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THE ANALYTIC STRUCTURE AND FUNDAMENTAL PROPERTIES OF SOLUTIONS OF EQUATIONS WITH ALGEBROID RIGHT-HAND SIDES IN THE CASE OF A FOCUS AND GENERALIZATIONS
A. N. ERUGIN
I. THE EQUATION OF THE INTEGRAL CURVES OF THE SYSTEM
1. Definitions. Notation
We shall call a \(T\)-function
\[ T(t)=\sum_{n=1}^{\infty}\left(a_n\sin nt+C_n\cos nt\right), \]
where
\[ \sum_{n=1}^{\infty}\left(|a_n|+|C_n|\right)<+\infty, \]
and a \(K\)-function \(a_0+T(t)\), \(a_0=\mathrm{const}\ne 0\).
We also note that, when speaking of the order of smallness of infinitesimals, we shall henceforth have in mind the definitions of equality and inequality given by us in [3]; the principal part of an infinitesimal \(\delta(\vartheta)\) will be denoted by \(G(\delta(\vartheta))\).
Referring to a formula, say, (2.5) of the article [3], we shall write (2.5) [3].
2. The equation of the integral curves in polar coordinates
Let a system be given
\[ \dot{x}=-y+X+\eta,\qquad \dot{y}=x+Y+\xi, \tag{2.1} \]
where
\[ X=\sum_{k=2}^{n+m}X_k(x,y),\qquad Y=\sum_{k=2}^{n+m}Y_k(x,y); \qquad X_k,\ Y_k, \]
are homogeneous polynomials in \(x,y\) of degree \(k\); \(\eta,\xi=0(r^{n+m})\).
Passing to polar coordinates, we obtain
\[ \dot{r}=P+p,\qquad \dot{\vartheta}=1+Q+q, \tag{2.2} \]
where
\[ P=X\cos\vartheta+Y\sin\vartheta =\sum_{k=2}^{n+m}\overline{P}_k(\vartheta)\,r^k; \]
\[ rQ=\overline{Q}=Y\cos\vartheta-X\sin\vartheta =\sum_{k=1}^{n+m-1}\overline{Q}_k(\vartheta)\,r^{k+1}; \]
\(\overline{P}_k,\ \overline{Q}_k\) are \(K\)-functions;
\[ \rho=\eta\cos\vartheta+\xi\sin\vartheta;\qquad \overline{q}=rq=\xi\cos\vartheta-\eta\sin\vartheta . \]
Hence we shall have the following equation of the integral curves
\[ r'=\sum_{k=2}^{n+m} r^k R_k(\vartheta)+R(\vartheta), \tag{2.3} \]
where
\[ R_2=\overline{P}_2,\qquad R_k=\overline{P}_k+U_k\quad (k>2), \]
\[ U_k=\sum_{s=s_0}^{k-2}\overline{P}_{k-s}\overline{U}_s \quad \bigl(s_0=\max(1,\ k-m-n)\bigr). \tag{2.4} \]
\[ \overline{U}_k=\sum_{s=\overline{s}_0}^{k}(-1)^s T_{k,s},\qquad \overline{s}_0=\max\left(1,\ E\left(\frac{k}{m+n-1}\right)\right), \]
\[ T_{k,s}=\sum_{\substack{\sum_{\sigma=1}^{s}\kappa_\sigma=k}} \prod_{\sigma=1}^{s}\overline{Q}_{k_\sigma}; \tag{2.5} \]
\[ R=\sum_{\sigma=1}^{3}\widetilde{R}_\sigma,\qquad \widetilde{R}_1=p\left(1+\sum_{k=1}^{\infty}(-1)^k(Q+q)^k\right); \tag{2.6} \]
\[ \widetilde{R}_2=P\sum_{k=1}^{\infty}(-1)^k\bigl((Q+q)^k-Q^k\bigr),\qquad \widetilde{R}_3=\sum_{k=n+m+1}^{\infty}U_k r^k . \]
Next we have
\[ G(\widetilde{R}_1)=p(r,\vartheta)\left(1+\sum_{\sigma=1}^{2} r^\sigma R_{p,\sigma}\right); \tag{2.7} \]
\[ R_{p,1}=-\overline{Q}_1,\qquad R_{p,2}=\overline{Q}_1^{\,2}-\overline{Q}_2. \]
\[ G(\widetilde{R}_2)=q(r,\vartheta)\sum_{\sigma=2}^{3} r^\sigma R_{q,\sigma};\qquad R_{q,2}=-\overline{P}_2,\quad R_{q,3}=2\overline{P}_2\overline{Q}_1-\overline{P}_3; \]
\[ G(\widetilde{R}_3)=U_{n+m+1}(\vartheta)\,r^{n+m+1}; \tag{2.8} \]
\[ u_{n+m+1}=\sum_{s=1}^{n+m+1}\sum_{k=1}^{s}(-1)^k P_{n+m+1-s} \sum_{\substack{\sum_{\sigma=1}^{k} C_{l_\sigma}=s}} \prod_{\sigma=1}^{k}\overline{Q}_{l_\sigma}; \tag{2.9} \]
\[ G(R)=\sum_{\sigma=1}^{3}G(\widetilde{R}_\sigma). \tag{2.10} \]
3. The equation of integral curves in canonical form
We shall seek a solution of equation (2.3) in the form
\[ r=C+\sum_{k=2}^{n+m}u_k(\vartheta)C^k+\xi(C,\vartheta), \tag{3.1} \]
where the \(u_k(\vartheta)\) are \(2\pi\)-periodic functions of \(\vartheta\), and \(C\) is an arbitrary constant; we shall use this notation also below. Substituting (3.1) into (2.3), we obtain, for determining the functions \(u_k\), a system of equations from which the functions \(u_k\) are successively determined. In doing so we shall have two cases:
1) the functions \(u_k\) \((k=1,2,\ldots,n+m)\) turn out to be \(2\pi\)-periodic;
2) the functions \(u_k\), beginning with some \(u_k\), say with \(u_m\), turn out to be nonperiodic.
In the first case it is not clear what the qualitative picture will be, since this will depend on \(R\), i.e., on \(\eta,\xi\), with respect to which we make no assumptions except the assumption on the order of smallness; therefore let us assume that, beginning with \(u_m\), the functions \(u_k\) will turn out to be nonperiodic, i.e., that \(u_k\) \((k=1,2,\ldots,m-1)\) are \(2\pi\)-periodic functions, \(u_m=g_0\vartheta+V(\vartheta)\), where \(g_0=\mathrm{const}\ne0\), and \(V(\vartheta)\) is a \(T\)-function [1]. In equation (2.3) make the change of variables
\[ r=\sum_{k=1}^{m}u_k(\vartheta)z^k\equiv\varphi(z,\vartheta); \qquad u_1\equiv1,\quad u_m\equiv V, \tag{3.2} \]
we obtain
\[ z'=\left[g_0z^m+\sum_{k=m+1}^{n}R_k\varphi^k+R(\varphi,\vartheta)\right]\tau(z,\vartheta), \]
where
\[ \tau(z,\vartheta)=(\varphi_z^{1})^{-1}=1+\sum_{s=1}^{\infty}G_s(\vartheta)z^s; \]
\[ G_s=\sum_{k=k_0}^{s}(-1)^k\widetilde{G}_{k,s} \quad \left(k_0=\max\left[1,\ E\left(\frac{s}{m-1}\right)+1\right]\right); \]
\[ \widetilde{G}_{k,s}= \sum_{\substack{l_\sigma=1\\ \sum_{\sigma=1}^{k}l_\sigma=s}}^{\prime} \prod_{\sigma=1}^{k} b_{l_\sigma}; \qquad b_l=(l+1)u_{l+1}\ (l=1,\ldots,m-2),\quad b_{m-1}=mV; \]
\[ \varphi^k=\sum_{s=k}^{mk}U_{k,s}(\vartheta)z^s; \qquad U_{k,s}=\sum_{\substack{\sum_{\sigma=1}^{k}l_\sigma=s}} \prod_{\sigma=1}^{k}u_{l_\sigma} \quad (u_k=0,\ k>m). \]
Hence we obtain the following equation
\[ z'=\sum_{q=0}^{n} Z_q(\vartheta) z^{m+q}+Z(z,\vartheta), \tag{3.3} \]
where
\[ Z_0=g_0=\mathrm{const}\ne 0; \]
\[ Z_q=\sum_{k=0}^{q} W_k G_{q-k}=\sum_{s=0}^{\tilde s} W_s G_{q-s} \]
\[ (\tilde s=\min(q,s),\ W_0\equiv g_0,\ W_q=0\ (q>n),\ G_0\equiv 1); \]
\[ W_q=\overline W_{q-m};\qquad \overline W_q=\sum_{s=s_0}^{q} R_s U_{s,q} \quad \left(s_0=\max\left(2,\ E\left(\frac{q}{m}\right)+1\right)\right); \]
\[ Z=Z_1+Z_2;\qquad Z_1=\sum_{q=n+1}^{\infty} Z_q z^{m+q};\qquad Z_2=\tau(z,\vartheta)R(\varphi(\vartheta),\vartheta); \tag{3.4} \]
\[ G(Z_1)=Z_{n+1}Z^{n+m+1}; \tag{3.5} \]
\[ G(Z_2)=p(\varphi,\vartheta)\left(1+\sum_{\sigma=1}^{2} Z_{p,\sigma}z^\sigma\right) +q(\varphi,\tau)\sum_{\sigma=2}^{3} Z_{q,\sigma}z^\sigma \simeq p(\varphi(\vartheta),\vartheta)^*; \tag{3.6} \]
\[ G(Z)=\sum_{\sigma=1}^{2}G(Z_\sigma). \tag{3.7} \]
II. ANALYTIC STRUCTURE AND FUNDAMENTAL PROPERTIES OF THE ASYMPTOTIC EXPANSION OF THE SOLUTION OF THE EQUATION OF INTEGRAL CURVES IN CANONICAL FORM
4. Analytic structure of the solution of the equation. Notation
We shall begin the study of the analytic structure of the solution of equation (3.3). Applying the method set forth in papers [3] and [4], we obtain
\[ z=\vartheta_1^{-1}(1+\gamma_2),\qquad \vartheta_1=\xi \vartheta^{\frac{1}{m-1}},\qquad \xi=g_0(1-m)^{\frac{1}{1-m}}; \tag{4.1} \]
\[ \gamma_2=\sum_{s=1}^{\infty}a_s\gamma_1^s,\qquad a_s=(-1)^s(s!)^{-1}\prod_{l=1}^{s}\left(l-(m-2)(m-1)^{-1}\right)(1-m)^3; \tag{4.2} \]
\[ \gamma_1(\vartheta)=\vartheta_1^{-(m-1)}(s_0+I),\qquad s_0=(1-m)^{-1}z_0^{1-m}-g_0\vartheta_0; \]
\[ I=\int_{\vartheta_0}^{\vartheta} \left[\sum_{q=1}^{n} Z_q z^q+z^{-m}Z\right]\,d\vartheta. \tag{4.3} \]
Next we shall have
\[ \gamma_1=\sum_{q=0}^{k}\sum_{s=q(m-1)}^{n} U_{q,s}(\vartheta)\vartheta_1^{-s}\ln^q\vartheta+\tilde\delta(\vartheta) \equiv \gamma_{1,n}+\tilde\delta; \tag{4.4} \]
\[ \text{* The sign } \simeq \text{ means that terms of higher order than those written are discarded.} \]
\[ \gamma_2=\sum_{q=0}^{k}\sum_{s=q(m-1)}^{n} H_{q,s}(\vartheta)\vartheta_1^{-s}\ln^q\vartheta+\delta(\vartheta)\equiv \gamma_{2,n}+\delta, \tag{4.5} \]
where
\[ U_{q,s}=u_{q,s}+\bar U_{q,s},\qquad H_{q,s}=h_{q,s}+\bar H_{q,s}; \]
\(u_{q,s}, h_{q,s}=\mathrm{const}\); \(\bar U_{q,s}, \bar H_{q,s}\) are \(T\)-functions, \(n=k(m-1)+x\). Let us note that the coefficients \(U_{q,s}, H_{q,s}\) can be obtained by the recurrence formulas obtained by us in [2—4]. Let us also note that the fundamental properties of the asymptotic expansion of the solution described by us in [2—6] are preserved. Denote
\[ \Theta(\vartheta)=\sum_{q=1}^{n} Z_q(\vartheta)\vartheta_1^{-q}(1+\gamma_{2,n}+\delta)^q; \]
\[ \tilde Z(\vartheta)=\vartheta_1^m(1+\gamma_{2,n}+\delta)^{-m}Z\bigl(\vartheta_1^{-1}(1+\gamma_{2,n}+\delta),\vartheta\bigr); \]
\[ \tilde Z_\sigma(\vartheta)=\vartheta_1^m(1+\gamma_{2,n}+\delta)^{-m}Z_\sigma\bigl(\vartheta_1^{-1}(1+\gamma_{2,n}+\delta),\vartheta\bigr)\quad(\sigma=1,2); \]
\[ \tilde\gamma_1(\vartheta)=\vartheta_1^{-(m-1)} \left[\left(s_0+\int_{\vartheta_0}^{\infty}\tilde Z\,d\vartheta\right)+ \int_{\vartheta_0}^{\vartheta}\Theta\,d\vartheta\right]\quad(n\ge m); \]
\[ \tilde\gamma_1(\vartheta)=\vartheta_1^{-(m-1)} \left(s_0+\int_{\vartheta_0}^{\vartheta}\Theta\,d\vartheta\right)\quad(n<m); \]
\[ \bar\delta_1(\vartheta)=\tilde\gamma_1-\gamma_{1,n}; \]
\[ \bar\delta_\sigma(\vartheta)=\vartheta_1^{-(m-1)}\int_{\infty}^{\vartheta}\tilde Z_{\sigma-1}\,d\vartheta \quad(\sigma=2,3;\ n\ge m); \]
\[ \bar\delta_\sigma(\vartheta)=\vartheta_1^{-(m-1)}\int_{\vartheta_0}^{\vartheta}\tilde Z_{\sigma-1}\,d\vartheta \quad(\sigma=2,3;\ n<m); \]
\[ f_l^{p,q}=-\frac{1}{(q-m+1)^{l+1}}\, \frac{(m-1)^{l+1}}{\xi^{m-1}}\, \frac{p!}{(p-l)!}; \]
\[ T_s^{q,\sigma}= \sum_{\sum_{\eta=1}^{s}q_\eta=q} \sum_{\sum_{\eta=1}^{s}\sigma_\eta=\sigma} \prod_{\eta=1}^{s} H_{q_\eta,\sigma_\eta}; \]
\[ G_s^{q,\sigma}= \sum_{\sum_{\eta=1}^{s}q_\eta=q} \sum_{\sum_{\eta=1}^{s}\sigma_\eta=\sigma} \prod_{\eta=1}^{s} U_{q_\eta,\sigma_\eta}. \]
5. The order of \(\delta(\vartheta)\) and \(G(\hat o)\)
Taking into account formulas (3.4)—(3.7); (4.2)—(4.5), we shall have
\[ \bar\delta(\vartheta)=\sum_{\sigma=1}^{3}\bar\delta_\sigma(\vartheta),\qquad \delta(\vartheta)=\sum_{\sigma=1}^{4}\delta_\sigma(\vartheta), \tag{5.1} \]
where
\[ \bar{\delta}_{\sigma}=\sum_{s=1}^{\infty} a_s(\gamma_{1,n}+\bar{\delta}_{\sigma})-\gamma_{2,n} \qquad (\sigma=1,2,3); \]
\[ \bar{\delta}_{4}=\sum_{s=1}^{\infty} a_s(\gamma_{1,n}+\bar{\delta})-\gamma_{2,n} -\sum_{\sigma=1}^{3}\bar{\delta}_{\sigma}. \]
The structure of \(G(\bar{\delta}_{\sigma}),\, G(\delta_{\sigma})\) \((\sigma=1,\ldots,4)\) depends on \(n,m,k,x\) and on the relations among them; therefore in item 5 we shall study only the principal cases, and subsequently we shall study special cases.
Substituting (4.1) and (4.5) into (4.3), integrating and changing the order of summation, we obtain
\[ G(\bar{\delta}_{1})=\vartheta_{1}^{-(n+1)} \sum_{q=1}^{\bar{s}} Q_{q,n+1}(\vartheta)\ln^{q}\vartheta \qquad (m>2,\ k\geqslant 1,\ x<m-2), \tag{5.2} \]
where \(\bar{s}=k\), \(Q_{q,n+1}\) are \(K\)-functions, and it is not difficult to show that \(Q_{k,n+1}=\mathrm{const}\). One can also obtain formulas for \(Q_{q,n+1}\). We give only the formula for \(Q_{k,n+1}\), since this formula is the simplest and, moreover, the most needed:
\[ Q_{k,n+1}= \sum_{\sigma=1}^{x+1}\sum_{s=1}^{\sigma} C_{\sigma}^{s}g_{\sigma}^{s}T_{s}^{k,n+1-\sigma}f_{0}^{k,n+1-\sigma}. \tag{5.3} \]
Next, substituting (4.1) and (4.5) into (3.4), we shall have
\[ G(\bar{\delta}_{2})= \sum_{\sigma=1}^{2}G_{0,n+\sigma}\vartheta_{1}^{-(n+\sigma)} \qquad (m\geqslant 3), \tag{5.4} \]
where
\[ G_{0,n+1}=g_{n+1}f_{0}^{0,n+1}; \qquad G_{0,n+2}=(g_{n+2}+(n+1)g_{1}a_{1}f_{0}^{0,1})f_{0}^{0,n+2}. \]
Taking into account formulas (3.7), (4.1), and (4.5), it is also not difficult to obtain that
\[ G(\tilde{Z}_{2})= p\vartheta_{1}^{m}\left(1+\sum_{\sigma=1}^{2}\vartheta_{1}^{-\sigma}S_{p,\sigma}\right) +q\vartheta_{1}^{m}\sum_{\sigma=2}^{3}\vartheta_{1}^{-\sigma}S_{q,\sigma} \qquad (m>3), \tag{5.5} \]
where
\[ S_{p,1}=Z_{p,1}-mh_{0,1}; \qquad S_{p,2}=Z_{p,2}+Z_{p,1}h_{0,1}(1-m)-h_{0,2}m+ \]
\[ +\frac{1}{2}h_{0,1}^{2}m(m+1); \qquad S_{q,2}=Z_{q,2}; \qquad S_{q,3}=Z_{q,3}+Z_{q,2}h_{0,1}(2-m). \]
Hence we have
\[ G(\bar{\delta}_{3})=\vartheta_{1}^{-(m-1)} \int_{\infty}^{\vartheta}G(\tilde{Z}_{2})\,d\vartheta \qquad (n\geqslant m); \]
\[ G(\bar{\delta}_{3})=\vartheta_{1}^{-(m-1)} \int_{\vartheta_{0}}^{\vartheta}G(\tilde{Z}_{2})\,d\vartheta \qquad (n<m). \tag{5.6} \]
Further we have
\[ \eta_{n}(\vartheta)=\sum_{s=1}^{\infty}a_s\gamma_{1,n}^{s}-\gamma_{2,n}, \tag{5.7} \]
\[ G(\eta_n)=\vartheta_1^{-(n+1)} \sum_{q=0}^{\tilde{s}} F_{q,n+1}(\vartheta)\ln^q\vartheta, \tag{5.8} \]
where \(\tilde{s}=k;\ F_{q,n+1}\) are \(K\)-functions, and it is not difficult to show that \(F_{k,n+1}\) is obtained by the formula
\[ F_{k,n+1}=\sum_{s=2}^{n+1} G_s^{k,n+1}. \tag{5.9} \]
Hence we have
\[ G(\delta_1)=a_1G(\bar{\delta}_1)+G(\eta_n) =\vartheta_1^{-(n+1)}\sum_{q=0}^{k} S_q(\vartheta)\ln^q\vartheta \simeq \]
\[ \simeq S_k\vartheta_1^{-(n+1)}\ln^k\vartheta, \tag{5.10} \]
where
\[ S_q=a_1Q_{q,n+1}+F_{q,n+1}\quad (q=0,\ 1,\ 2,\ldots,\ k-1,\ k), \]
\[ G(\delta_\sigma)=\left(a_1+2a_2u_{0,1}\vartheta_1^{-1}\right)G(\bar{\delta}_\sigma) \quad (\sigma=2,\ 3); \tag{5.11} \]
\[ G(\delta)=\sum_{\sigma=1}^{3}G(\delta_\sigma). \tag{5.12} \]
It should be noted that the most difficult task is the determination of \(G(\delta_3)\), but this is very essential, since for a large class of equations \(G(\delta)=G(\delta_3)\). The order \(\delta_3\) and \(G(\delta_3)\) depend on \(p\), \(q\), \(S_{p,\sigma}\), \(S_{q,\sigma}\), and on the relations between them. In order to obtain \(G(\delta_3)\), one must study special cases. Problems of this type are studied in [7].
6. Special cases
Let us indicate how the special cases differ from the principal one.
Special cases for \(\delta_1\):
1) \(m=2;\ \bar{s}=n;\ \tilde{s}=n+1,\ s=n+1;\ F_{n+1,n+1}=a_{n+1}h_{1,1}^{n+1}=a_{n+1}g_1^{n+1}\times\)
\[ \times g_0^{2(n+1)};\quad S_{n+1}=F_{n+1,n+1},\quad S_q=a_1Q_{q,n+1}+F_{q,n+1};\quad \xi=1; \]
2) \(m>2,\ k=0,\ x<m-2;\) this case is analogous to the principal case;
3) \(m>2,\ k=0,\ x=m-2;\ \bar{s}=1,\ \tilde{s}=0,\ s=1;\) moreover \(Q_{0,m-1}=C\);
\[ Q_{1,m-1}=\sum_{\sigma=1}^{m-1}\sum_{s=1}^{\sigma} g_\sigma C_\sigma^s T_s^{0,m-1-\sigma}\xi^{1-m}=\mathrm{const}; \]
4) \(m>2,\ k\geqslant 1,\ x=m-2;\ \bar{s}=k,\ \tilde{s}=k+1,\ s=k+1;\ F_{n+1,n+1}=\)
\[ =a_{k+1}h_{1,1}^{k+1}. \]
Special case for \(\bar{\delta}_2,\ \delta_2\): for \(m=2\) we have
\[ G(\bar{\delta}_2)=\sum_{s=0}^{1}\sum_{\sigma=s+1}^{2}G_{s,n+\sigma}\vartheta^{-(n+\sigma)}\ln^s\vartheta, \]
where
\[ G_{0,n+1}=g_{n+1}f_0^{0,n+1},\quad G_{0,n+2}=\left(g_{n+2}-(n+1)g_0^{(-1)}C\right)f_0^{0,n+2}+ \]
\[ +(n+1)g_1g_0^{-2}f_1^{1,n+2};\quad G_{1,n+2}=(n+1)g_1g_0^{(-2)}f_0^{1,n+2}. \]
Special cases for $\tilde\delta_3,\delta_3$:
1) for $m=2$ we have
\[ \tilde Z_2 = p\vartheta^2 g_0^2 \left( 1+\sum_{\sigma=1}^{2}\sum_{s=0}^{\sigma} S_{p,\sigma,s}\vartheta^{-\sigma}\ln^s\vartheta \right) + q\vartheta^2\sum_{\sigma=2}^{3} S_{q,\sigma,0}\vartheta^{-\sigma}\ln^s\vartheta, \]
where
\[ S_{p,1,0}=-g_0^{-1}Z_{p,1}-2h_{0,1}; \qquad S_{p,2,0} = g_0^{-2}Z_{p,2}+g_0^{-1}Z_{p,1}H_{0,1} + \]
\[ +3h_{0,1}^2-2H_{0,2}; \qquad S_{p,1,1}=-2h_{1,1}; \qquad S_{p,2,1}=(g_0^{-1}Z_{p,1}+Ch_{0,1})h_{1,1}-2h_{1,2}, \]
\[ S_{p,2,2}=3h_{1,1}^2-2h_{2,2}; \qquad S_{q,2,0}=Z_{q,2,0}; \qquad S_{q,3,0}=-g_0^{-1}Z_{q,3}; \]
2) for $m=3$ we have
\[ \tilde Z_2 = p\vartheta_1^3 \left( 1+\sum_{\sigma=1}^{2}\sum_{s=0}^{\sigma-1} S_{p,\sigma,s}(\vartheta)\vartheta_1^{-\sigma}\ln^s\vartheta \right) + q\vartheta_1^3\sum_{\sigma=2}^{3} S_{q,\sigma}(\vartheta)\vartheta_1^{-\sigma}, \]
where
\[ S_{p,1,0}=S_{p,1}; \qquad S_{p,2,0}=Z_{p,2}-2Z'_{p,1}h_{0,1}-3H_{0,2}+3h_{0,1}^2; \qquad S_{p,2,1}=-3h_{1,2}. \]
We also note that for $G(\hat\vartheta_\sigma)$ $(\sigma=2,3)$ the formulas are the same.
III. Generalizations
Similar assertions can also be made for more general cases of system (2.1) and equation (3.3). For example, in system (2.1) the functions $X,Y$ may be regarded as generalized polynomials, i.e., one may study a system analogous to system (1) [9].
Further, in paper [5] we indicate that the method set forth in papers [3, 4], strictly speaking, pertains not to system (2.1), but to an equation of type (3.3), and that this method can be applied to the study of equations of a more general type than those equations which are obtained from system (2.1). For example, one may study an equation whose right-hand side is a holomorphic function with respect to $z$, whose coefficients have series of type (1.1) [4], and show that the equation likewise has a solution of the form (2.5) [3], where $\gamma_2$ has the form (1.1) [4], and that recurrent formulas can be obtained for the coefficients of the asymptotic expansion of $\gamma_2$, similar to the corresponding recurrent formulas in [3] and [4]; in what follows we shall call these formulas the formulas ($\therefore$). We shall not formulate this theorem more precisely in the present paper, but shall investigate equation (3.3), where $Z_q$ are functions of type (4.5). Let equation (3.3) be given, where $n=k(m-1)+\varkappa$, $Z(z,\vartheta)=O(z^n)$, and the coefficients $Z_q$ have the form
\[ Z_q(\vartheta) = \sum_{s=0}^{s_q} \sum_{\sigma=s(m-1)}^{\sigma_q} Z_{q,s,\sigma}(\vartheta)\vartheta_1^{-\sigma}\ln^s\vartheta + \tilde Z_q(\vartheta), \tag{7.1} \]
where
\[ \sigma_q=k_q(m-1)+\varkappa_q; \qquad \tilde Z_q(\vartheta)=O(\vartheta_1^{-\sigma_q}); \]
\[ Z_{q,s,\sigma}=g_{q,s,\sigma}-Z_{q,s,\sigma}; \qquad g_{q,s,\sigma}=\mathrm{const},\quad \overline{Z}_{q,s,\sigma} \]
are $T$-functions, $g_{0,0,0}\ne0$; henceforth we shall call this equation the equation ($\therefore$).
Applying the same method, one can show that a solution of equation ($\therefore$) has the form
\[ z=\vartheta_1^{-1}(1+\gamma_2), \qquad \gamma_2=\gamma_{2,\eta}+\delta, \tag{7.2} \]
where the function \(\gamma_{2,\eta}\) is obtained analogously to \(\gamma_{2,n}\); \(\delta=0(\vartheta_1^{-\eta})\). Equations with more general right-hand sides can also be studied in an analogous way.
7. Notation
Denote
\[ \Theta_\sigma(\vartheta) = \overline Z_0(\vartheta) + \sum_{q=1}^{n} \overline Z_q(\vartheta)\, \vartheta_1^{-q}(1+\gamma_{2,\sigma}+\delta_\sigma)^q, \]
\[ \overline Z_0=Z_0-g_{0,0,0}-\widetilde Z_0; \qquad \overline Z_q=Z_q-\widetilde Z_q \quad (\sigma=1,2,\ldots,\eta); \]
\[ \widetilde\Theta_\sigma(\vartheta) = \sum_{q=1}^{n} \widetilde Z_q(\vartheta)\, \vartheta_1^{-q}(1+\gamma_{2,\sigma}+\delta_\sigma)^q \quad (\sigma=1,2,\ldots,\eta). \]
\[ \widetilde Z_\sigma(\vartheta) = \vartheta_1^{m}(1+\gamma_{2,\sigma}+\delta_\sigma)^{-m} Z\!\left(\vartheta_1^{-1}(1+\gamma_{2,\sigma}+\delta_\sigma),\vartheta\right) \quad (\sigma=1,2,\ldots,\eta). \]
\[ -\overline\delta_{1,\sigma}(\vartheta) = \vartheta_1^{-(m-1)} \left[ \left( s_0+\int_{\vartheta_0}^{\infty} (\widetilde Z_\sigma+\widetilde\Theta_\sigma)\,d\vartheta \right) + \int_{\vartheta_0}^{\vartheta} \Theta\,d\vartheta \right]\gamma_{1,\sigma}; \]
\[ (n\ge m;\quad \mu=\min(q+\sigma_q)\ge m;\quad \sigma=1,2,\ldots,\eta); \]
\[ q\in[0,n]. \]
\[ \overline\delta_{2,\sigma}(\vartheta) = \vartheta_1^{-(m-1)} \int_{\infty}^{\vartheta} \widetilde Z_\sigma\,d\vartheta \quad (n\ge m;\quad \sigma=1,2,\ldots,\eta); \]
\[ \overline\delta_{3,\sigma}(\vartheta) = \vartheta_1^{-(m-1)} \int_{\infty}^{\vartheta} \widetilde\Theta_\sigma\,d\vartheta \quad (\mu\ge m;\quad \sigma=1,2,\ldots,\eta); \]
\[ \overline\delta_{1,\sigma}(\vartheta) = \vartheta_1^{-(m-1)} \left( s_0+\int_{\vartheta_0}^{\vartheta} \Theta_\sigma\,d\vartheta \right) -\gamma_{1,\sigma} \quad (n<m;\quad \mu<m;\quad \sigma=1,2,\ldots,\eta); \]
\[ \overline\delta_{1,\sigma}(\vartheta) = \vartheta_1^{-(m-1)} \left[ \left( s_0+\int_{\vartheta_0}^{\infty} \widetilde\Theta_\sigma\,d\vartheta \right) + \int_{\vartheta_0}^{\vartheta} \Theta_\sigma\,d\vartheta \right] -\gamma_{1,\sigma} \]
\[ (n<m;\quad \mu\ge m;\quad \sigma=1,2,\ldots,\eta); \]
\[ \overline\delta_{1,\sigma}(\vartheta) = \vartheta_1^{-(m-1)} \left[ \left( s_0+\int_{\vartheta_0}^{\infty} \widetilde Z_\sigma\,d\vartheta \right) + \int_{\vartheta_0}^{\vartheta} \Theta_\sigma\,d\vartheta \right] -\gamma_{1,\sigma} \]
\[ (n\ge m;\quad \mu<m;\quad \sigma=1,2,\ldots,\eta); \]
\[ \overline\delta_{2,\sigma} = \vartheta_1^{-(m-1)} \int_{\vartheta_0}^{\vartheta} \widetilde Z_\sigma\,d\vartheta \quad (n<m;\quad \sigma=1,2,\ldots,\eta); \]
\[ \overline\delta_{3,\sigma} = \vartheta_1^{-(m-1)} \int_{\vartheta_0}^{\vartheta} \widetilde\Theta_\sigma\,d\vartheta \quad (\mu<m;\quad \sigma=1,2,\ldots,\eta). \]
\[ \sigma_0 = \min_{q\in[0,n]}(k_q(m-1)+x_q+q) = k_{q_0}(m-1)+x_{q_0}+q_0 = k_0(m-1)+x_0; \]
\[ k_0=k_{q_0}+\widetilde k; \qquad x_0=x_{q_0}+q_0-\widetilde k; \qquad \widetilde k=E\!\left((x_{q_0}+q_0)(m-1)^{-1}\right). \]
Further, there are two main cases I: \(n<\sigma\), II: \(n\geqslant \sigma_0\); moreover it can be shown that \(\eta=\min(n,\sigma_0)\). We shall also denote \(\tilde{\delta}_{\sigma}=\tilde{\delta}_{\sigma,\eta}\) \((\sigma=1,2,3)\) and define \(\tilde{\delta}_{\sigma}\) \((\sigma=1,2,3)\) by formulas (5.1). By the same method one can show that \(\tilde{\delta}_1,\tilde{\delta}_{\sigma},\tilde{\delta}\) are obtained by formulas (5.2), (5.7)—(5.12), respectively. The special cases for \(\tilde{\delta}_1,\delta_1\) can be studied in an analogous way, but the coefficients \(S_{\sigma}\) are obtained, of course, by formulas different from the old ones. Further, about \(\delta_{\sigma},G(\delta_{\sigma})\) \((\sigma=2,3)\) one can say the same as in item 5 about \(\delta_3,G(\delta_3)\), and also refer to the article [7]. Let us note that for the present case this is important, since very often \(G(\delta)=G(\delta_{\sigma})\) \((\sigma=2,3)\), and here it is essential to obtain a classification of the functions \(Z,Z_q\) in this aspect; moreover, the classification of the functions \(L(\vartheta)\) in the article [7] should serve as the basis, and it is therefore necessary to apply it to this investigation.
8. Equation with nonperiodic coefficients
We shall indicate the properties of the asymptotic expansion of the solution of equation \((**)\), and we shall indicate only those properties that differ from the corresponding properties of the asymptotic expansion of equation (3.3).
Let \(n<\sigma_0,\ \eta=n\).
-
If \(n+1<\sigma_0\), then the coefficients \(H_{q,\sigma}\) are obtained by formulas \((*)\); the rest is analogous to the preceding case, and the structure of the coefficients \(Z_q\) is not substantially affected. We also have
\[ s=\bar{k}+E\bigl((x+1)(m-1)^{-1}\bigr),\qquad G(\delta)=\sum_{\sigma=1}^{2}\delta_{\sigma}. \] -
The subcase \(n+1=\sigma_0\) is analogous to the first subcase, but we have
\[ G(\delta)=\sum_{\sigma=1}^{3}\delta_{\sigma_0}, \]
i.e., there are cases when \(G(\delta)=G(\delta_3)\).
Let \(n\geqslant\sigma_0\); then the coefficients \(H_{q,\sigma}\) are obtained by formulas \((*)\) for
\[
q\leqslant k_{q_0},\quad \sigma\leqslant k_{q_0}(m-1)+x_{q_0}\equiv\sigma_0,
\]
and for
\[
q\in(k_{q_0},k_0],\quad \sigma\in(\sigma_0,\sigma_0]
\]
by other formulas, which cannot be obtained in the general case; in order to obtain them, one must study particular cases, applying the general method. The rest is analogous to the preceding case; let us note only that
\[
\bar{S}=q(k_0,x_0)=k_0-\varkappa(x_0),\qquad
x_0=
\begin{cases}
1&(x_0=0),\\
0&(x_0>0);
\end{cases}
\]
\[
s=\bar{S}=k_0+E\bigl((x_0+1)(m-1)^{-1}\bigr).
\]
Literature
- Lyapunov A. M. The General Problem of the Stability of Motion. GITTL, Moscow—Leningrad, 1950.
- Erugin A. N. Izv. AN BSSR, No. 4, 1958, pp. 23—32.
- Erugin A. N. Izv. AN BSSR, No. 1, 1960, pp. 27—42.
- Erugin A. N. Izv. AN BSSR, No. 2, 1960, pp. 16—26.
- Erugin A. N. DAN BSSR, vol. IV, 1960, pp. 273—275.
- Erugin A. N. Izv. AN BSSR, No. 3, 1960, pp. 17—20.
- Erugin A. N. Izv. AN BSSR, No. 1, 1961, pp. 37—41.
- Erugin A. N. DAN BSSR, vol. V, No. 5, 1961, pp. 191—193.
- Erugin A. N. DAN BSSR, vol. V, No. 7, 1961, pp. 284—285.
Received by the editors
November 1, 1964
Leningrad Branch of the V. A. Steklov Institute of Mathematics