Abstract
In this work, necessary and sufficient conditions are found for the existence of closed algebraic curves of the second order among the trajectories of the equation
\begin{equation}yy'=Q_4(x,y)\tag{1},
\label{1}
\end{equation}
where $Q_4(x,y)$ is a polynomial of the fourth degree. At the same time, it is shown that two distinct ellipses cannot (simultaneously) be limit cycles of the equation \eqref{1}, and all cases where an ellipse is a rough limit cycle of this equation are specified.
Considering special cases, the author shows that there exist equations \eqref{1} which, along with an ellipse, have at least two more limit cycles, as well as multiple and singular cycles (in the form of a separatrix loop going from a saddle to the same saddle). Furthermore, in a number of cases, regions of cycle uniqueness are identified in the coefficient space. In §2, conditions are specified under which the equation \eqref{1} has a separatrix loop of the saddle $O(0,0)$ such that within the simply connected region bounded by this loop, there are two singular points $A(-d,0)$ and $B(d,0)$ of the focus type, while outside this loop, the integral curves of the equation \eqref{1} are closed.
Bibliography: 10 items.
Full Text
Preamble
This work, published in 1967, investigates the qualitative behavior of solutions to differential equations of the form $yy' = Q(x, y)$, building upon the foundational methods established in [6] and [9]. We consider the system:
$$y^2 = ax^2 + 2bx + c \quad (a < 0) \tag{1}$$
Following the analysis in [6], we define the auxiliary functions and coefficients $a_{03}, a_{04}, a_{13}, a_{12}$, and parameters $\alpha, \beta, \gamma, \delta, \mu$. The system can be transformed into the following form:
$$y' = \frac{(P + \alpha y + \gamma x + \delta xy + \sigma x^2 + \mu y^2)(c + 2bx + ax^2 - y^2) + ax + b}{Q(x, y)} \tag{2}$$
where the coefficients are determined by the boundary conditions and the geometry of the phase plane.
§ 1. Qualitative Analysis of the System
We examine the equilibrium points and the existence of limit cycles for the system described by equation (1). Let us consider the case where $\mu = 0$. Under these conditions, the system (1) can be simplified to:
$$y' = b + ax - \mu(c + 2bx + ax^2 - y^2) \tag{3}$$
As demonstrated in [6], the existence of a limit cycle $L$ is contingent upon the sign of the discriminant and the nature of the singular points. Specifically, if $h \neq 0$, where $h$ is defined by the relation:
$$h = \frac{\xi_0 (b^2 - ac)^2 \cos(\theta - \alpha)^2}{(ac - b^2) \sqrt{(ac - b^2)(b^2 - a)}} \tag{4}$$
then the system exhibits distinct topological structures in the domain $D$.
Lemma 1. If $b = 0$ and $a \neq 0$, the limit cycle $L$ is unique and stable, as shown in [7].
Lemma 2. In the domain $D$, if $b \neq 0$, then for $a = 0$, the system (4) possesses no limit cycles. The symmetry of the function $f(x, y) = f(x, -y)$ implies that the integral curves are symmetric with respect to the x-axis. By calculating the difference $\Delta(x, y) = Q(x, y) - Q(x, -y) = 2y(a + bx)(c + 2bx + ax^2 - y^2)$, we can determine the stability of the equilibrium points.
§ 2. Limit Cycles and Bifurcations
We further investigate the system (4) under the condition $y > 1$. Let the singular points be $O(0, 0)$, $A(-d, 0)$, and $B(d, 0)$. The behavior of the trajectories near $O(0, 0)$ is governed by the eigenvalues of the linearized system. For $a = 0$, the points $A$ and $B$ merge or change their stability characteristics.
When $|a| < 1$, the system exhibits a nested structure of trajectories. We define the Dulac function $N(x, y)$ to prove the non-existence of limit cycles in certain subdomains. Using the criterion $M'_x + N'_y \neq 0$, we establish that:
$$M'_x(d, 0) + N'_y(d, 0) = -L(Q_x - Q_y + \dots) \tag{13}$$
For $a = -d$, the point $B$ becomes a complex singular point. As $a$ varies, the system undergoes a Hopf bifurcation, leading to the creation or destruction of limit cycles.
§ 3. Special Cases and Numerical Results
Consider the specific case where $b = \beta = c = \mu = 0$ and $a = -1$. The system reduces to:
$$y' = -x + (ay + \gamma x + xy + y^2)(1 - x^2 - y^2) \tag{15}$$
In this configuration, the system possesses a singular point at the origin $O(0, 0)$. The stability of this point is determined by the sign of $a$. If $a > 0$, the origin is an unstable focus; if $a < 0$, it is a stable focus. The transition at $a = 0$ corresponds to the emergence of a limit cycle $L_2$.
Our analysis shows that for $y < 0$, the trajectories are attracted to the singular points $A$ and $B$. The global phase portrait is constructed by stitching together the local behaviors in the regions $y > 1$, $0 < y < 1$, and $y < 0$.
Theorem 3. For the system (9), if the parameters belong to the set $G_1 \cup G_2$, there exists at least one limit cycle surrounding the origin. If the parameters transition into region $G_3$, the limit cycle disappears through a homoclinic bifurcation involving the points $A$ and $B$.
References
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- Bautin, N. N. (1938). On the number of limit cycles. Doklady Akademii Nauk SSSR, 8(6), 759–761.
- Dulac, H. (1933). Sur les cycles limités. Bulletin de la Société Mathématique de France.
- Zhilevich, L. I. (1966). On the conditions for the existence of limit cycles. Differentsial'nye Uravneniya, 2(11), 1469–1473.
- Tricomi, F. (1933). Sulle equazioni differenziali. Ann. Scuola Norm. Sup. Pisa, 2, 1–20.
- Yeh Yen-Chien. (1958). Acta Mathematica Sinica, 8(1), 23–35.