Abstract
Two-dimensional autonomous systems of the form
\begin{equation}
\dot{x}=P(x,y),\quad\dot{y}=Q(x,y)\tag{1}
\label{1}
\end{equation}
are considered, where the right-hand sides have isolated singular points $O_j$ of the pole type. The concept of a "quasi-residue" $J_j$ of a singular point $O_j$ is introduced as the limit value of the integral
\begin{equation}
J_j=\frac1{2\pi}\lim_{r\to0}\oint Q\,dx-Pdy\tag{2}
\end{equation}
taken over a contour $|\gamma|$ surrounding the singular point $O_j$ as the contour $|\gamma|$ deforms into a point (without crossing the singular point $O_j$). In terms of quasi-residues $J_j$, sufficient conditions for the absence of periodic trajectories for the system \eqref{1} are established. The obtained negative criteria are a known generalization of the Bendixson and Dulac criteria to the case of multiply connected domains, since the phase plane is considered here with excluded points $O_j$ where the poles are located.
Full Text
Preamble
This work, following the foundational principles established in [1, 2], examines the qualitative behavior of autonomous systems of differential equations. We consider a system of the form:
$$ \dot{x} = P(x, y), \quad \dot{y} = Q(x, y) \tag{1.1} $$
where $P(x, y)$ and $Q(x, y)$ are continuous functions. Associated with this system is the differential form:
$$ \omega = Q(x, y)dx - P(x, y)dy \tag{1.2} $$
The equilibrium points $O_j(x_j, y_j)$ of the system (1.1) are defined by the simultaneous vanishing of the functions $P$ and $Q$, such that $P(x_j, y_j) = Q(x_j, y_j) = 0$. For each such singular point, we define a characteristic index $J_j$ using the line integral of the form (1.2) along a small closed contour $\gamma_j$ surrounding the point $O_j$:
$$ J_j = \lim_{r \to 0} \frac{1}{2\pi} \oint_{\gamma_j} \frac{Q(x, y)dx - P(x, y)dy}{r^2} \tag{1.3} $$
where $r = \sqrt{(x - x_j)^2 + (y - y_j)^2}$. This integral characterizes the local rotation and divergence properties of the vector field near the singularity.
By applying Green's theorem to the region $D$ bounded by a contour $C$ and containing several singular points $O_j$, we can relate the line integral along the boundary to the sum of the indices and the integral of the divergence over the domain:
$$ \oint_C \omega = \iint_D \left( \frac{\partial Q}{\partial x} + \frac{\partial P}{\partial y} \right) dxdy \tag{1.4} $$
This relationship allows for a global qualitative analysis of the phase portrait of the system (1.1).
2. Stability and Divergence Analysis
Let us consider the closed trajectories (limit cycles) of the system (1.1). If a region $G$ contains singular points $O_j$, the integral along the boundary $\Gamma$ of the region can be expressed as the sum of the integrals along the small contours $\gamma_i$ surrounding each point:
$$ \oint_\Gamma \omega = \oint_C \omega + \sum \oint_{\gamma_i} \omega $$
Using the divergence of the vector field $\mathbf{V} = P\mathbf{i} + Q\mathbf{j}$, we obtain:
$$ \iint_D \text{div} \mathbf{V} \, dxdy = \oint_C (Q dx - P dy) \tag{2.1} $$
The behavior of the system can be classified based on the value of the integral $J_j$. Specifically, for a singular point $O_j$, the following conditions are of interest:
a) $J < 0$, b) $J = 0$, c) $J > 0$. \tag{2.3}
These conditions, combined with the sign of the divergence $\text{div} \mathbf{V} = P_x + Q_y$, determine the stability and the nature of the equilibrium points (e.g., nodes, foci, or saddles).
As an example, consider a linear system:
$$ \dot{x} = ax + by, \quad \dot{y} = cx + dy \tag{2.5} $$
For the origin $O(0,0)$, the index $J$ is related to the trace of the matrix, $-\frac{1}{2}(a+d)$. The divergence is given by $\text{div} \mathbf{V} = a + d$. In the specific case where $a=d=1$ and $b=c=0$, we find $\text{div} \mathbf{V} = 2$, indicating a source. Conversely, if $a=d=0$ and $b=-c$, then $J=0$ and $\text{div} \mathbf{V} = 0$, which corresponds to a center.
3. Generalized Weight Functions
The analysis can be extended by introducing a weight function $X(x, y)$ into the differential form. We define a generalized index $J^$ as:
$$ J^ = \lim_{r \to 0} \oint_{\gamma} X(x, y) (Q dx - P dy) \tag{3.1} $$
A common choice for the weight function is:
$$ X(x, y) = [(x - x_j)^2 + (y - y_j)^2]^\mu \tag{3.2} $$
where $\mu$ is a parameter chosen to ensure the convergence of the integral near the singularity $O_j$. The generalized divergence of the weighted vector field $\mathbf{V}^ = X\mathbf{V}$ is then:
$$ \text{div} \mathbf{V}^ = \nabla \cdot (X\mathbf{V}) = X \text{div} \mathbf{V} + \mathbf{V} \cdot \nabla X \tag{3.5} $$
By analyzing the sign of $\text{div} \mathbf{V}^$ within a domain $D$, one can establish criteria for the non-existence of closed trajectories (Bendixson's criterion and its generalizations). If $\text{div} \mathbf{V}^$ does not change sign in a simply connected region, the system (1.1) cannot have periodic solutions entirely contained within that region.
References
- Bendixson, J. "Sur les courbes définies par des équations différentielles," Acta Mathematica, 24, 1–88, 1901.
- Dulac, H. "Sur les cycles limites," Bulletin de la Société Mathématique de France, 51, 45–188, 1923.
- Trudy Matematicheskogo Instituta im. V.A. Steklova, 811–822.
- Rendiconti del Circolo Matematico di Palermo, 204, 1703–1706, 1937.
- Lefschetz, S. Differential Equations: Geometric Theory, Moscow, 1961.