0,c_{s+1},\ldots,c_n)=\xi_{i1}(0,a_1, \]
\[ c_2,\ldots,c_s)=\eta_{i1}(0,a_1,c_2,\ldots,c_s)=\xi_{i2}(0,a_1,c_2,\ldots,c_n)=\eta_{i2}(0,a_1, \]
\[ c_2,\ldots,c_n)=0\quad (i=s+1,\ldots,n), \]
and, in addition,
\[ \xi_{i2}(\varphi_1,c_2,\ldots,c_s,0,\ldots,0)=\eta_{i2}(\varphi_1,a_1,c_2,\ldots,c_s,0,\ldots,0)=0 \quad (i=s+1,\ldots,n). \]
Substituting (3.17) into system (3.16), we obtain a system of equations for determining \(\xi_{i0},\eta_{i0}\):
\[ \frac{d\xi_{i0}}{d\varphi_1}=-\lambda_i\eta_{i0},\qquad \frac{d\eta_{i0}}{d\varphi_1}=\lambda_i\xi_{i0} \quad (i=s+1,\ldots,n). \tag{3.18} \]

therefore,

\[ \xi_{i0}=c_i\cos\lambda_i\varphi_1,\quad \eta_{i0}=-c_i\sin\lambda_i\varphi_1 \quad (i=s+1,\ldots,n). \tag{3.19} \]

From formulas (3.15) and from the condition of the theorem it follows that the functions
\(Q_i(\rho_1,\ldots,\rho_s,\varphi_1,\ldots,\varphi_s,0,\ldots,0,a_1,c_2,\ldots,c_s)\)
\((i=s+1,\ldots,n)\) have order in \(a_1\) not lower than the \((k-1)\)-st; therefore the functions \(\xi_{i1}\) and \(\eta_{i1}\), determined by equalities (3.17), have order not lower than the \((k-2)\)-nd.

We have already noted that \(P_i\to0\) \((i=1,\ldots,s)\) as \(a_1\to0\); therefore, from the first \(s\) equations of system (3.16), it follows that the quantities \(\rho_i\) are small together with \(a_1\). Since, by assumption, \(h_i>0\) \((i=2,\ldots,s)\), it follows from formulas (3.3), (3.5), and (3.11) that, for sufficiently small \(a_1,c_2,\ldots,c_s\), the quantities \(y_i\) \((i=1,\ldots,s)\) vanish for the first time after \(t=0\) when \(\varphi_i=\pi\) \((i=1,\ldots,s)\).

Let us show that the initial data \(\xi_i\) \((i=s+1,\ldots,n)\), i.e. the quantities \(c_i\) \((i=s+1,\ldots,n)\), can be chosen in such a way that, for \(\varphi_i=\pi\) \((i=1,\ldots,s)\), one has \(y_i=0\) \((i=s+1,\ldots,n)\). Consider the system of equations \(y_i=0\) \((i=s+1,\ldots,n)\). From (3.12), (3.17), and (3.19) we obtain

\[ c_i\sin\lambda_i\pi+a_1\eta_{i1}(\pi,a_1,c_2,\ldots,c_s)+ \]

\[ +a_1\eta_{i2}(\pi,a_1,c_2,\ldots,c_n)=0 \quad (i=s+1,\ldots,n). \tag{3.20} \]

By assumption, the \(\lambda_i\) \((i=s+1,\ldots,n)\) are not integers; hence \(\sin\lambda_i\pi\ne0\). Therefore, at
\(a_1=c_2=c_s=\ldots=c_n=0\), the system of equations (3.20) has a nonzero Jacobian with respect to \(c_i\) \((i=s+1,\ldots,n)\), and consequently this system has a unique analytic solution

\[ c_i=\bar p_i(a_1,c_2,\ldots,c_s) \quad (i=s+1,\ldots,n). \tag{3.21} \]

As was noted above, the functions \(\eta_{i1}\) have order in \(a_1\) not lower than the \((k-2)\)-nd; consequently, the solution (3.21) of system (3.20) can be represented in the form

\[ c_i=a_1^{\,k-1}p_i(a_1,c_2,\ldots,c_s) \quad (i=s+1,\ldots,n), \tag{3.22} \]

where the functions \(p_i\) are analytic in \(a_1,c_2,\ldots,c_s\) for sufficiently small values of these variables.

We shall consider only those solutions of system (3.16) whose initial data satisfy conditions (3.22). From the equations

\[ \frac{d\xi_i}{d\varphi_1}=-\lambda_i\eta_i(1+\Phi_1)^{-1} \]

it follows that then the functions \(\xi_i\) are representable in the form

\[ \xi_i(\varphi_1,a_1,c_2,\ldots,c_n) = a_1^{\,k-1}\bar\xi_i(\varphi_1,a_1,c_2,\ldots,c_n) \]

\[ (i=s+1,\ldots,n), \]

where \(\bar\xi_i\) are analytic functions. Substituting these values of the functions \(\xi_i\) into the equations for \(\varphi_i\) and taking into account formulas (3.7) and (3.14), we obtain

\[ \frac{d\varphi_i}{d\varphi_1} = 1+W_1(1,h_2+c_2,\ldots,h_s+c_s)a_1^{\,k-1}\cos^{k+1}\varphi_1- \]

\[ -\frac{W_i(1,h_2+c_2,\ldots,h_s+c_s)}{h_i+c_i} a_1^{\,k-1}\cos^{k+1}\varphi_1+\ldots \tag{3.23} \]

\[ (i=2,\ldots,s). \]

In these equations the unwritten terms have order no lower than the \(k\)-th with respect to \(a_1\). Integrating equations (3.23) with respect to \(\varphi_1\) from 0 to \(\pi\), we obtain

\[ \begin{aligned} \varphi_i(\pi)-\pi ={}&\left[ W_1(1,h_2+c_2,\ldots,h_s+c_s) -\frac{W_i(1,h_2+c_2,\ldots,h_s+c_s)}{h_i+c_i} \right] a_1^{k-1}\times \\ &\times \int_0^\pi \cos^{k+1}\varphi\,d\varphi+\ldots \qquad (i=2,\ldots,s), \end{aligned} \tag{3.24} \]

where the unwritten terms have order no lower than the \(k\)-th with respect to \(a_1\).

Equating the right-hand sides of these equalities to zero, canceling by \(a_1^{k-1}\), and multiplying by

\[ \frac{h_i+c_i}{\displaystyle\int_0^\pi \cos^{k+1}\varphi\,d\varphi}, \]

we obtain the equations

\[ (h_i+c_i)W_1(1,h_2+c_2,\ldots,h_s+c_s) -W_i(1,h_2+c_2,\ldots,h_s+c_s) +a_1R(a_1,c_2,\ldots,c_s)=0 \quad (i=2,\ldots,s), \tag{3.25} \]

where \(R\) is an analytic function of \(a_1,c_2,\ldots,c_s\) for sufficiently small values of these variables.

By assumption, the left-hand side of system (3.25) vanishes for \(a_1=c_2=\cdots=c_s=0\), and the Jacobian of this system with respect to \(c_i\) \((i=2,\ldots,s)\) is different from zero for \(a_1=c_2=\cdots=c_s=0\). Thus system (3.25) has the unique analytic solution

\[ c_i=p_i(a_1)\qquad (i=2,\ldots,s). \tag{3.26} \]

Formulas (3.22) and (3.26) give the system of initial data of a family of symmetric periodic solutions.

From the equation for \(\varphi_1\) (the \((s+1)\)-st equation of system (3.13)) it follows that the periods of the solutions of this family depend analytically on \(a_1\) and tend to \(2\pi\) as \(a_1\to0\). The theorem is proved.

1. We now turn to the case when the expansions of the functions \(g_i(x_1,\ldots,x_s,0,\ldots,0)\) \((i=1,\ldots,s)\) in a series in powers of \(x_1,\ldots,x_s\) begin with forms of even order.

The following is true.

Theorem 3.2. Suppose that the functions \(g_i(x_1,\ldots,x_s,0,\ldots,0)\) are represented in the form

\[ g_i(x_1,\ldots,x_s,0,\ldots,0) =V_i(x_1,\ldots,x_s)+ \]
\[ +\,W_i(x_1,\ldots,x_s)+\ldots \quad (i=1,\ldots,s), \tag{3.27} \]

where \(V_i\) are forms of degree \(k-1\), \(W_i\) are forms of degree \(k\), \(k\) is an odd number and \(k\geqslant5\), and the unwritten terms have order no lower than \((k+1)\).

Suppose that the system of equations (3.8), where the functions \(U_i\) are still defined by formulas (3.9), has a solution \(z_i=h_i,\ h_i\ne0\) \((i=2,\ldots,s)\), and that inequality (3.10) is satisfied.

Suppose, moreover, that the expansions of the functions \(g_i(x_1,\ldots,x_s,0,\ldots,0)\) \((i=s+1,\ldots,n)\) in a series in powers of \(x_1,\ldots,x_s\) begin with terms of order no lower than the \(k\)-th.

Then system (3.1) has an analytic family of symmetric periodic solutions with periods close to \(2\pi\).

Proof. Exactly as in the proof of the preceding theorem, we obtain equations (3.20), and from them the relations (3.22), determining the initial data \(x_i\) for \(i=s+1,\ldots,n\).

As in the proof of the preceding theorem, we verify that the relations
\[ \xi_i(\varphi_1,\ a_1,\ c_2,\ldots,c_n) = a_1^{k-1}\,\bar{\xi}_i(\varphi_1,\ a_1,\ c_2,\ldots,c_n) \quad (i=s+1,\ldots,n), \]
hold, where \(\bar{\xi}_i\) are analytic functions. Substituting these relations into the equations for \(\varphi_i\) from system (3.16) and taking into account formulas (3.14) and (3.27), as well as the inequality \(k\ge 5\), we obtain the equations
\[ \begin{aligned} \frac{d\varphi_i}{d\varphi_1} &= 1 + V_1(1,h_2+c_2,\ldots,h_s+c_s)a_1^{k-2}\cos^k\varphi_1 \\ &\quad - \frac{V_i(1,h_2+c_2,\ldots,h_s+c_s)}{h_i+c_i}\, a_1^{k-2}\cos^k\varphi_1 \\ &\quad + W_1(1,h_2+c_2,\ldots,h_s+c_s)a_1^{k-1}\cos^{k+1}\varphi_1 \\ &\quad - \frac{W_i(1,h_2+c_2,\ldots,h_s+c_s)}{h_i+c_i}\, a_1^{k-1}\cos^{k+1}\varphi_1+\ldots \end{aligned} \tag{3.28} \]
\[ (i=2,\ldots,s), \]
where the terms not written out have order not lower than the \(k\)-th in \(a_1\). Integrating equations (3.28) from \(0\) to \(\pi\) and taking into account that \(\int_0^\pi \cos^k\varphi\,d\varphi=0\) because \(k\) is odd, we obtain equations (3.24), and the proof is completed exactly as above. The theorem is proved.

Remark 3.1. From the method of proof of Theorems 3.1 and 3.2 it follows that the functions \(p_i\) \((i=2,3,\ldots,n)\), defined by equalities (3.22) and (3.26), can be constructed by the method of undetermined coefficients; i.e., under the conditions of these theorems, the families of periodic solutions of system (3.1) can be effectively found.

References

1. Lyapunov A. M. The general problem of the stability of motion. Collected Works, vol. II. Moscow—Leningrad, 1956.
2. Ryabov Yu. A. On a theorem of A. M. Lyapunov. Scientific Notes of Moscow State University, vol. VII, issue 165, 1954.
3. Krasnosel’skii M. A. DAN SSSR, 111, 283–286, 1956.
4. Krasnosel’skii M. A. Positive solutions of operator equations. Moscow, 1962.

Received by the editors
June 21, 1965

Leningrad State
University