Preamble

DIFFERENTIAL EQUATIONS
DECEMBER 1967, VOLUME III, No. 12

Consider the equation $\ddot{x} + R(x, \dot{x}) = f(x)$, which is equivalent to the system:

$$\begin{aligned} \dot{x} &= y \ \dot{y} &= -R(x, y) + f(x) \end{aligned}$$

where $f(x)$ is defined such that:
1) The functions $R(x, y)$ and $f(x)$ are continuous for all real values and are continuously differentiable in the neighborhood of the singular points of equation (1). To avoid critical cases, it is also assumed that the derivatives of the functions $f(x)$ and $R(x, y)$ do not vanish at the zeros of these functions.
2) The functions $f(x)$ and $R(x, \dot{x})$ are periodic with period $2\pi$. The function $f(x)$ has a finite number of zeros within one period:

2 N

The function has zeros within its period: $n + m\tau$, where $n$ and $m$ are any finite integers. 4) $\Delta(x)$ changes sign—the first when passing through the point $x$, and the second when passing through...

2 R +

5) $\int_{0}^{2\pi} U(x)dx > 0$. 6) $R(x, x)$ is increasing with respect to $x$. 7) $R(x, 0) = 0$ for any $x$. 8) $\lim R(x, x) > f_1(x)$, $\lim R(x, x) < f_2(x)$.

Equation (1) can be interpreted as describing the motion of a material point along a closed curve under the influence of forces depending on position and velocity—specifically, under the action of a driving (pushing) force. Studies of this equation under similar constraints have been conducted in works \cite{1, 2, 3, 4} and others. The presence of a driving force determines the following relative arrangement of the zeros of the functions $f_1(x)$ and $f_2(x)$:

$x_0 < \eta_0 < \eta_1 < x_1 < x_2 < \dots < x_{2k} <$

$< \eta_{2k} < \eta_{2k+1} < \dots < \eta_{2m+1} < x_{2m+1} < \dots < x_{2n-1}$. In particular, when $n = m$, we have:

$x_0 < \eta_0 < \eta_1 < x_1 < x_2 < \dots < x_{2k-1} < x_{2k} <$

$< \eta_{2k} < \eta_{2k+1} < \dots < \eta_{2m-1} < x_{2n-1}$.

For simplicity of presentation, we shall henceforth assume that $n = m$. The possibility of the existence of limit cycles in the phase plane in this case, which encompass one or several instability segments of the type $[\eta_{2i+1}, x_{2i+1}]$, has been demonstrated in \cite{3, 4}. Let us consider in more detail the questions regarding the existence and relative positioning of "all-encompassing" limit cycles. We define an all-encompassing limit cycle of equation (1) in domain $D$ as a limit cycle that encloses all instability segments of the type $[\eta_{2i+1}, x_{2i+1}]$ lying within that domain.

Theorem. If in domain $D$, for which one of the sufficient conditions for the existence of a limit cycle is satisfied, the instability segments $[\eta_{2i+1}, x_{2i+1}]$ are located inside regions $D_1$ and $D_2$ bounded by separatrices and segments of the $Ox$ axis, and phase trajectories can only exit the regions $D_1$ and $D_2$, then there exists at least one all-encompassing limit cycle in domain $D$.

Proof. By the assumption that the regions $D_1$ and $D_2$ contain all unstable segments $[\eta_{2i+1}, x_{2i+1}]$, these regions will not contain only the single unstable segment of type $[\eta_{2i}, x_{2i}]$ lying in domain $D$. That is, region $D_1$ will be bounded by separatrices adjacent to point $(x_0, 0)$, and region $D_2$ will be bounded by separatrices adjacent to point $(x_{2n}, 0)$ [FIGURE:1]. Phase trajectories only exit regions $D_1$ and $D_2$, while they only enter domain $D$. Thus, phase trajectories only enter the region $D \setminus (D_1 \cup D_2)$. Since there are no stable singular points within this region, by the Bendixson theorem, there must exist a stable limit cycle in this region. This cycle will be all-encompassing, as it cannot enclose only region $D_1$ or $D_2$ due to the fact that it cannot cross an instability segment.

Remark 1. In each of the regions $D_1$ and $D_2$, either no all-encompassing limit cycle exists at all, or there is an even number of them.

Remark 2. If integral curves only exit the regions $D_1$ and $D_2$, then an all-encompassing limit cycle in domain $D$ is either absent or there is an even number of them.

Remark 3. If the region $D$ is bounded by separatrices for which the following equality holds:

x{R% k ) = x{S 2k ),

...or by separatrices for which the equality holds:

X(R2s) =* x(S'2s),

Consequently, we observe the degeneracy of the separatrices themselves into an all-encompassing limit cycle. Let $\eta_1, \eta_2$ denote the abscissas of the intersection points of the separatrices with the $x$-axis that are closest to the origin. Suppose that $|f_1(x) - f(x)| < \delta$ within the domain $G$.

Theorem 2

In a domain $D$ containing the instability intervals $[\eta_3, \eta_4]$, there exist all-encompassing limit cycles. If $\frac{dR(x, y)}{dx} > m > 0$, then the distance between any pair of these cycles exceeds a specific magnitude. Here, $\eta_3$ and $\eta_4$ represent the abscissas of the left and right intersection points of the boundary of domain $D$ with the $x$-axis, respectively.

The validity of this theorem follows directly from Theorem 1 \cite{5}. Indeed, by considering the interval $[\omega_2, \omega_1]$ as an interval known not to contain any points lying on the all-encompassing limit cycles, we satisfy the conditions of the aforementioned theorem.

Theorem 3

If $n$ instability intervals $[\eta_3, \eta_4]$ of the system are located within a certain number of limit cycles $V_k$, and the cycles $T_k$ are, in turn, enclosed by a limit cycle $V$, then the cycle $V$ cannot be separated from the cycles $T_k$ by a distance greater than...

• ' = (k+ l)dp 0

To prove this, let us assume the existence of a cycle $\Gamma'$ distinct from $\Gamma$. For a closed contour consisting of limit cycles (Fig. 2), we have, by virtue of system (2):
$$J = \oint [R(x, y) + f(x)] dx + y dy = \oint [R(x, y) + f(x)] dx \quad \text{(4)}$$
We now represent the integral in expression (4) over the closed contour by isolating the boundaries of the $\epsilon$-neighborhoods of the segments of the $x$-axis located between cycles $\Gamma'$ and $\Gamma$ as separate contours. This approach is necessary due to the non-uniqueness of the function $f(x)$ on the $x$-axis; we consider $\epsilon$ to be arbitrarily small. Applying Green's theorem, we can write:
$$\oint f(x) dx + \oint f(x) dx + \dots + \oint f(x) dx$$

Obviously, as $\epsilon \to 0$, we have:
$$\iint_G \frac{\partial R(x, y)}{\partial y} dx dy + \oint [f(x) - h(x)] dx + \oint [f(x) - h(x)] dx$$
Furthermore, we obtain $(k+1)d\rho$, where $G$ denotes the region lying between $\Gamma'$ and $\Gamma$, and $S$ denotes the area of this region. [FIGURE:2] It is not difficult to see that:

$$S > 2\rho (x_{2n-1} - x_1)$$

Moreover, we replace the segment $[a_{i-1}, a_i]$ of the axis between two adjacent limit cycles $\Gamma_{i-1}$ and $\Gamma_i$ with a larger axis segment between the end of the rightmost instability segment lying inside $\Gamma_{i-1}$ and the beginning of the leftmost instability segment lying inside $\Gamma_i$. We then take $\rho$ to be the length of the largest of these segments. Then, according to the condition of the theorem, we have:

J < — 2 m p ( x 2 r t _ ! — rix) + (k + l)rfp 0 < 0,

This leads us to a contradiction. Thus, we have demonstrated that the distance $\rho$ between any two encompassing limit cycles does not exceed the values determined by Theorem 2 and Theorem 3. However, when determining $\rho$, the value $\rho^$ generally cannot be calculated directly. Therefore, instead of using the exact separatrices that bound the region $D$, one may employ their respective upper or lower approximations \cite{5}. Furthermore, the abscissas of the intersection points between these approximating curves and the axis may be taken as $\eta^$. In this manner, the value of $\rho$ can be effectively computed.

References

1. Amerio, L. Ann. Sc. norm. Sup. di Pisa, III, 19–57, 1957.
2. Tabueva, V. A. Izv. vuzov. Matematika, No. 4 (5), 248–264, 1958.
3. Barbashin, E. A., Tabueva, V. A. PMM, Vol. XXIII, No. 5, 826–835, 1959.
4. Vdovina, E. V. Matem. zapiski UrGU, Vol. III, No. 2, 9–16, 1962.
5. Barbashin, E. A., Vdovina, E. V. Izv. vuzov. Matematika, No. 3 (16), 43–47, 1960.

Received by the editorial office on June 5, 1967.
A. M. Gorky Ural State University