ON SINGULAR PERIODIC SOLUTIONS OF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Ya. V. BYKOV

It is known that the conditions for the existence of holomorphic periodic solutions of systems of differential equations depending on a parameter have been studied quite well. To a lesser extent, the conditions for the existence of periodic solutions expandable in positive fractional powers of the parameter have been studied [1, 2, 3, etc.].

In [4] it is shown that, under particular assumptions concerning the coefficients, a linear system may have periodic solutions for which \(\mu=0\) is a pole of the first, second, and higher orders.

In the present paper we study the existence of a periodic solution \(v(t,\mu)\), representable in the form of the series

\[ v(t,\mu)=\sum_{k=-p}^{\infty} v_k(t)\mu^{k/q}\,*, \]

where \(p\) and \(q\) are positive integers. We shall call this solution singular if at least one of the coefficients \(v_{-p}(t),\ldots,v_{-1}(t)\) is different from zero.

In what follows: 1) when speaking of a periodic function, we shall everywhere mean periodicity with period \(\omega\); 2) all known functions in the equations considered below are assumed to be periodic in the argument \(t\) and continuous; 3) when speaking of the solvability of a certain equation, we mean the existence of a periodic solution.

Let \(v,u,w_1,\ldots,w_k,\psi\) be \(n\)-dimensional vector functions; \(A(t)\) a matrix of order \(n\times n\); \(f_k(t,w_1,\ldots,w_k)\) an operator mapping the collection of periodic vectors \(w_1(t),\ldots,w_k(t)\) into an \(n\)-dimensional periodic vector, linear with respect to each of the vector arguments \(w_1(t),\ldots,w_k(t)\). Such an operator in functional analysis is called a \(k\)-linear form or a \(k\)-linear operator and is conventionally written in the form

\[ f_k(t,w_1,\ldots,w_k)\equiv f_k(t)w_1\ldots w_k, \]

where the right-hand side should not be understood as simple multiplication of vectors. Here \(f_k(t)\) denotes an operator under whose action on the vector \(w_1(t)\) we obtain the operator \(F_2(t,w_1)\equiv f_k(t)w_1(t)\). Then, by acting with the operator \(F_2(t,w_1)\) on the vector \(w_2(t)\), we obtain the operator \(F_3(t,w_1,w_2)\), and so on.

Consequently, \(f_k(t)w_1\ldots w_k\) denotes the vector obtained

* As applied to a system of \(n\) differential equations with \(n\) unknown functions, \(v(t,\mu)\), \(v_k(t)\) are \(n\)-dimensional vectors.

by successively applying to the vectors \(w_1(t), \ldots, w_k(t)\), respectively, the operators \(f_k(t), F_2(t,w_1), \ldots, F_k(t,w_1,\ldots,w_{k-1})\):

\[ f_k(t) w_1 \ldots w_k = \{\ldots [(f_k(t) w_1) w_2] w_3 \ldots \} w_k . \]

In the simplest case

\[ f_{km}(t,w_1,\ldots,w_k) = \sum_{\alpha_1=1}^{n} \cdots \sum_{\alpha_k=1}^{n} \psi_{m\alpha_1\ldots\alpha_k}(t)\, w_1^{\alpha_1}\ldots w_k^{\alpha_k}, \tag{A} \]

where: 1) \(\psi_{m\alpha_1\ldots\alpha_m}(t)\) is a scalar periodic function; 2) \(w_i^1,w_i^2,\ldots,w_i^n\) are the components of the vector \(w_i\); 3) \(f_{km}(t,w_1,\ldots,w_k)\) is the \(m\)-th component of the vector \(f_k(t,w_1,\ldots,w_k)\) \((m=1,2,\ldots,n)\).

In the general case, each term on the right-hand side of equality (A) should be understood as a \(k\)-linear form, in the sense of functional analysis, of the scalar functions \(w_1^{\alpha_1}(t),\ldots,w_k^{\alpha_k}(t)\). The vector \(f_k(t,u,\ldots,u)\), obtained from the vector \(f_k(t,w_1,\ldots,w_k)\) when \(w_1=\cdots=w_k=u\), will, for simplicity, be denoted by \(f_k(u)\). In deriving working formulas this vector is sometimes written in the form \(f_k(u,u,\ldots,u)\), omitting the argument \(t\).

Introduce the notation

\[ f(t,u)=\sum_{k=0}^{m} f_k(u). \]

The paper considers the existence of complex-valued periodic solutions of the system of differential equations

\[ L(v) \equiv \frac{dv(t)}{dt} - A(t)v(t) = \psi(t) + \mu f(t,v), \tag{1} \]

where \(\mu\) is a complex parameter and \(t\) is a real variable. If \(\mu\) is a real parameter, then some of these solutions may be real (if the corresponding algebraic equations have real roots).

The scalar equation

\[ x^{(n)}(t)+a_1(t)x^{(n-1)}(t)+\cdots+a_n(t)x(t)= \]

\[ = p(t)+\mu \sum_{j=0}^{m} H_{\alpha_1\ldots\alpha_n}(t) (x(t))^{\alpha_1}(\dot{x}(t))^{\alpha_2}\ldots (x^{(n-1)}(t))^{\alpha_n} \tag{1_1} \]

is reduced to the form (1). Therefore all propositions concerning equation (1) are transferred in the corresponding way to equation \((1_1)\). These propositions are not given here.

I

Let us consider the existence of periodic solutions of system (1) representable in the form of the series

\[ v(t,\mu)=\sum_{k=-1}^{\infty} v_k(t)\,\mu^{k/m}, \tag{2} \]

where \(v_k(t)\) is a periodic vector.

By the substitution $\mu=\lambda^m,\ v=\dfrac{u}{\lambda}$, equation (1) is transformed into the form

\[ L(u)=\lambda\psi(t)+\sum_{k=0}^{m}\lambda^{m+1-k} f_k(u). \tag{3} \]

Let

\[ f_k(u,w,\ldots,w)+f_k(w,u,w,\ldots,w)+\cdots+f_k(w,\ldots,w,u)\equiv B_k(w)u. \]

Obviously, $B_k(w)u$ may be regarded as the result of the action of a certain linear operator $B_k(w)$ on the vector $u(t)$. Let us note some properties of this operator:
$B_k(c\varphi)=c^{k-1}B_k(\varphi)$, where $c$ is a constant, and
$B_k(\varphi)\varphi=k f_k(\varphi)$.

Formally substituting the series

\[ u(t,\lambda)=\sum_{k=0}^{\infty} u_k(t)\lambda^k, \tag{4} \]

where the $u_k(t)$ are as yet unknown vectors, into equation (3) and equating coefficients of like powers of the parameter $\lambda$, we obtain the following recurrence relations:

\[ L(u_0)=0, \tag{$5_0$} \]

\[ L(u_1)=\psi(t)+f_m(u_0), \tag{$5_1$} \]

\[ L(u_2)=B_m(u_0)u_1+f_{m-1}(u_0), \tag{$5_2$} \]

\[ L(u_k)=B_m(u_0)u_{k-1}+r_k(u_0,u_1,\ldots,u_{k-2}), \tag{$5_k$} \]

where $r_k(u_0,u_1,\ldots,u_{k-2})$ is a known function of the vectors $u_0,u_1,\ldots,u_{k-2}$.

It is known that in the nonresonant case (the equation $L(u)=0$ has only the trivial periodic solution) equation (1) has a solution holomorphic with respect to the parameter $\mu$. One can show that this solution is unique also in the class of functions representable in the form of the series (2). Indeed, in this case $u_0(t)\equiv0$, and, taking this into account, the first $m+1$ equations of the system $(5_k)$ take the form

\[ L(u_1)=\psi,\qquad L(u_k)=0\quad (k=2,\ldots,m). \]

Hence

\[ u_k(t)\equiv0\quad (k=2,\ldots,m); \]

\[ L(u_{m+1})=\sum_{k=0}^{m} f_k(u_1), \]

\[ L(u_k)=0\quad \text{for } k=m+2,\ldots,2m. \]

Hence

\[ u_k(t)\equiv0\quad \text{for } k=m+2,\ldots,2m. \]

Taking these identities into account, $L(u_{2m+1})=r_{2m+1}(u_1,u_{m+1})$, and so on.

Consequently, all coefficients of the series (4) are determined uniquely, and moreover
$u_k(t)\equiv0$ for $k\ne nm+1$ $(n=0,1,2,\ldots)$, i.e. the series (2) is a series in positive integral powers of the parameter $\mu$.

Suppose that the equation $L(u)=0$ has nontrivial periodic solutions. For simplicity we restrict ourselves to the case in which, up to a constant factor, this equation has only one nontrivial periodic solution $\varphi(t)$. The general case is considered analogously.

Let \(z(t)\) denote a nontrivial periodic solution of the system adjoint to \(L(u)=0\).

Suppose that 1) the functional \(Y[x]\) is defined by the rule
\[ Y[x]=\int_0^\omega (z(t);x(t))\,dt, \]
where \((z;x)\) is the ordinary scalar product of the vectors \(z\) and \(x\); 2) \(p=Y[\psi(t)]\), \(q=Y[f_m(\varphi)]\).

From equation \((5_0)\) we obtain that \(u_0(t)=c_0\varphi(t)\), where \(c_0\) is a constant. It is known that, for the solvability of equation \((5_1)\), it is necessary and sufficient that the vectors \(z(t)\) and \(\psi(t)+f_m(u_0)\) be orthogonal on the interval \([0,\omega]\), i.e., that the equation
\[ p+c_0^m q=0 \tag{6} \]
have a solution with respect to \(c_0\). Hence it follows that in the case \(p\ne 0=q\), equation (1) has no periodic solutions representable in the form of the series (2).

Suppose \(p\ne 0\ne q\). Then equation (6) has \(m\) distinct solutions \(c_0=c_{0k}\) \((k=1,2,\ldots,m)\). Fix some solution \(c_{0k}\), and also set \(u_{0k}(t)=c_{0k}\varphi(t)\). Then \(u_{1k}=c_1\varphi(t)+\psi_{1k}(t)\), where \(c_1\) is some constant, and \(\psi_{1k}(t)\) is a particular periodic solution of the equation
\[ L(u)=\psi(t)+f_m(u_{0k}). \]

For equation \((5_2)\) to be solvable, it is necessary and sufficient that the equation
\[ m q c_1+Y\,[f_{m-1}(\varphi)+B_m(\varphi)\psi_{1k}]=0 \]
have a solution with respect to \(c_1\). Since \(q\ne 0\), this equation has a solution, and \(c_1\) is uniquely determined by \(c_{0k}\). Let \(c_1=c_{1k}\); \(u_{1k}=c_{1k}\varphi+\psi_{1k}\). Then \(u_2(t)=c_2\varphi+\psi_{2k}\), where \(c_2\) is some constant and \(\psi_{2k}\) is a particular solution of system \((5_2)\), in which \(u_0,u_1\) have been replaced respectively by \(u_{0k},u_{1k}\).

System \((5_3)\) is solvable if and only if
\[ m c_{0k}^{m-1}c_2q+Y\,[B_m(u_{0k})\psi_{2k}+r_3(u_{0k},u_{1k})]=0. \]

Hence \(c_2=c_{2k}\) is uniquely determined by \(c_{0k}\). Continuing such reasoning, we obtain that the coefficients of the series (4) are uniquely determined by \(c_{0k}\).

Thus, in the case \(p\ne 0\ne q\), equation (1) has \(m\) distinct periodic solutions representable in the form of the series (2); moreover, if \(\mu\) is a real number, then: 1) for \(m\) odd, only one of these periodic solutions is real; 2) for \(m\) even, (1) has no real periodic solutions if \(p\) and \(q\) have the same signs, and has two real solutions if \(p\) and \(q\) have different signs.

Example 1. The scalar equation
\[ \frac{dv}{dt}=b+\mu\sum_{k=0}^{m}a_kv^k, \tag{A_1} \]
where \(b,a_k\) are constants, with \(b\ne 0\ne a_m\), on the basis of the above has \(m\) distinct periodic solutions representable in the form of the series
\[ v=\sum_{k=-1}^{\infty}v_k\mu^{k/m}. \]

This proposition can be proved by another method. After the substitution \(\mu=\lambda^m,\; v=\dfrac{u}{\lambda}\), equation \((A_1)\) takes the form

\[ \frac{du}{dt}=\lambda F(\lambda,u), \tag{11} \]

where

\[ F(\lambda,u)\equiv b+a_m u^m+\lambda a_{m-1}u^{m-1}+\cdots+\lambda^m a_0 . \]

It is obvious that every root \(u=u(\lambda)\) of the equation \(F(\lambda,u)=0\) is a periodic solution of equation \((A_{11})\). From the implicit-function existence theorem it follows that the equation \(F(\lambda,u)=0\) has \(m\) distinct solutions

\[ u_n(\lambda)=\sum_{k=0}^{\infty} u_{kn}\lambda^k \quad (n=1,2,\ldots,m), \]

where \(u_{0n}\) are the roots of the equation \(b+a_m x^m=0\), and these series have nonzero radii of convergence.

Example 2. The scalar equation

\[ \frac{dv}{dt}=\psi(t)+\mu\sum_{k=0}^{m} f_k(t)v^k, \tag{2} \]

where \(\psi(t), f_k(t)\) are periodic functions, has no periodic solutions representable in the form (2) if

\[ p_1=\int_{0}^{\omega}\psi(t)\,dt\ne 0;\qquad q_1=\int_{0}^{\omega} f_m(t)\,dt=0. \]

If \(p_1\ne 0\ne q_1\), then equation \((A_2)\) has \(m\) periodic solutions representable in the form (2).

Case \(p=0\ne q\). Then \(c_0=0\), and, taking into account the identity \(u_0(t)\equiv 0\), the recurrence relations \((5_k)\) take the form

\[ L(u_1)=\psi(t), \tag{1} \]

\[ L(u_k)=0 \quad (k=2,\ldots,m), \tag{k} \]

\[ L(u_{m+1})=\sum_{k=0}^{m} f_k(u_1), \tag{m+1} \]

\[ L(u_{m+2})=\sum_{k=1}^{m} B_k(u_1)u_2, \tag{m+2} \]

\[ L(u_{m+i})=\sum_{k=1}^{m} B_k(u_1)u_i+r_{m+i}(u_1,u_2,\ldots,u_{i-1}), \tag{m+i} \]

where \(r_{m+i}(u_1,\ldots,u_{i-1})\) is a known function of its arguments. From equations \((7_k)\) \((k=1,2,\ldots,m)\) we find \(u_1=c_1\varphi+\psi_1,\; u_n(t)=c_n\varphi\) \((n=2,\ldots,m)\), where \(c_i\) are constants, and \(\psi_i(t)\) is a particular periodic solution of equation \((7_1)\).

The solvability condition for equation \((7_{m+1})\) leads to the algebraic equation

\[ Y\left[\sum_{k=0}^{m} f_k(c_1\varphi+\psi_1)\right]=0. \tag{8} \]

If equation (8) has no solutions with respect to \(c_1\), then equation (1) has no solutions representable in the form (2) (if (8) has no real solutions, then (1) has no real periodic solutions representable in the form (2)).

Let 1) \(c_{1a}\) be a solution of equation (8) and \(u_{1a}=c_{1a}\varphi+\psi_1\); 2)

\[ b_a=Y\left[\sum_{n=1}^{m} B_n(u_{1a})\varphi\right]. \]

Let \(b_a\ne0\). Then from the solvability conditions for equations \((7_k)\) \((k=2,\ldots,m)\) it follows that \(u_k(t)\equiv0\) \((k=2,\ldots,m)\). Taking these identities into account, the following equations of the recurrent system take the form

\[ L(u_k)=0\quad (k=m+2;\ldots;2m);\qquad L(u_{2m+1})=\sum_{k=1}^{m} B_k(u_1)u_{m+1}, \]

\[ L(u_{2m+i})=\sum_{k=1}^{m} B_k(u_1)u_{m+i}\quad (i=2,3,\ldots,m). \]

From the solvability conditions for the last equations it follows that \(u_k(t)\equiv0\) \((k=m+2;\ldots;2m)\). Continuing analogous reasoning, we show that, if \(b_a\ne0\), then

\[ u_{ka}(t)\equiv0\quad (k\ne 2n+1;\; n=0,1,2,\ldots). \]

Consequently, in this case there exists a periodic solution holomorphic with respect to the parameter \(\mu\) in a neighborhood of the point \(\mu=0\).

Thus, if 1) \(p=0\ne q\); 2) equation (8) has \(k\) such distinct roots that \(b_a\ne0\) \((a=1,2,\ldots,k)\), then system (1) has \(k\) distinct periodic solutions holomorphic with respect to the parameter \(\mu\).

Remark. If \(p=0\ne q\), \(b_a=0\), then there may exist periodic solutions of system (1) expandable in integral positive or positive fractional powers of the parameter \(\mu\). This case is not considered here.

The case \(p=q=0\). Then the equations

\[ L(u)=\psi(t),\qquad L(u)=f_m(\varphi) \]

have periodic solutions.

Let \(\alpha(t)\), \(\beta(t)\) be particular periodic solutions of these equations, \(u_0(t)=c_0\varphi(t)\). Then the general solution of equation \((5_1)\) can be written in the form

\[ u_1(t)=c\varphi+\alpha(t)+c_0^m\beta(t), \]

where \(c,c_0\) are some constants.

Equation \((5_2)\) is solvable if and only if

\[ Y\left[B_m(\varphi)(\alpha(t)+c_0^m\beta(t))-f_{m-1}(\varphi)\right] =a+c_0^m b=0. \tag{9} \]

If \(a\ne0=b\), then (9) has no solutions. Consequently, system (1) has no periodic solutions representable in the form (2).

If \(a=0\ne b\), then, as in the case \(p=0\ne q\), one can obtain conditions for the existence of holomorphic periodic solutions of system (1).

The arguments given above lead us to the following theorem.

Theorem 1. Suppose that, up to a constant factor, the system \(L(u)=0\) has a unique nontrivial periodic solution.

Then:

1) if \(p\ne 0=q\), then equation (1) has no periodic solutions representable in the form of the series (2);

2) if \(p\ne 0\ne q\), then system (1) has \(m\) distinct periodic solutions representable in the form of the series (2), and for all these solutions \(v_{-1}(t)\ne0\);

3) if: a) \(p=0\ne q\); b) equation (8) has \(k\) distinct solutions \(c_a\) \((a=1,\ldots,k)\), for which

\[ b_a\ne 0\quad (a=1,\ldots,k), \]

then equation (1) has \(k\) distinct periodic solutions representable in the form of a series in positive integral powers of the parameter \(\mu\);

4) if a) \(p=0=q\); b) \(a\ne0=b\), then system (1) has no periodic solutions representable in the form of the series (2);

5) if \(p=0=q\); \(a=0\ne b\), then system (1) has no singular periodic solutions representable in the form (2).

Remarks. 1. By a certain transformation, equation (1) was brought to the form (3), and then the existence of a formal solution of equation (3) in the form of the series (4) in positive integral powers of the parameter \(\lambda\) was considered. The proof of the assertion that the series (4) have a nonzero radius of convergence, and the estimate of the radii of convergence, are cumbersome. These questions were considered in numerous works of A. M. Lyapunov, Yu. A. Ryabov, and others. In [5] a certain list of literature on this question is given.

2) If \(a\ne0\ne b\), then a periodic solution of system (1), representable in the form (2), exists only when an enumerable number of conditions is satisfied. These conditions are not given here.

In what follows it will be shown that, in the case \(p=q=0\), there may exist a periodic solution of system (1) representable in the form of a series in other fractional powers.

II

There exist differential equations of the form (1) which have periodic solutions representable in the form of the series

\[ v(t,\mu)=\sum_{k=-1}^{\infty} v_k(t)\mu^{\frac{1}{m-1}} . \tag{10} \]

For example, the scalar equation \(u''+u=1+\mu u^2\) has the periodic solution

\[ u=\frac{1}{2\mu}\left(1+\sqrt{1-4\mu}\right), \]

representable in the form of the series (10).

By applying the implicit function theorem, it is easily shown that the scalar differential equation

\[ a_0u''+a_1u'+a_2u=a_3+\mu\sum_{k=0}^{m} b_k u^k, \]

where \(a_2, a_3, b_0,\ldots,b_m\) are constants, \(a_2\ne0\ne b_m\), and, for odd \(m\), \(b_m\) and \(a_2\) have the same signs, has a real periodic solution representable in the form (10), with \(v_{-1}(t)\ne0\).

Let us now consider the existence of a periodic solution of system (1) representable in the form of the series (10).

After the substitution \(\mu=\lambda^{m-1}\), \(v=\dfrac{u}{\lambda}\), equation (1) takes the form

\[ L(u)\lambda\psi+\sum_{k=0}^{m}\lambda^{m-k} f_k(u). \tag{11} \]

In order that, in what follows, it be possible to apply known theorems from the theory of linear differential equations with periodic coefficients, we shall assume that

\[ f_m(w)=\sum_{\sum \alpha_k=m} f_{m\alpha_1\ldots\alpha_n}(t)w_1^{\alpha_1}\ldots w_n^{\alpha_n}, \]

where \(w_1,\ldots,w_n\) are the components of the vector \(w\); \(f_{m\alpha_1\ldots\alpha_n}(t)\) is an \(n\)-dimensional vector*.

We shall seek a solution of equation (11) in the form of the series (4). Then, by the usual method, one obtains the following recurrent system of equations:

\[ L(u_0)=f_m(u_0), \tag{12_0} \]

\[ M(u_1)=\psi(t)+f_{m-1}(u_0), \tag{12_1} \]

\[ M(u_k)=r_k(u_0,u_1,\ldots,u_{k-1})\quad (k=1,2,\ldots), \tag{12_k} \]

where \(r_k(u_0,u_1,\ldots,u_{k-1})\) is a known function of the vectors \(u_0,u_1,\ldots,u_{k-1}\);

\[ M(u)\equiv \frac{du}{dt}-F(t)u;\qquad F(t)\equiv A(t)+C(t); \]

the matrix \(C(t)\equiv C(t,u_0)\) is determined from the equality

\[ f_m(u,u_0,\ldots,u_0)+f_m(u_0,u,u_0,\ldots,u_0)+\cdots+f_m(u_0,\ldots,u_0,u)\equiv C(t)u. \]

The following holds.

Theorem 2. If equation \((12_0)\) has \(k\) distinct periodic solutions, \(u_{0\alpha}(t)\) \((\alpha=1,\ldots,k)\), for which the corresponding systems

\[ \frac{du(t)}{dt}=[A(t)+C(t,u_{0\alpha})]u(t) \]

have only trivial periodic solutions, then system (1) has \(k\) distinct periodic solutions representable in the form of the series (10), and at least \(k-1\) of them are singular periodic solutions.

* Analogues of Theorems 2 and 3 hold without this assumption, but under other restrictions. For this it is necessary to state a number of auxiliary propositions, which would require a considerable increase in the length of the paper.

Proof. From the recurrence relations \((12_k)\), by virtue of the conditions of the theorem, the periodic vectors \(u_1(t), u_2(t), \ldots\) are uniquely determined through \(u_0(t)\).

Thus, the series

\[ u(t)=\sum_{k=0}^{\infty}\lambda^k u_k(t) \tag{41} \]

formally satisfies system (11).

It is obvious that the series (10), where \(v_k(t)\equiv u_{k+1}(t)\), formally satisfies system (1), and moreover \(v_{-1}(t)\ne 0\), if \(u_0(t)\ne 0\).

We now show that the series \((41)\) has a nonzero radius of convergence.

After the substitution \(u=w+u_0\), equation (11) can be written in the form

\[ M(w)=\lambda\psi+\sum_{k=2}^{m}x_k(t)w^k+ \sum_{k=0}^{m-1}\lambda^{m-k}f_k(w+u_0), \tag{11_1} \]

where

\[ \sum_{k=2}^{m}x_k w^k=f_m(w+u_0)-f_m(u_0)-C(t,u_0)w. \]

On the basis of one of the author’s formulas [6], equation \((11_1)\) in the class of periodic functions is equivalent to the integral equation

\[ w(t)=-W(t)B^{-1}D\int_t^{t+\omega}W^{-1}(s)r(s)\,ds, \tag{11_2} \]

where

\[ r(t)=\lambda\psi(t)+\sum_{k=2}^{m}x_k(t)w^k(t)+ \sum_{k=0}^{m-1}\lambda^{m-k}f_k(w+u_0); \]

\(W(t)\) is the fundamental matrix of the system \(M(u)=0\); \(D=W(\omega)\); \(B=D-E\); \(E\) is the identity matrix.

Suppose that on the segment \(0\le t\le \omega\) the inequalities

\[ \left\|W(t)B^{-1}D\int_t^{t+\omega}W^{-1}(s)\,ds\right\|\le T;\qquad \|u_0(t)\|\le z_0;\qquad \|\psi\|\le \psi_0; \]

\[ \|x_k(t)w^k\|\le a_k\|w\|^k;\qquad \|f_k(w+u_0)\|\le b_k(\|w\|+z_0)^k \]

hold. Then the equation

\[ \Phi(h,z)\equiv z-T\left\{ h\psi_0+\sum_{k=2}^{m}a_k z^k+ \sum_{k=0}^{m-1}h^{m-k}b_k(z+z_0)^k \right\}=0, \]

where \(h=|\lambda|\), is a majorant of equation \((11_2)\).

The equation \(\Phi(0,z)=0\) has the solution \(z=0\), and moreover

\[ \left.\frac{d\Phi(h,z)}{dz}\right|_{z=0;\ h=0}\ne 0. \]

Since, in addition, \(\Phi(h,z)\) is a holomorphic function of \(h\) and \(z\), by the implicit-function existence theorem the equation \(\Phi(h,z)=0\) has a unique solution \(z(h)\) satisfying the condi-

\(z(0)=0\) and holomorphic with respect to the parameter \(h\) in a neighborhood of the point \(h=0\)

\[ z(h)=\sum_{k=1}^{\infty} h^k z_k . \tag{4_2} \]

The radius of convergence of the series \((4_1)\) is not smaller than the radius of convergence of the series \((4_2)\).* Suppose that \(u_0(t)\equiv 0\). Then \(C(t)\equiv 0\). Since, by assumption, the equation \(M(u)\equiv 0\) has no nontrivial periodic solutions, the equation \(L(u)=0\) also has no such solutions. It is known that in this case equation (1) has a unique periodic solution, holomorphic with respect to the parameter \(\mu\) in a neighborhood of the point \(\mu=0\). It can be shown that this solution is unique also in the class of periodic solutions representable in the form of the series (10). For this it is enough to note that the system \((12_k)\) in this case takes the form

\[ M(u_1)=\psi,\qquad M(u_k)=0\quad (k=2,\ldots,m-1), \]

\[ M(u_m)=\sum_{k=0}^{m} f_k(u_1),\ldots . \]

Let us consider an example. Let \(\omega=2\pi\); let \(\psi(t)\), \(f_k(t)\) \((k=0,1,2)\) be arbitrarily fixed periodic functions. The scalar equation

\[ L_1(v)\equiv v''+v=\psi(t)+\mu\left[v^3+\sum_{k=0}^{2} f_k(t)v^k\right] \tag{B} \]

has two real periodic solutions representable in the form of the series

\[ v(t,\mu)=\sum_{k=-1}^{\infty} v_k(t)\mu^{k/2}, \tag{B_1} \]

where \(v_{-1}(t)\equiv 1;\,-1\); since 1) the equation \(L_1(u_0)=u_0^3\) has the periodic solutions \(u_{00}=1\), \(u_{01}=-1\), \(u_{00}=0\); 2) the analogues of the operator \(M\), constructed for \(u_0=1;\,-1\), have one and the same form

\[ M_1(u)\equiv u''-2u; \]

3) the equation \(M_1(u)=0\) has no nontrivial periodic solutions.

The existence of a holomorphic periodic solution of equation \((B)\) does not follow from Theorem 2, since for \(u_0=0\) the operator \(M_1(u)\equiv L_1(u)\), and the equation \(L_1(u)=0\) has a nonzero periodic solution. It is easy to see that if the function \(\psi(t)\) is not orthogonal to the functions \(\sin t,\cos t\), then equation \((B)\) has no holomorphic periodic solutions.

As a second example let us consider the scalar equation

\[ L(v)\equiv v''+a(t)v'+v=\psi(t)+\mu\left[v^3+\sum_{k=0}^{2} f_k(t)v^k\right], \tag{\(\Gamma\)} \]

where \(a(t)\) is also a periodic function. If the equation \(M(u)\equiv u''+a(t)u'-2u=0\) has only the zero periodic solution,

\[ \text{* By another, more complicated device one can show that if equation (11) has a formal solution in the form of a series in integral positive powers of the parameter \(\lambda\), then this series has a nonzero radius of convergence.} \]

then the equation \((\Gamma)\) has two real periodic solutions representable in the form of the series \((B_1)\).

The case where the equation \(M(u)=0\) has a nontrivial periodic solution. For simplicity we shall restrict ourselves to considering the case where this equation, up to a constant factor, has only one nontrivial periodic solution \(\varphi(t)\). Let \(z(t)\) be a nontrivial periodic solution of the system adjoint to \(M(u)=0\). As in item 1, define the functional

\[ T(x)=\int_0^\omega (z(t);\,x(t))\,dt . \]

Let \(u_0(t)\equiv 0\). Then the recurrence system \((12_k)\) takes the form

\[ L(u_1)=\psi, \tag{13_1} \]

\[ L(u_k)=0 \quad (k=2,\ldots,m-1), \tag{13_k} \]

\[ L(u_m)=\sum_{k=0}^{m} f_k(u_1), \tag{13_m} \]

\[ L(u_{m+1})=\sum_{k=1}^{m} B_k(u_1)u_2, \tag{13_{m+1}} \]

\[ L(u_n)=\sum_{k=1}^{m} B_k(u_1)u_{n-m+1}+r_n[u_1,\ldots,u_{n-m}]. \tag{13_n} \]

Let

\[ P=T(\psi)=0. \]

Since \(u_1=c\varphi+\psi_1\), the solvability condition for equation \((13_m)\) leads to an algebraic equation \((8_1)\) of the form (8) with respect to \(c\) (see item I, the case \(p=0\ne q\)).

If equation \((8_1)\) has \(k\) distinct solutions \(c_1,\ldots,c_k\) such that \(b_a\ne 0\) \((a=1,2,\ldots,k)\), then in the class of functions representable in the form of the series (10), equation (1) has \(k\) distinct periodic solutions, and all these solutions are representable as series in integral positive powers of the parameter \(\mu\).

Let \(u_0(t)\not\equiv 0\) be a periodic solution of equation \((12_0)\);

\[ \varphi_k(w_1,\ldots,w_k)=\Sigma f_k[w_{n_1},\ldots,w_{n_k}], \]

where the summation is carried out over all distinct permutations \([n_1,\ldots,n_k]\) of the numbers \([1,2,\ldots,k]\);

\[ \Phi(u,v,u_0)=\varphi_k(u,v,u_0,\ldots,u_0). \]

We now write the recurrence system \((12_k)\) in the form

\[ M(u_1)=\psi+f_{m-1}(u_0), \tag{12_1^*} \]

\[ M(u_2)=\Phi(u_1,u_1,u_0)+B_{m-1}(u_0)u_1+f_{m-2}(u_0), \tag{12_2} \]

\[ M(u_3)=\Phi(u_1,u_2,u_0)+B_{m-1}(u_0)u_2+r_3(u_0,u_1), \tag{12_3} \]

\[ M(u_k)=\Phi(u_1,u_{k-1},u_0)+B_{m-1}(u_0)u_{k-1}+r_k(u_0,\ldots,u_{k-2}), \tag{12_k} \]

where \(r_k(\ )\) is a known function of \(u_0,\ldots,u_{k-2}\).

Equation (12) has a periodic solution if and only if the equality

\[ P+Q=0, \]

is satisfied, where

\[ Q=T[f_{m-1}(u_0)]; \qquad P=T[\psi(t)]. \]

Suppose that this equality is satisfied. Then \(u_1=c\varphi(t)+\psi_1(t)\), where \(c\) is a certain constant, and \(\psi_1\) is a particular periodic solution of equation \((12_1^*)\). The solvability condition for equation \((12_2)\) leads to the algebraic equation

\[ a_0c^2+a_1c+a_2=0, \tag{13} \]

where

\[ a_0=T[\Phi(\varphi,\varphi,u_0)], \]

\[ a_1=T[B_{m-1}(u_0)\varphi+2\Phi(\varphi,\psi_1,u_0)], \]

\[ a_2=T[B_{m-1}(u_0)\psi_1+\Phi(\psi_1,\psi_1,u_0)+f_{m-2}(u_0)]. \]

If equation (13) has no solutions, then equation (1) has no periodic solutions representable in the form of the series (10).

Suppose that (13) has a solution \(c=c_n\) \((n=1,2)\). Let \(u_{1n}=c_n\varphi+\psi_1\); \(u_{2n}=h\varphi+\psi_2\), where \(\psi_2\) is a particular solution of equation \((12_2)\), when in the right-hand side it is assumed that \(u_1=u_{1n}\).

Let

\[ P_n=T[B_{m-1}(u_0)\varphi+\Phi(u_{1n},\varphi,u_0)], \qquad P_1\ne 0. \]

Then, from the solvability condition for equation \((12_3)\), the constant \(h\) is uniquely determined through \(u_{11}\), \(u_0\); consequently, \(u_{21}(t)\) is uniquely determined (through \(u_0,u_{11}\)). Further, from the solvability conditions of the subsequent equations one determines \(u_{31}(t)\), \(u_{41}(t)\), \(\ldots\).

Thus, the following has been proved.

Theorem 3. Let 1) \(u_0(t)\ne 0\) be a periodic solution of equation \((12_0)\); 2) up to a constant factor, the system \(M(u)=0\) has only one nontrivial periodic solution.

Then:

1) equation (1) has no singular periodic solutions representable in the form of the series (10), if at least one of the conditions is satisfied:
a) \(P+Q\ne 0\); b) equation (13) has no solutions;

2) if: a) \(P+Q=0\); b) equation (13) has a solution \(c=c_1\), for which \(P_1\ne 0\), then equation (1) has a singular periodic solution representable in the form (10). If these conditions are satisfied, and in addition equation (13) has two distinct solutions and \(P_1\ne 0\ne P_2\), then the system has two singular periodic solutions representable in the form of the series (10).

III

Let us consider the existence of a periodic solution of system (1), representable in the form

\[ v(t)=\sum_{k=-1}^{\infty} v_k(t)\mu^{\frac{k}{q}}, \]

where \(q=m+c\); \(c\) is a positive integer.

We shall restrict ourselves to the consideration of the case \(c=1\), i.e., we shall study the existence of a periodic solution representable in the form of the series

\[ v(t)=\sum_{k=-1}^{\infty} v_k(t)\,\mu^{\frac{k}{m+1}} . \tag{14} \]

The general case is studied similarly.

Next suppose that the system \(L(u)=0\), up to a constant factor, has only one nontrivial periodic solution \(\varphi(t)\); the functional \(Y(x)\) is defined in the same way as in Section I.

After the substitution \(\mu=\lambda^{m+1}\), \(v=u:\lambda\), system (1) takes the form

\[ L(u)=\lambda\psi+\lambda^2 f_m(u)+\lambda^3 f_{m-1}(u)+\cdots+\lambda^{m+2} f_0(t). \tag{15} \]

We shall seek a solution of system (15) in the form of series (4). Then

\[ L(u_0)=0, \tag{16_0} \]

\[ L(u_1)=\psi(t), \tag{16_1} \]

\[ L(u_2)=f_m(u_0), \tag{16_2} \]

\[ L(u_3)=B_m(u_0)u_1+f_{m-1}(u_0), \tag{16_3} \]

\[ L(u_4)=B_m(u_0)u_2+\frac{1}{2}\Phi(u_1,u_1,u_0)+B_{m-1}(u_0)u_1+f_{m-2}(u_0), \tag{16_4} \]

\[ \begin{aligned} L(u_k)&=B_m(u_0)u_{k-2}+\Phi(u_1,u_{k-3},u_0)+B_{m-1}(u_0)u_{k-3}\\ &\quad + r_k(u_0,u_1,\ldots,u_{k-4}). \end{aligned} \tag{16_k} \]

Obviously, \(u_0(t)=c_0\varphi\). Equations \((16_1)\), \((16_2)\) are solvable if and only if

\[ p=Y[\psi(t)]=0; \qquad q=Y[f_m(\varphi)]=0. \]

Let these conditions be satisfied. Then \(u_1=c_1\varphi+\psi_1\), \(u_2=c_2\varphi+c_0^m\varphi_2=c_2\varphi+\psi_2\), where \(\varphi_2\) is a particular periodic solution of the system \(L(u)=f_m(\varphi)\). Taking into account \(q=0\), system \((16_3)\) is solvable if and only if

\[ P_1=Y[B_m(\varphi)\psi_1+f_{m-1}(\varphi)]=0. \]

Let

\[ P_2=Y[\Phi(\psi_1,\varphi,\varphi)+B_{m-1}(\varphi)\varphi]. \]

The solvability condition for system \((16_4)\) is written in the form

\[ c_0^{m+1}Y[B_m(\varphi)\psi_2]+P_2c_1+Y\left[\frac{1}{2}\Phi(\psi_1,\psi_1,\varphi)+\right. \]

\[ \left.{}+B_{m-1}(\varphi)\psi_1+f_{m-2}(\varphi)\right]\equiv a_{41}c_0^{m+1}+P_2c_1+a_{42}=0. \]

If \(P_2\ne0\), then from this \(c_1\) is uniquely determined (in terms of \(c_0\)).

Let \(u_3=c_3\varphi+\psi_3\). The constant \(c_2\) is determined from the solvability condition of equation \((16_5)\)

\[ P_2c_2+Q_5=0, \]

where

\[ Q_5=Y[c_0B_m(\varphi)\psi_3+\Phi(\psi_1,\psi_2,\varphi)+c_1\Phi(\varphi,\psi_2,\varphi)+ \]

\[ {}+B_{m-1}(\varphi)\psi_2+c_0^{-m+1}r_5(u_0,u_1)]. \]

The vectors \(u_3(t), \ldots\) are determined from the solvability conditions for the subsequent equations of system \((16_k)\).

Thus the following has been established.

Theorem 4. Suppose that the system \(L(u)=0\), to within a constant factor, has only one nontrivial periodic solution. Then:

1) system (1) has no singular periodic solutions representable in the form (14), if at least one of the following conditions is satisfied:
a) at least one of the numbers \(p,q\) is different from zero; b) \(p=-q=0\ne P_1\);

2) if \(p=q=P_1=0\ne P_2\), then system (1) has a singular periodic solution representable in the form of the series (14).

IV

Let us now consider the condition for the existence of a periodic solution of system (1) representable in the form

\[ v(t)=\sum_{k=-p}^{\infty} v_k(t)\mu^{\frac{k}{q}}, \]

where \(p\) and \(q\) are integers. After the substitution \(\mu=\lambda^q,\; v=u:\lambda^p\), system (1) takes the form

\[ L(u)=\lambda^p\psi+\sum_{k=0}^{m}\lambda^{q+(1-k)p}f_k(u). \]

Then, putting \(q+p>mp\), one seeks a periodic holomorphic solution of the last equation. As an example, consider the equation

\[ L(v)=\psi(t)+\mu[f_3(v)+f_2(v)+f_1(v)+f_0(v)], \tag{17} \]

whose periodic solution is sought in the class of functions representable in the form of the series

\[ v(t)=\sum_{k=-2}^{\infty} v_k(t)\mu^{\frac{k}{4}}, \tag{18} \]

taking into account that \(p=2,\; q=4,\; m=3\). By the substitution indicated above, system (17) is reduced to the form

\[ L(u)=\lambda^2\psi+f_3(u)+\lambda^2 f_2(u)+\lambda^4 f_1(u)+\lambda^6 f_0(u). \tag{19} \]

We shall seek the solution of system (19) in the form of the series (4). Then

\[ L(u_0)=f_3(u_0), \tag{20_0} \]

\[ L(u_1)=B_3(u_0)u_1, \tag{20_1} \]

\[ L(u_2)=B_3(u_0)u_2+\Gamma(u_1,u_1,u_0)+\psi+f_2(u_0), \tag{20_2} \]

where

\[ \Gamma(u_1,u_1,u_0)=f_3(u_1,u_1,u_0)+f_3(u_1,u_0,u_1)+f(u_0,u_1,u_1), \]

\[ L(u_3)=B_3(u_0)u_3+\Phi(u_1,u_2,u_0)+f_3(u_1,u_1,u_1), \tag{20_3} \]

\[ L(u_k)=B_3(u_0)u_k+\Phi(u_1,u_{k-1},u_0)+r_k(u_0,u_1,\ldots,u_{k-2}), \tag{20_k} \]

where

\[ \begin{aligned} \Phi(u,v,w)={}&f_3(u,v,w)+f_3(v,w,u)+f(w,u,v)\\ &+f_3(u,w,v)+f_3(w,v,u)+f(v,u,w). \end{aligned} \]

Suppose that system \((20_0)\) has a nontrivial periodic solution \(u_0(t)\).

Let

\[ M(u)\equiv \frac{du}{dt}-\bigl(A(t)+B_3(u_0)\bigr)u . \]

If the system \(M(u)=0\) has only the trivial periodic solution, then one obtains a solution of system (17) representable in the form of the series

\[ v(t)=\sum_{k=-1}^{\infty} v_k(t)\mu^{\frac{k}{2}} . \]

This case was considered in Section II.

Let the equation \(M(u)=0\), up to a constant factor, have only one nontrivial periodic solution \(\varphi(t)\); let \(z(t)\) be a nontrivial periodic solution of the corresponding adjoint system; the functional \(T(x)\) is defined by the same rule as in Section I.

\[ u_1(t)=c_1\varphi(t). \]

The solvability condition for equation \((20_2)\) leads to the algebraic equation

\[ a_0c_1^2+a_1=0, \tag{21} \]

where

\[ a_0=T[\Gamma(\varphi,\varphi,u_0)],\qquad a_1=T[\psi+f_2(u_0)]. \]

Suppose that equation (21) has a solution \(c_1=c_{1n}\) for which

\[ p_n=T[\Phi(u_{1n},\varphi,u_0)]\ne 0, \]

where

\[ u_{1n}=c_{1n}\varphi(t). \]

Then from the solvability conditions of the subsequent equations \((20_k)\) the vectors \(u_2(t), u_3(t),\ldots\) are uniquely determined through \(c_{1n}\).

Thus, we have proved

Theorem 5. Let 1) equation \((20_0)\) have a nontrivial periodic solution \(u_0(t)\); 2) the system \(M(u)=0\), up to a constant factor, have only one nontrivial periodic solution \(\varphi(t)\).

Then:

1) system (17) has no singular periodic solutions representable in the form (18), if equation (21) has no solutions;

2) if equation (21) has a solution \(c=c_{11}\) for which \(p_1\ne 0\), then system (17) has a singular periodic solution representable in the form of the series (18).

V

As an example, consider the scalar equation

\[ L(y)\equiv \frac{d^2y}{dt^2}+y=\psi(t)+\mu\sum_{k=0}^{m} f_k(y)^{*}\quad (m>1), \tag{22} \]

where \(f_k(y)\) \((k=1,2,\ldots,m-1)\) is a \(k\)-linear operation on \(y(t)\), mapping—

\[ {}^{*}\ \text{The more general case } \frac{d^2y}{dt^2}+k^2y=F(t,y) \text{ is reduced to the form (22) by the substitution} \]

\[ \tau=dt. \]

which maps a periodic function \(y(t)\) of period \(\omega=2\pi\) into a periodic function with the same period; \(f_m(y)=a_m y^m\), where \(a_m\) is constant, and for odd \(m\), \(a_m>0\), \(\mu>0\), \(\mu\) is a small parameter; \(\psi(t+2\pi)=\psi(t)\).

We shall show that equation (22) has at least one singular real periodic solution representable in the form

\[ y=\sum_{k=-1}^{\infty} y_k(t)\mu^{\frac{k}{m-1}} . \tag{23} \]

By the substitution \(\mu=\lambda^{m-1}\), \(y=x\lambda\), equation (22) is brought to the form

\[ L(x)=\lambda\psi+a_m x^m+\lambda f_{m-1}(x)+\lambda^2 f_{m-2}(x)+\cdots+\lambda^m f_0(t). \tag{24} \]

We shall seek a solution of this equation in the form

\[ x(t)=\sum_{k=0}^{\infty} x_k(t)\lambda^k . \tag{25} \]

Then

\[ L(x_0)=a_m x_0^m, \tag{26_0} \]

\[ L(x_1)=m a_m x_0^{m-1}x_1+\psi(t), \tag{26_1} \]

\[ L(x_k)=m a_m x_0^{m-1}x_k+r_k(x_0,x_1,\ldots,x_{k-1}). \tag{26_k} \]

Equation \((26_0)\) has the nontrivial periodic solution

\[ x_0=\left(\frac{1}{a_m}\right)^{\frac{1}{m-1}}. \]

Then the recurrence relations \((26_k)\) may be written in the form

\[ M(x_1)=\psi, \tag{27_1} \]

\[ M(x_k)=r_k(x_0,x_1,\ldots,x_{k-1}), \tag{27_k} \]

where

\[ M(u)\equiv \frac{d^2u}{dt^2}-(m-1)u. \]

Since the equation \(M(u)=0\) has only the trivial periodic solution, the recurrence relations \((27_k)\) uniquely determine the periodic functions \(x_1(t), x_2(t), \ldots\). Consequently, equation (22) has a real singular periodic solution representable in the form

\[ y=\sum_{k=-1}^{\infty} x_{k+1}(t)\mu^{\frac{k}{m-1}}, \]

where \(x_0\ne0\).

The Duffing equation

\[ \frac{d^2y}{dt^2}+k^2y-\gamma y^3=\psi(t)+\gamma_1 f(t), \]

where \(\gamma\) and \(\gamma_1\) are assumed small; \(\psi(t+2\pi)=\psi(t)\); \(f(t+2\pi)=f(t)\), in the resonant case, when \(k^2=1\) or differs little from it, reduces to an equation of the form (22).

Indeed, putting \(1-k^2=\mu a\); \(\gamma=\mu a_3\); \(\gamma_1=\mu a_0\), the Duffing equation can be written in the form

\[ \frac{d^2y}{dt^2}+y=\psi(t)+\mu(a_3y^3+ay+a_0f(t)). \]

Consequently, the Duffing equation (for small values of the parameter \(\mu\)) has a singular periodic solution.

References

1. Moulton F. R., Periodic orbits. Washington, 1920.
2. Plotnikov S. V. PMM, 26, No. 4, 1962.
3. Proskuryakov A. P. Periodic solutions of quasi-linear autonomous systems with one degree of freedom in the form of series in integral and fractional powers of a parameter. Analytical methods in the theory of nonlinear oscillations, 1. Proceedings of the International Symposium on Nonlinear Oscillations. Kiev, 1963.
4. Shimanov S. N. PMM, 16, issue 2, 1952.
5. Ryabov Yu. A. On estimating the range of applicability of the small-parameter method in problems of the theory of nonlinear oscillations, 1. Proceedings of the International Symposium on Nonlinear Oscillations. Kiev, 1963.
6. Bykov Ya. V., Imanaliev M. Collection “Studies on integro-differential equations in Kirghizia.” Frunze, issue 1, 1961, p. 146.

Received by the editors
February 26, 1965

Institute of Physics and Mathematics
Academy of Sciences of the Kirghiz SSR