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Qualitative Investigation in the Large of Integral Curves of Isochronous Systems of Two Differential Equations
A. P. VOROB'EV
Consider the system of differential equations
\[ \frac{dx}{dt}=-\frac{\partial}{\partial y}(u^2+v^2)/2\Delta(x,y),\quad \frac{dy}{dt}=\frac{\partial}{\partial x}(u^2+v^2)/2\Delta(x,y), \tag{1} \]
where \(u\) and \(v\) are functions of the variables \(x\) and \(y\), and
\[ \Delta(x,y)=\frac{\partial u}{\partial x}\frac{\partial v}{\partial y} -\frac{\partial u}{\partial y}\frac{\partial v}{\partial x}. \tag{2} \]
Everywhere below we shall consider system (1) in a domain \(D\), containing the origin, in which the following conditions are satisfied:
a) the functions \(u(x,y)\) and \(v(x,y)\) are defined and continuous together with their first partial derivatives, and
\[ u(0,0)=v(0,0)=0,\quad \Delta(0,0)=1; \]
b) the determinant (2) satisfies the strict inequality
\[ \Delta(x,y)>0. \]
It can be shown, as in the case of holomorphic \(u(x,y)\) and \(v(x,y)\) [1], [2], that the origin for system (1) is an isochronous center, i.e. that the period of every closed trajectory enclosing the point \((0,0)\) is equal to \(2\pi\).
Theorem 1. All singular points of system (1) in the domain \(D\) are isochronous centers.
Proof. From the representation of the right-hand sides of system (1) it follows that all its singular points in the domain \(D\) are determined from the system of equations
\[ u(x,y)=0,\quad v(x,y)=0. \tag{3} \]
Suppose that system (3) has a solution \(x=x_0,\ y=y_0\) in the domain \(D\), distinct from \(x=0,\ y=0\). To determine the character of this singular point, transfer the origin to it by the substitution \(\xi=x-x_0,\ \eta=y-y_0\). Putting \(\Delta(x_0,y_0)=\Delta_0,\ u_x(x_0,y_0)=a,\ u_y(x_0,y_0)=b,\ v_x(x_0,y_0)=c,\ v_y(x_0,y_0)=d\), we write out the linear terms of system (1) in a neighborhood of the point \(\xi=0,\ \eta=0\):
\[ \frac{d\xi}{dt}=-\frac{1}{\Delta_0}\left[(ab+cd)\xi+(b^2+d^2)\eta+\ldots\right], \tag{4} \]
\[ \frac{d\eta}{dt}=\frac{1}{\Delta_0}\left[(a^2+c^2)\xi+(ab+cd)\eta+\ldots\right]. \tag{4} \]
It is easy to verify that the characteristic equation for system (4) has the following form:
\[ \lambda^2+1=0. \tag{5} \]
Thus, the roots of the characteristic equation (5) are purely imaginary. Taking into account, moreover, that \(u(x,y)\) and \(v(x,y)\) are defined throughout the entire domain \(D\) and that system (1) has the integral
\[ u^2(x,y)+v^2(x,y)=c^2, \]
we arrive at the conclusion: the singular point \(x=x_0,\ y=y_0\) is a center. We now show that it is an isochronous center. Indeed, system (4) can be written as follows:
\[ \begin{aligned} \frac{d\xi}{dt} &=-\frac{\partial}{\partial \eta} \left[u^2(\xi+x_0,\eta+y_0)+v^2(\xi+x_0,\eta+y_0)\right]/2\Delta(\xi+x_0,\eta+y_0),\\ \frac{d\eta}{dt} &=\frac{\partial}{\partial \xi} \left[u^2(\xi+x_0,\eta+y_0)+v^2(\xi+x_0,\eta+y_0)\right]/2\Delta(\xi+x_0,\eta+y_0). \end{aligned} \tag{6} \]
By direct verification one can see that the transformation
\[ u=u(\xi+x_0,\eta+y_0),\qquad v=v(\xi+x_0,\eta+y_0) \]
in a neighborhood of the point \(\xi=0,\ \eta=0\) brings system (6) to the form
\[ \frac{du}{dt}=-v,\qquad \frac{dv}{dt}=u. \]
This proves that the singular point \(x=x_0,\ y=y_0\) is an isochronous center. The theorem is proved.
Theorem 2. All closed trajectories of system (1) in the domain \(D\) have positive orientation.
Proof. Under the stated assumptions on the functions \(u(x,y)\) and \(v(x,y)\), the closed trajectories around the point \(x=0,\ y=0\) are oriented positively, i.e., the motion of the representative point along them as the parameter \(t\) increases takes place counterclockwise. We shall show that the other closed trajectories, if they exist, also have positive orientation. Suppose that in the domain \(D\) there are other closed trajectories, distinct from those which enclose the origin. Then in the domain \(D\) there is at least one singular point, distinct from \(x=0,\ y=0\), which, according to Theorem 1, will be an isochronous center. Let this point have coordinates \(x=x_0,\ y=y_0\). Then, transferring the origin of coordinates to it, we obtain system (4). Introduce polar coordinates in system (4). The term in the derivative \(\dfrac{d\varphi}{dt}\) that does not depend on \(\rho\) will have the following form:
\[ \varphi(\theta)=\frac{1}{\Delta_0}\left[(a^2+c^2)\cos^2\theta+2(ab+cd)\cos\theta\sin\theta+(b^2+d^2)\sin^2\theta\right]. \]
Since \(\Delta_0>0\), the function \(\varphi(\theta)\) is strictly positive for all \(\theta\in[0,2\pi]\), and this means that the closed trajectories around the singular point \(x=x_0,\ y=y_0\) are oriented positively. The theorem is proved.
It follows from Theorems 1 and 2 that the qualitative picture of the behavior of the integral curves of system (1) in the domain \(D\) can be described as follows: if system (1) has the origin as its only singular point, then either the closed integral curves fill the entire domain \(D\), or there is a separatrix separating the domain of the center and adjoining, at both ends, the boundary of the domain \(D\), while the remaining part of the domain is filled with integral curves adjoining, at both ends, the boundary of the domain \(D\); if, however, there are other singular points besides the origin, then they are also centers, and the domain of each of the centers is separated by a separatrix adjoining, at both ends, the boundary of the domain \(D\), while the remaining part of the domain is filled with integral curves (in this case there are infinitely many of them) adjoining, at both ends, the boundary of the domain \(D\).
In conclusion we give examples.
Example 1. Let \(u=x\), \(v=y+f(x)\), where \(f(x)\) is a holomorphic function in a neighborhood of the point \(x=0\), with radius of convergence \(|x|<a>0\), vanishing together with \(x\). In this case we shall take the strip \(-a<x<a\) as the domain \(D\). It is easy to verify that for these functions \(\Delta(x,y)\equiv 1\), and the canonical system (1) in the domain \(D\) has the unique singular point \(x=0\), \(y=0\), which is an isochronous center.
Example 2. Consider the functions
\[ u=(y+1)\left(\frac{1}{6}y^{2}+y+1-\sqrt{1-2x}\right)-\frac{1}{18}y^{3}-\frac{1}{2}y^{2}-y, \]
\[ v=\frac{1}{6}y^{2}+y+1-\sqrt{1-2x}. \]
In this case we shall take the half-plane \(x<\frac{1}{2}\) as the domain \(D\).
It is easy to verify that for these functions \(\Delta(x,y)=1\), and the canonical system (1) in the domain \(D\) has two singular points \(x=0\), \(y=0\) and \(x=0\), \(y=-6\), which are isochronous centers with positive orientation.
References
- Vorob'ev A. P. DAN BSSR, 7, No. 3, 1963.
- Vorob'ev A. P. DAN BSSR, 7, No. 8, 1963.
Received by the editors
January 28, 1965
Institute of Mathematics
Academy of Sciences of the BSSR