Preamble

DIFFERENTIAL EQUATIONS 1967, Vol. III, No. 3
SYSTEMS OF DIFFERENTIAL EQUATIONS WITH FIXED CRITICAL SINGULAR POINTS

Consider a system of differential equations given by:

$$\frac{dx}{dz} = P_1(x, y, z), \quad \frac{dy}{dz} = P_2(x, y, z)$$

where $Q_i(x, y, z)$ are polynomials in the variables $x$ and $y$ with coefficients that are analytic with respect to $z$ and holomorphic in a certain domain. The expressions $\frac{P_1}{Q_1}$ and $\frac{P_2}{Q_2}$ are irreducible rational functions. The objective is to determine the conditions that the system must satisfy such that both components of the solution, $x(z)$ and $y(z)$, do not contain movable critical singular points. Let us first assume and represent the denominators respectively in the form:

$$Q_i(y, z) = \prod [y - \alpha_k(z)]^{m_k}$$

Y1 + Y2 + - + Ys = Af > 1 , Y1Y2 ••• Y s ¥ = 0 ;

$\sigma_1 + \sigma_2 + \dots + \sigma_r = m > l, \quad \sigma_1 \dots \sigma_r = \mu \neq 0;$

$P_1(y, z) = \sum P_k(z) y^k, \quad P_2(x, z) = \sum f_k(z) x^k$

where these functions are holomorphic in some domain $(z) \in D$. Taking the functions $y - a(z)$ and $x - b(z)$ as the new unknowns, respectively, and retaining the original notation, we can write the system in the form

/=2 /=2

In the following sections, we shall restrict ourselves to system transformations such that the existence of movable critical singular points in the transformed system necessarily implies their existence in the original system.

Suppose that $P \neq 0$,

$y_i > 0, \quad \chi > 1$. (3)

We introduce a parameter into system (1) (see p. 160):

$x = a^{\chi-1} \tau, \quad y = a^{\chi-1} w, \quad z = z_0 + \lambda y^{\chi-1} u,$

$\frac{dz}{d\tau} = 0$ ($i=2, k=2, \dots$). In the resulting system, we set $\lambda = 0$; then

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For the case where $l=2$ ($k=0$), we have $a \neq 0$ and $b \neq 0$. This system possesses a one-parameter family of solutions:
$$x = L(t + C)^{k_1}, \quad y = K(t + C)^{k_2}$$
where $L$ and $K$ are constant numbers, and $C$ is an arbitrary parameter. It is evident that if at least one exponent $k_i$ in (5) is not an integer, then (5) contains critical movable singular points. Consequently (as noted in the footnote), the system (2) also possesses movable critical singularities. However, from (3) it follows that...

0 , < 0 .

Thus, equation (3) ensures the existence of critical moving singular points for system (2). Now, let us assume that

$Y_i = \dots$ and $b_1 > i$: (6)

We write system (2) in the form

J k =0 k =0

where $Q \neq 0$, and all functions are holomorphic in some domain. We introduce a parameter $\lambda$ into system (7).
Remark: In all subsequent discussions, we shall take values from a certain sub-domain of the domain in which the functions possess the required properties.
Differential equations

we obtain the solution to this system, which we seek in the form. It is easy to see that $s = 0, 1, 2, \dots$. To determine these values, we have a system. From the first equation of this system,

$$s_0(z_0) = C_1 \ln(A(f_0) \ln(z + C_2)).$$

It follows that condition (6) also ensures the presence of moving critical singular points in system (2). We formulate the obtained results as follows:
Lemma 1. If $\frac{\partial Q}{\partial z} \neq 0$, $\frac{\partial P}{\partial z} \neq 0$, $Q \neq 0$, and $P \neq 0$, then for system (2) to have only single-valued moving singular points, it is necessary that $\gamma_1 = \dots = \gamma_n = \delta_1 = \dots = \delta_n = 0$.
Suppose the conditions of Lemma 1 are satisfied; then system (2) can be written in the form $\frac{dy}{dz} = \dots$. We introduce a parameter into (8)

$$x = \tau, \quad y = \lambda w, \quad z = z_0 + \lambda t$$

and set $\lambda = 0$ in the resulting system; then

$dT = r_J dW_0 P_2(T_0, z_0)$

By eliminating from this system, we obtain:

In order for the system to be free of movable critical singular points, it is necessary ([1], p. 166) that $Q$ decomposes with respect to no more than four factors, and the degree must be lower. Consequently, the decomposition of $z) = \dots$ into partial fractions must contain no polynomial part.

$P_2(x, z_0)$

Remark. In the lemma, it was assumed that $z \neq 0$. This property is utilized in the reasoning regarding (9); therefore, even when $z = 0$, the absence of a polynomial part in the expansion is necessary. In this case, the result depends on the following:

Lemma. For the system to lack movable critical singular points, it is necessary that the system can be represented in one of the following forms:
1. $y - a(z) \phi a_i(z)$, where $(z) \neq \beta_i(z)$ and $(z) \neq 0$;
2. $(z) y^1$, where any rational function contains no linear factors;
3. $y = P(z)$, where $P(z)$ is an arbitrary rational function.

Suppose we have a system of the form (10); we introduce a parameter into it.

x—fa(z) = KT, y-aj(z)=lW,

For $K = 0$, we obtain:

The one-parameter family of solutions for this system is given by $Z(t) = K(t+C)$ and $Z(t) + B = L(t+C)$. In system (10), the roles can be interchanged, as was done in the formulation of the lemma relative to the proof.

$T_0 = \exp K_1(t+C)$, $H_0 = \exp L_1(t+C)$,

if $\frac{K}{L} = \frac{K_1}{L_1}$. Here $K, K_1, L, L_1$ are non-zero constants, and $C$ is an arbitrary parameter. From this, it is evident that to ensure the absence of moving critical singularities, it is necessary that $n_j$ are integers and $m_j = 1 - n_j$. We now introduce the parameter $\lambda$ into the system as follows:

$x = T, \quad y - a_1(z) = \lambda W, \quad z = z_0 + U$.

For $\lambda = 0$, we obtain:

Eliminating the variables, we obtain the following. It is evident that, as can be easily seen, we then have $s > \dots$. Suppose that $(M - 1)(m - 1) \geq 0$, and there exists at least one pair for which the value is $-1$. It is clear that one can always set the values such that $N_k(z)$ can be written as:

$k = 1 \dots n_k, \quad k = 1, 2, \dots, m$

$A_k = \pm \frac{t-1}{A-A_k}, \quad k = 2, \dots, M$

Furthermore, some values of $m$ may be equal to $\infty$ (this corresponds to cases where the variables are unbounded). Taking this into account, the system can be transformed by a change of the independent variable into the following form:

$y - a$. Taking into account (16), equation (14) can be written as:

T O — Pfe K = R + L

Now, let us introduce the parameter $\lambda$ into equation (16):

$$x' = \lambda x, \quad y' = y, \quad z' = z_0 + kt$$

By setting $\lambda = 0$ and reducing the system to a single second-order differential equation with respect to $x$, we obtain $\frac{d^2x}{dt^2} + 2\gamma \frac{dx}{dt} + \omega_0^2 x = f(t)$. For equations (17) and (18), we respectively derive the first integrals:

$$n=1, \quad n=r+1$$

{v,-a fa>-a*).

£ = 2 &=s+l

By virtue of the problem being solved, equations (19) and (20) are Briot-Bouquet equations ([1], p. 114); consequently, the exponents of the binomials can only take on specific sets of values. If $< 0$, then in (20), and if $< 0$, then in (19), without loss of generality, one may assume that system (16) can be written in the form:

where $h_x > 2$, and it is easily seen [1] that $h < 3$.

I. YABLONSKY

If we assume $r < m$ and rewrite (19) as

then we obtain either $2$ or $\dots > 2m$. The first case is excluded, while the second indicates the presence of critical moving singularities. Consequently, only $r = m$ is possible. By performing the substitution $z = \dots$ in system (21) and constructing formulas analogous to (19) and (20) for any new system, we conclude that either $= 1$ or one of the equations (19), (20) cannot be a Briot-Bouquet equation. Thus, we have proven Lemma 3: if system (10) is such that $(M - 1) \neq 0$ and there exist such $A_j(z)$, then this system contains moving critical singular points. Assuming $-\dots A = \dots = -B$ in (19) and (20), we obtain the corresponding results. It is evident that under the identified conditions, the system admits a first integral:
$$(z - \alpha)(z_1 - \alpha)(z - \beta)(z_1 - \beta) = -1$$
After reducing (10) (taking the above into account) to a single equation using (22), we have:
$$+ 4C$$

$$\frac{dz}{dJ} = C(K - \alpha^2)$$

Equation (23) can be integrated using elliptic functions; consequently, it is a single-valued function. From this, it is easily seen that the variable $y$ is also a single-valued function. This leads to the following theorem:

Theorem: Among the systems that can be reduced to the form $L$, there is only one system that possesses no critical moving singular points.

Finally, let us consider the case for system (10) where $l(m - 1) = 0$. Without loss of generality, we may assume:

1. Then the system (10) can be

reduced to the form

where some values may be equal. System (25) possesses a first integral $\left(1 - \mu\right) y - a$. By reducing the order of system (25), we obtain the equation

$= P(x)$ (where $P$ is a polynomial),

which must belong to the class of Briot-Bouquet equations. The possible values of $m$ and their corresponding $\lambda_i$ will be determined. Specifically:
I. For $m = 1$, $\lambda$ can take the values: $-1$ or any integer.
II. For $m = 2$, the pairs $(\lambda_1, \lambda_2)$ can take the values: $(n, -n)$ where $n$ is an integer; $(\infty, \infty)$; $(2, 2)$; $(2, 2, 3)$; $(2, 6)$; $(3, 6)$; $(2, 4)$; $(4, 4)$; $(3, 3)$.
III. For $m = 3$, the triplets $(\lambda_1, \lambda_2, \lambda_3)$ can take the values: $(2, 3, 6)$; $(2, 4, 4)$; $(3, 3, 3)$; $(2, 2, 2)$.
IV. For $m = 4 \dots$
It is evident from (25) that the absence of movable multi-valued singular points in the components implies their absence in $z$. In this case, $x$ and $y$ are expressed in terms of elementary and elliptic functions.

Theorem 2. For system (10) to have no movable critical singular points, it is necessary and sufficient that, through a transformation of the independent variable and a first integral of the form (26), the system reduces to a Briot-Bouquet equation; specifically, that $m$ and the corresponding $\lambda_i$ in (25) take the specified values.

Let us consider a system of the form (11), and assume first that $\lambda \neq 0$. Then $z_i = \psi_i(x, z_j)$ for $i \neq j$. It can always be assumed that

Introduction

Into the system, the parameters $y = W$ and $z = z$ are introduced, and at...

0 we obtain

/(2o)f / = 2

a ^ 0 , p k (z 0 )=£0.

By utilizing the first integral

$1 - \beta \sum p_k(z) W_{k+l} = C$

we arrive at the equation $\left( - \delta_i + \mathcal{J}_k \right) W_0$

L k =0

Jablonskii established that for the absence of movable critical singular points, it is necessary that $n=2$; however, this condition may not be sufficient.
L e m m a: For system (11) to be free of movable critical singular points, it is necessary that $n=2$.
Suppose the lemma holds; that is, $n=2$. If $n=N=1$, the system is linear, which implies it contains no movable singular points at all.

Suppose $nN \neq 1$. We introduce the parameter $\lambda$:

• — -,n +i ~ • # ~ , J V + i ™ > 2 — ' z o + ^

For $X = 0$, we obtain:

Consider a one-parameter family of solutions for the system $A(t + C)B(t + \dots)$, where $C$ is a constant and an arbitrary parameter. To ensure uniqueness, it is necessary to require that $N, n-1, s$, and $l$ be integers. It is easily seen that the latter is possible only for the following values:

$n = 2, n = 3, N = 3$. Suppose we have $n = 2$ and $s = 1$; that is:

-f'Po ~ - =

P 2 ( z ) # 0 , 4l (x)=£0.

This system can be reduced via the linear transformation $z = y + p(x)y$ to the form $z' = x$, which is clearly equivalent to a second-order equation.

J^=j 2 (z)y* + b(z)y. (33) dz 2

Theorem 3. Equation (33), and consequently the system, contains movable critical singular points if and only if the coefficients are constant, or if by the transformation $y = \lambda(z) W + \phi(z)$, where

$$\lambda(z) = C \rho^2(z), \quad \phi(z) = \int \rho^2(z) dz,$$

$v = -\lambda(z) - b \rho^2(z) \int \rho(z) dz$, equation (33) is reduced to

$$\frac{dW}{dz} = 6W^2 + az + b,$$

where $a$, $b$, and $C$ are constant numbers. In this case, the solution of the system is expressed through elementary functions, elliptic functions, or the Painlevé transcendents (specifically, solutions to the first Painlevé equation). Theorem 3'. In order for equation (33), and therefore system (32), to be free of movable critical singular points when $\rho_1 \neq \text{const}$ and $\rho_2 \neq \text{const}$, it is necessary and sufficient that $f(z) = az + b$, where $a$ and $b$ are certain constant numbers. Theorem 3' follows from Theorem 3 and the form of equation (33) after applying transformation (34).

Suppose we have $n=3, N=1$, i.e.,

The equation $\text{P}_3(z)y''' + \text{P}_2(z)y'' + \text{P}_1(z)y' = q_i(z)x$ (35) can be reduced to the form $y''' = \text{P}_3(z)y'' + \text{P}_2(z)y' + \text{P}_1(z)y$, which is equivalent to the equation $\text{P}_2(z)y'' + \text{P}_1(z)y'$. Analogous to the case where $n = 2$ and $m = 1$, we have the following:

Theorem 4. Equation (37), and consequently system (36), do not contain movable critical singular points if and only if the coefficients are constant. This can be achieved via the transformation $y = y_1(z)W + v(z)$. If $\delta \neq 0$, then by choosing the appropriate parameters, one can ensure $\delta = 0$, and by choosing $\alpha = 1$.

A. I. YABLONSKII

7 T '

$T(z)dz$, where $C$ and $a$ are constant values. In this case, the solutions of the system are expressed through elementary functions, elliptic functions, and Painlevé transcendents (specifically, solutions to the second Painlevé equation). Theorem 4: For equation (37), and consequently system (36), to be free of movable critical singular points ($const, const, const$), it is necessary and sufficient that $P_3(z) - 12P_1(z)P_2(z) - \dots + 6P_1(z)P_2(z) - \dots$

$-8/3 - 36P_3(z) \left[ aC_1 \int P_3(z) dz + bC_2 \right] = 0;$

$9P_1(z) P(z) P_1'(z) + 2P_1^3(z) P(z) - P(z) P_1(z)$

2 (z) + p 2 (z) p7 (z) - 27a C 3 p f / 6 (z) = 0,

$p_3(z) p$, where $a, b, c$ are constant numbers. This case was previously examined in \cite{2}. Suppose system (1) is written in the form (12); it can then be reduced to $Q(x, z)$. The general solution of this system is:

$$x = C \int dz + C$$

If $Q = \Pi - P/W_1$ in the case where $\frac{dQ}{dz} \neq 0$, and among the values of $\delta$ there are equal values for which $p_y(z) = \text{const}$, then the solution $y = y(z, C)$ contains moving critical singular points of a logarithmic nature.

Theorem 5

For system (39) to be free of moving critical singular points, it is necessary and sufficient that one of the following conditions be satisfied: $\frac{dQ}{dz} = 0$. If $\frac{dQ}{dz} \neq 0$, then for the constant $\delta_i > 2$.

The validity of this theorem can be verified by direct calculation of the quadrature in (40). Ultimately, Theorems 1, 2, 3, 4, 5, and the theorem presented in \cite{2} provide the necessary and sufficient conditions for the absence of critical moving singular points in nonlinear systems of the form (1). Consequently, it is possible to conclude...

References

References

Golubev, V. V. Lectures on the Analytical Theory of Differential Equations. GITTL, 1950.

Yablonskii, A. I. Differential Equations, Vol. 2, No. 6, pp. 752–762, 1966.

Received by the Editorial Board
Institute of Mathematics
May 12, 1966