UDC 517.917

THE BEHAVIOR OF INTEGRAL CURVES OF A SYSTEM OF DIFFERENTIAL EQUATIONS CLOSE TO A HOMOGENEOUS ONE

I. Z. MANEVICH

We shall give a picture of the integral curves in a neighborhood of the singular point \(O\) for a system of the form

\[ \frac{dx}{dt}=f(x)+F(x), \tag{1} \]

where \(f(x)\) satisfies the conditions:
\[ \left. \begin{array}{ll} 1) & x \text{ and } f(x) \text{ are } n\text{-dimensional vectors},\\ 2) & f(x) \text{ is a continuous vector function for all } x,\\ 3) & f(x)\ne 0 \text{ for } x\ne 0,\\ 4) & f(cx)=c^m f(x), \end{array} \right\} \tag{A} \]
where \(m=\dfrac{p}{q}>0\), and \(F(x)\) is an \(n\)-dimensional vector function such that \(F(x)=o(\|x\|^m)\), and the right-hand side of system (1) satisfies conditions ensuring uniqueness of solutions of this system.

We obtain the behavior of the integral curves of system (1) by comparison with the picture of the integral curves for the system

\[ \frac{dx}{dt}=f(x). \tag{2} \]

In systems (1) and (2) we introduce generalized polar coordinates \(\rho,\varphi\) by the formula

\[ \rho=\|x\|,\qquad \varphi=\frac{x}{\rho}; \]

we replace the parameter \(t\) by the formula

\[ \rho^{m-1}\,dt=d\tau. \]

It is clear from this that the geometric picture of the integral curves will not change.

Denote:

\[ (\varphi,f(\varphi))=R(\varphi),\qquad \frac{(F(\rho\varphi),\varphi)}{\rho^m}=r(\rho,\varphi), \]

\[ f(\varphi)-(\varphi,f(\varphi))\varphi=B(\varphi),\qquad \frac{F(\rho\varphi)-(\varphi,F(\rho\varphi))\varphi}{\rho^m}=b(\rho,\varphi). \]

Then system (1) takes the form

\[ \rho^{-1}\frac{d\rho}{d\tau}=R(\varphi)+r(\rho,\varphi), \tag{3.a} \]

\[ \frac{d\varphi}{d\tau}=B(\varphi)+b(\rho,\varphi), \tag{3.b} \]

and system (2) will be

\[ \rho^{-1}\frac{d\rho}{d\tau}=R(\varphi), \tag{4.a} \]

\[ \frac{d\varphi}{d\tau}=B(\varphi). \tag{4.b} \]

By virtue of condition (A) and the notation,

\[ \lim_{\rho\to 0} r(\rho,\varphi)=0,\qquad \lim_{\rho\to 0} b(\rho,\varphi)=0. \]

Systems (3.b) and (4.b) are defined on the sphere \(S^{n-1}\) of unit radius \(\|\varphi\|=1\). It can be shown that \(x=0\) is the only singular point for system (2), and consequently also for system (1) when \(\rho\) is sufficiently small. In what follows we put \(n=3\).

We shall call an invariant line of system (2) an isocline of the vector field determined by the function \(f(x)\), which is a solution of this system. It is not hard to see that the vector \(\varphi\) of an invariant line satisfies the equation \(B(\varphi)=0\).

The integral curves of system (2), \(x(\tau)=\varphi(\tau)\rho(\varphi)\), will be called, according to the form of \(\varphi(\tau)\) in a neighborhood of the singular point on \(S^2\) corresponding to the invariant line, and, moreover, if \((\varphi,f(\varphi))<0\) along the invariant line, parabolic integral curves of the first kind; if \((\varphi,f(\varphi))>0\), then of the second kind.

Definition 1. An invariant line of system (2) will be called parabolic of the first kind (of the second kind) if all integral curves of this system in a sufficiently small neighborhood of the invariant line*) are parabolic integral curves of the first (second) kind.

Definition 2. An invariant line of system (2) will be called hyperbolic if the integral curves of system (2) in a sufficiently small neighborhood of the invariant line are hyperbolic, with the exception of a set of integral curves lying on a finite number of surfaces passing through the invariant line and homeomorphic to half-planes, on which the integral curves of system (2) are parabolic of the first or second kind.

Definition 3. An invariant line of system (2) will be called focal if all integral curves \(x(\tau)\) of this system in a sufficiently small neighborhood of the invariant line are spirals tending with one end to the point \(O\), and the singular point \(S\) on the unit sphere corresponding to the invariant line is a center-focus (for the definition of a center-focus see [1]).

Definition 4. Invariant lines that are parabolic of the first and second kind, hyperbolic, and focal will be called rough.

*) By a sufficiently small neighborhood of an invariant line we mean a conical neighborhood of the invariant line, bounded by the sphere \(\rho=\bar{\rho}\) for \(\bar{\rho}\) sufficiently small.

It turns out that, in order to construct the picture of the integral curves of system (1) in a neighborhood of rough invariant straight lines, it is sufficient to know the type of this invariant straight line and the direction of the vector field defined on it by system (2).

Theorem 1. If an invariant straight line of system (2) is parabolic of the first kind, then there exists a sufficiently small conical neighborhood of the invariant straight line in which all integral curves of system (1) enter the singular point \(O\) as \(\tau \to +\infty\).

Proof. In a neighborhood of the point \(S\) corresponding to the given invariant straight line, on the unit sphere we draw a closed curve \(c\) of sufficiently small diameter, such that \(c\) intersects the integral curve \(\varphi(\tau)\) of system \((4.b)\) at the point \(N\) at some angle \(\alpha>0\). This is evidently always possible, since \(\varphi(\tau)\) is a parabolic integral curve. We project a neighborhood of the point \(S\) and the curve \(c\) onto a sphere \(D\) of such small diameter that \(|b(\rho,\varphi)|<\delta\), where \(\delta\) is some sufficiently small number, whose magnitude will be specified below. We draw the curve \(\varphi(\tau)\) of system \((3.b)\), passing through the point \(N_1\) corresponding to \(N\) on the sphere \(D\). We shall show that the curve \(\varphi(\tau)\) does not leave a sufficiently small neighborhood \(c_1\) of the points \(S_1\) corresponding to \(c\) and \(S\) on the sphere \(D\). Indeed, if \(\varphi(\tau)\) leaves, at \(\bar{\tau}\), the neighborhood \(c_1\) through the point \(M_1\), then, by virtue of the continuous dependence of the solutions of the system on the right-hand side [2], for \(\delta\) sufficiently small the solution \(\varphi(\tau)\) leaves the neighborhood \(c_1\) through the point \(\overline{M}_1\), where \(|\overline{M}_1-M_1|<\varepsilon\), \(\varepsilon\) being an arbitrarily small positive number, which is impossible because \(\varphi(\tau)\) is a parabolic integral curve in a neighborhood of the point \(S_1\).

For each integral curve \(\varphi(\tau)\) of system \((4.b)\) we find such a point \(N_1\). This set of points forms a certain curve \(\gamma\), which all curves of system \((3.b)\) intersect from outside inward. We construct the cone \(K(\gamma)\) with directrix \(\gamma\). We bound \(K(\gamma)\) by the hyperplane \(\rho=\bar{\rho}\) in such a way that all integral curves of system (1) intersect \(\rho=\bar{\rho}\) in the direction of \(O\). This is possible for sufficiently small \(\rho\), since \(\rho,(\varphi)\to0\) by virtue of the fact that \((\varphi,f(\varphi))<0\) along a parabolic invariant straight line of the first kind. The constructed cone \(K(\gamma,\bar{\rho})\) is the desired one, which all integral curves of system (1) intersect from outside inward and tend to the point \(O\), since

\[ \rho(\tau)=\rho_0 \exp \int_{\tau_0}^{\tau}(\varphi,f(\varphi))\,d\tau \]

tends to zero as \(\varphi(\tau)\to\varphi_0(\tau)\), where \(\varphi_0(\tau)\) is the invariant straight line and \((\varphi_0(\tau), f(\varphi_0))=-c\,(c>0)\).

Theorem 2. If an invariant straight line of system (2) is parabolic of the second kind, then there exists a sufficiently small conical neighborhood of the invariant straight line in which a set of trajectories intersects its lateral surface from outside inward and exits through the bounding surface \(\rho=\bar{\rho}\), but there is at least one trajectory which enters the point \(O\) as \(\tau\to-\infty\).

The proof is analogous to the proof of Theorem 1*).

Theorem 3. If an invariant straight line of system (2) is hyperbolic, then there is a sufficiently small neighborhood of the invariant straight line,

*) Theorems 1 and 2 are a direct generalization of the theorems of L. E. Reizin [3], proved for equations with polynomial right-hand sides.

in which there are trajectories of system (1) issuing from it in both directions, and there is a set of trajectories entering the singular point \(O\) as \(\tau \to +\infty\).

For the proof, a system is constructed whose trajectories on the sphere \(S^2\) form a certain angle \(\alpha\) with the trajectories of system \((4.b)\), and then arguments analogous to those given in Theorem 1 are used.

Theorem 4. If the invariant line of system (2) is focal, then all integral curves of system (1) in a sufficiently small neighborhood of the invariant line are focal and enter the point \(O\) as \(\tau \to +\infty\) \((\tau \to -\infty)\).

For the proof, closed curves \(\underline{c_\alpha}\) and \(\underline{\bar c_\alpha}\) are constructed on the surface of the sphere \(S^2\), which are curves without contacts for the curves \(\varphi(\tau)\), and then the cones \(K(c_\alpha)\) and \(K(\bar c_\alpha)\). Thus we obtain the theorem.

Theorem 5. The pattern of the integral curves of system (1) in a sufficiently small neighborhood of a rough invariant line of system (2) is completely determined by the type of this invariant line.

In conclusion, the author expresses his deep gratitude to his scientific adviser, Professor I. P. Makarov, for guidance and assistance in the work.

References

1. Nemytskii V. V., Stepanov V. V. Qualitative Theory of Differential Equations. Moscow–Leningrad, 1949.

2. Petrovskii I. G. Lectures on the Theory of Ordinary Differential Equations. Moscow–Leningrad, 1952.

3. Reizin L. E. Izv. AN LatvSSR, No. 2, 1951.

Received by the editors
July 22, 1965

Tambov Pedagogical
Institute