Preamble

This section investigates the existence and stability of periodic solutions for a class of differential equations. We consider the equation:
$$\frac{d\phi}{dt} = 1, \quad \frac{dz}{dt} = f_1(\phi)z + f_0(\phi)z^2 + \dots$$
where $f_i(\phi)$ are periodic functions with period $2\pi$. Following the methodologies established in \cite{3, 11}, we analyze the behavior of the system near $z=0$. The functions $f_1(\phi)$ and $f_0(\phi)$ are assumed to be sufficiently smooth, and we specifically examine cases where $f_1(\phi) = \cos \phi$ or involves higher-order trigonometric terms such as $\sin(2k+1)\phi$ and $\cos 2k\phi$.

1. Transformation and Expansion

To analyze the trajectories, we introduce the transformation $z = \rho u(\phi, \rho)$, where $u(\phi, 0) = 1$. Substituting this into the governing equations, we derive a power series expansion for the solution in terms of the initial displacement $\rho_0$. The displacement function after a full period $2\pi$ is given by:
$$F(\rho, 2\pi, a) - F(\rho, 0, a) = \sum_{k=2}^{\infty} \alpha_k(2\pi, a) \rho^k$$
where the coefficients $\alpha_k$ depend on the parameters $a = (a_1, a_2, \dots, a_n)$. The vanishing of these coefficients determines the existence of periodic solutions (limit cycles) in the vicinity of the equilibrium point.

We define the auxiliary functions $u_k(\phi)$ and $v_{kl}(\phi)$ to satisfy the recursive system of linear differential equations:
$$\begin{aligned} u_1'(\phi) &= 0, \quad u_1(0) = 1 \ u_l'(\phi) &= (l-1) u_{l-1}(\phi) f_1(\phi), \quad (l=2, 3, \dots) \end{aligned}$$
For the specific case where $f_1(\phi) = \cos \phi$, the solutions take the form $u_k(\phi) = \sin^{k-1} \phi$. This allows for the explicit calculation of the focal values and the determination of the stability of the origin.

2. Analysis of Focal Values

The coefficients $\alpha_k(2\pi, a)$ are calculated using the properties of the functions $v_{kl}(\phi)$. For $m=1, 2, \dots$, we consider the case where $f_1(\phi) = \cos \phi$ and $f_0(\phi)$ is a trigonometric polynomial. The integration over the period $2\pi$ yields:
$$\alpha_{2m+2}(2\pi, a) = \frac{(-1)^m \pi}{2^{2m-2}} a_{2m+1} + \dots$$
The coefficients $\alpha_k$ for $k < 2m+2$ vanish under specific parameter constraints, indicating a higher-order focus. By varying the parameters $a_k$, we can bifurcate multiple limit cycles from the origin. The relationship between the coefficients $a_k$ and the resulting displacement function is analyzed to ensure the independence of the bifurcating cycles.

3. Special Cases and Trigonometric Polynomials

We further examine the scenario where $f_1(\phi) = \cos \phi$ and $f_{0k}(\phi) = \cos(2m-2)\phi$. Using the recursive relations (12) and (14), we determine the conditions under which $\alpha_k(2\pi, a) = 0$. For $m \geq 2$, the functions $v_{kl}(\phi)$ are expressed as:
$$v_{31}(\phi) = a_{30} \cos(2m-1)\phi + \dots$$
The resulting algebraic system for the coefficients $b_{k0}$ allows us to identify the maximum number of periodic solutions. Specifically, we demonstrate that for a given degree of the trigonometric polynomial $f_0(\phi)$, there exists a neighborhood of the origin in the parameter space where the system exhibits exactly $n$ limit cycles.

4. Conclusion and Stability

The stability of the periodic solutions is determined by the sign of the first non-vanishing coefficient $\alpha_k(2\pi, a)$. If $\alpha_k < 0$, the limit cycle is stable; if $\alpha_k > 0$, it is unstable. The results obtained here extend the classical theorems of Lyapunov and Poincaré regarding the center-focus problem to non-autonomous systems with periodic coefficients. The construction of the displacement function $F(\rho, \phi, a)$ provides a robust framework for studying bifurcations in higher-dimensional parameter spaces.

References

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2. Bautin, N. N. On the number of limit cycles appearing with the variation of coefficients from an equilibrium point of focus or center type. Mat. Sb., 30, 1952.
3. Cherkas, L. A. On the number of limit cycles of a certain differential equation. Differentsial'nye Uravneniya, 1966.
4. Sansone, G., and Conti, R. Non-linear Differential Equations. Pergamon Press, 1964.
5. Shilov, G. E. Mathematical Analysis: Functions of One Variable. Moscow, 1953.