Abstract
In problems related to determining the number of limit cycles bifurcating from a singular point of the second group, it is sometimes (RZhMat, 1965, 7B199) essential to establish the fact that for a system close to a Hamiltonian one, depending on a parameter $\mu$, under certain additional conditions, the displacement function $\rho(\rho_0,2\pi,\mu)-\rho_0$ has a zero of order higher than the first with respect to $\mu$ at $\mu=0$. This note demonstrates how this fact can be established for systems of a fairly general form, using the ideas presented in the well-known work of L. S. Pontryagin (ZhETF, 1934, 4, no. 9). Bibliography: 2.
Full Text
Introduction
This section examines the behavior of solutions for the differential equation system discussed in the work of L. S. Pontryagin \cite{1}. We consider the system:
$$ \frac{dp}{d\phi} = M(p, \phi, \mu) $$
with the initial condition $r(p, 0) = 0$. Here, the function $r(p, \phi, \mu)$ is defined in the neighborhood $|p - p_0| < \epsilon$ for $0 < \phi < 2\pi$. We define $H(p, \mu)$ as the integral of the function $r(p, \phi, \mu)$ such that $p = p(p_0, \phi, \mu)$ represents the solution starting at $p(p_0, 0, \mu) = p_0$.
Following the methodology established in \cite{1}, we analyze the displacement function $p(p_0, 2\pi, \mu) - p_0$. The stability and existence of periodic solutions depend on the derivative of this displacement with respect to the parameter $\mu$. Specifically, we examine the condition:
$$ \frac{\partial p(p_0, 2\pi, 0)}{\partial \mu} \neq 0 $$
Consider a system of the form:
$$ \begin{aligned} \frac{dx}{dt} &= -y + p(x, y, \mu) \ \frac{dy}{dt} &= x + q(x, y, \mu) \end{aligned} $$
where $p(x, y, 0) = 0$ and $q(x, y, 0) = 0$. We assume the functions $p$ and $q$ are sufficiently smooth in the domain $D$. In polar coordinates, where $x = \rho \cos \phi$ and $y = \rho \sin \phi$, the system can be transformed to analyze the radial distance $\rho$. Let $\rho = \rho(\rho_0, \phi, \mu)$ be the solution such that $\rho(\rho_0, 0, \mu) = \rho_0$. The closed trajectory condition for $\mu = 0$ implies that $\rho(\rho_0, 2\pi, 0) = \rho_0$.
To determine the bifurcation of periodic solutions, we evaluate the successor function $h(h_0, \mu)$. For $x > 0$ and $y = 0$, we have $h = H(p, 0)$. The relationship between the displacement in the original coordinates and the transformed system is given by:
$$ \frac{dh(h_0, 2\pi, 0)}{d\mu} = H(p_0, 0) \frac{\partial p(p_0, 2\pi, 0)}{\partial \mu} $$
As an application, consider the Hamiltonian $H = -\gamma(x^2 + y^2) + ax^4 + bx^3y + cx^2y^2 - dy^3$. Let the perturbations be defined as:
$$ \begin{aligned} p &= \mu(x^3 - 2xy^2) \ q &= \mu(2x^2y - y^3) \end{aligned} $$
By applying the criteria for the existence of limit cycles as developed by N. N. Bautin \cite{2}, we can determine the conditions under which the equilibrium point at the origin loses stability and generates a periodic orbit. The results obtained here align with the qualitative theory of differential equations as presented in the cited literature.
References
- Pontryagin, L. S. (1934). On the dynamical systems close to Hamiltonian systems. Zh. Eksp. Teor. Fiz., 4(9), 1–3.
- Bautin, N. N. (1965). On the number of limit cycles appearing with the variation of coefficients from an equilibrium state of focus or center type. Mat. Sb., 1(53–66).