An example of a composite stationary point

Preamble

DIFFERENTIAL EQUATIONS 1967, Vol. III, No. 3
BRIEF COMMUNICATIONS

An Example of a Complex Equilibrium Point

A. D. MYSHKIS, L. E. REIZIN

In the article \cite{1}, the question was raised regarding the possibility of constructing an example of an isolated equilibrium point that belongs to the limit set of some other trajectory, but does not serve as either a positive or negative limit set for any trajectory other than the point itself. In the present article, such an example is constructed. Consider the system of differential equations:

$$\frac{d\zeta}{dt} = P(\zeta, \eta, \zeta),$$

$$\frac{d\eta}{dt} = Q(\zeta, \eta, \zeta),$$ (1)

$$\frac{d\zeta}{dt} = R(\zeta, \eta, \zeta),$$ where

$P(\zeta, \eta, \zeta) = \dots$, $Q(\zeta, \eta, \zeta) = \dots$. The functions are defined as $P(\zeta, \eta, \zeta) = \sum (2 \zeta_i \dots)$ and $Q(\zeta, \eta, \zeta) = \sum (S_i \dots)$, where the sums are taken over all integer values of $k$.

$f_k = f(\rho; 2^{-k} \zeta, 2^{-k} \eta)$,

£ (((6 cos - : ( - l ) 0, sin (*/36)), 3-2-W+i))— + 2 ) , - 0 ,

$$ \exp \left( 2 - 2 (b - a)^2 (t - a)^{-1} (b - t)^{-1} + 1 \right), \text{ if } a < t < b, $$

$$ f(t; a, b) = \dots, \text{ if } t \in (a, b), $$

$$ g(t; a, b) = \int_a^t f(\tau; a, b) \, d\tau \bigg/ \int_a^b f(\tau; a, b) \, d\tau, $$

ft a cp = £(a 2 /p 2 ; sin 2 (2^/9), 1/2),

P = (£ 2 + ^ 2 + C 2 ) 1 / 2 ,

As shown in [FIGURE:N], it is easy to see that at any given point, no more than two terms under the summation sign in expressions (2) are non-zero. Consequently, the functions $P(\xi, \eta, \zeta)$ and $Q(\xi, \eta, \zeta)$ possess continuous derivatives of all orders throughout the entire space. This ensures that the theorems regarding the existence and uniqueness of solutions for system (1) are applicable at every point in the space. Furthermore, it is straightforward to verify that this system possesses a unique equilibrium point at the origin $O(0, 0, 0)$.

In the region defined by $a^2 / p^2 > 1/2$, system (1) transforms into the following system:

$$d\xi/dt = -\eta \exp(-1/p),$$

$$d\eta/dt = \xi \exp(-1/p). \quad (3)$$

d Vdt = 0.

An Example of a Single Complex Equilibrium Point: The trajectories in this region are circles centered on the axis. Within the cone $a (2\pi/9)$ for $\rho = 0$, the system takes the following form:

$$d\xi/dt = \xi \exp(-1/\rho),$$

$$d\eta/dt = \eta \exp(-1/\rho),$$

$$d\zeta/dt = \zeta \exp(-1/\rho),$$

That is, its trajectories intersect the sphere along the normals. The graphs of the functions $f$ and $g$ are provided. For $\rho = (z + 2)$, the system is expressed as:
$$\begin{aligned} d\xi/dt &= ((-1)^k \eta - \xi(1-\zeta)) \exp(-1/\rho), \\ d\eta/dt &= (\xi \cos(\pi/18) - (-1)^k \eta \sin(\pi/18) + \xi(1-\zeta)) \exp(-1/\rho), \end{aligned}$$

$$d\zeta/dt = ((-1)^k \eta \sin(\pi/18) + \zeta \eta) \exp(-1/\rho),$$

From this, since $q = 1$ when $p \to 0$, we obtain $\frac{dq}{dt} = 2 \frac{d \ln t}{dt} \frac{dq}{dt} \exp(1/p)$, where $\frac{dp}{dt}$ is non-zero only in specific regions. If we perform a change of variables,

$l_i = 5 \cos(m\pi/18) - (-1)^k C \sin(m\pi/18)$,

$(-1)^k S \sin(\pi/18) + \cos(\pi/18)$, when $a = (\pi/36)$. The only region where the system trajectories can pass through the sphere is $1/2$. We shall show that between the spheres $\rho = 3 \cdot 2^{k+1}$ and $\rho = 2^k$, the trajectories of the system are contained within circular cones with a common axis. If $c < \sin(\alpha)$, the system transforms into the following system within this region:

$\frac{d \ln d}{dt} = ((-m_{ij} + b_i g_j) f_t + \delta_{ij} h_j) \exp(-1/p)$,

$\frac{d M_i}{dt} = ((\xi_{ij} + \eta_j g_j) f_t + q_{ij}) \exp(-1/p)$,

d k/dt = (C; ft/,- + C; A/) exp ( - 1 / p ) ,

$y = i$, if $3 \cdot 2^{-(i+2)} < 2^{-i}$, and $j = i + 1$, $(i+1)(i+2)$. Consequently, we obtain $a = \text{const}$. From points 2° and 3°, it follows that those trajectories passing through the sphere $r = 3 \cdot 2^{-i}$ reach the sphere and fall into the region defined on it.

a? + ,/p<sin(*/36), (5)

A. D. MYSHKIS and L. E. REIZIN demonstrated that those trajectories passing through the sphere $\rho = 3 \cdot 2^{-k}$ reach the sphere $\rho = 2^{-k}$ and enter a specific region upon it defined by

$a^2 / \rho^2 < \sin^2(\pi/36)$. (6)

If there were trajectories that passed through both spheres, they would necessarily have to fall into both regions (5) and (6) simultaneously. However, this is impossible because the simultaneous fulfillment of both inequalities (5) and (6), in view of (4), leads to the contradictory inequality

$c^2 \cos^2(\pi/18) + c^2 \sin^2(\pi/18) < \rho^2 \sin^2(\pi/36)$.

Thus, no trajectory can cross two spheres of the form $\rho = 3 \cdot 2^{-k}$ in succession. [This

2. Distribution of various regions for system (1)

This demonstrates that no trajectory of system (1) can contain the origin in its limit set [FIGURE:2]. Finally, it should be noted that system (1) remains invariant under the transformation $\xi = -\xi'$, $\eta = -\eta'$, and $\zeta = -\zeta'$, as all functions $P_i$ are odd with respect to these variables. In particular, the field directions at symmetric points are opposite. We now construct the system:

$$d\xi/dt = \sum_{k} P(\xi, \eta, \zeta - 2k\pi) F_k,$$

$$d\eta/dt = \sum_{k} Q(\xi, \eta, \zeta - 2k\pi) F_k, \quad d\zeta/dt = R(\xi, \eta, \zeta),$$

where the sums are taken over all integers $k$. The right-hand sides of system (7) possess continuous derivatives of all orders throughout the entire space; therefore, the existence and uniqueness of a trajectory passing through any point in space is guaranteed for system (7). System (7) is periodic with respect to $\zeta$ with a period of $2\pi$. Indeed, if $\zeta$ is replaced by $\zeta + 2\pi$ and $k$ is replaced by $k + 1$, the form of the system remains unchanged. It follows from (8) that in the regions $-2\pi/3 < \zeta - 2k\pi < 2\pi/3$, each of the sums in (7) contains only one term, and the entire system (7) coincides with system (1). Consequently, it follows from Section 4 that the points $\zeta = \pi$ are isolated equilibrium points that do not belong to the limit sets of any other trajectories. System (7) possesses other isolated equilibrium points as well, specifically $\zeta = (2k + 1)\pi$. However, these other equilibrium points will not be of interest to us in the following discussion.

Among the properties of system (7), it should be noted that for large $a$, it takes the form (3). Finally, we construct the system:
$$d\xi/dt = (2\delta \sin(\zeta/2)(\sin(\zeta/2) + \cos(\zeta/2)) H/(2a),$$
$$d\eta/dt = (2\delta \sin(\zeta/2)(\sin(\zeta/2) + \cos(\zeta/2)) - a H/(2a),$$

$$d\zeta/dt = \sum_{k} R(\xi, \eta, \zeta - 2k\pi) F_k,$$

$(a, \xi, \eta)$ exp $(2\delta (\zeta/2))$ if $a > 2\delta \sin(\zeta/2)$, and $a < 2\delta \sin(\zeta/2)$. [FIGURE:3] Distribution of various regions for system (9). The right-hand sides of system (9), like those of system (7), have continuous partial derivatives of all orders throughout the space. For $|2\delta \sin(\zeta/2)|$ and for $a > 7$, system (9) coincides with system (7), and specifically for $a > 7$, it coincides with system (3). The surface

$\dot{\phi} = 2 \zeta \sin^2(\zeta/2)$ (11)

The phase space is filled with closed trajectories where $\text{const}$, which degenerate into points on the $\zeta$-axis as they approach the limit. For $5|\sin(\zeta/2)| < \delta$, we have $\dot{\zeta} = 2 \zeta \sin(\zeta/2) \cos(\zeta/2)$. In this equation, $\alpha(\zeta/2)$ serves as the solution toward which all other solutions converge as $\zeta \to \phi$ [FIGURE:3]. Indeed, if we denote $(\zeta/2) = \dots$

$= |A|$, then

(—1/fx), d-Udt = exp (—1/p.)

dfx/dC = —

From equation (12), it is evident that as $C \to \infty$, the value of $\phi$ also approaches infinity, although the rate of increase decreases as $C$ grows. As $t \to \infty$, the variable $\mu$ increases and eventually exceeds the limit $36-25(C/2)$. It is easy to see that while $C \to \infty$, the term $\mu$ tends toward a finite limit by virtue of the expression for the factor in (10).

DIFFERENTIAL EQUATIONS
A. D. MYSHKIS, L. E. REIZIN
Let us define the mapping:

$$x = \left( 8 + \phi + (S^2 + 16)^{1/2} \right) \cos C$$

$$y = \left( 8 + \phi + (S^2 + 16)^{1/2} \right) \sin C$$

The inverse mapping is given by:

g = ( r - 8 ) / ( 2 r ) 1 / 2 ,

$z = \zeta / (2\Gamma)^{1/2}$

$\arctan(y/x) = \phi$. The general equilibrium points are $(0, 0, 0)$, from which it is evident that the mapping is topological and even analytic in both directions, provided that in the $(x, y, z)$ space we restrict ourselves to the domain $x^2 + y^2 > 0, -\infty < z < +\infty$, and in the $(\xi, \eta, \phi)$ space we identify points with coordinates $\phi$ and $\phi + 2\pi$. Due to the periodicity of the right-hand sides of system (9) with respect to $\phi$, the trajectories of the system will transform into the trajectories of the system obtained from (9) after substitution (13). For $a > \gamma$ (i.e., outside the region $S$), the system after mapping (13), according to Section 7, yields

$dx/dt = -2xz(z + 8)^{-1}$

• dy/dt = —2yz(r^8)- 1 ,

$$\frac{dz}{dt} = - \delta - \phi z - z^2 \frac{z + \delta + (2z)^{1/2}}{2z(z + \delta)}$$

From this, it is evident that for $z > 0$, the right-hand sides of the derivatives are continuous. It would have been possible to maintain the continuity of the derivatives of all orders by multiplying the right-hand sides by $\exp(-1/z)$; however, this is not essential for our purposes. Under the mapping (13), the $\zeta$-axis is transformed into the circle $x^2 + y^2 = \delta^2$, and the equilibrium point $O$ is mapped to the equilibrium point $x = \delta, y = 0$.

0. All

Trajectories passing through the points of the circle $\theta = 0, -a < r < a$ (excluding the point $r = 0, \theta = 0$), according to Section 8°, have the surface defined by Eq. (11) as their positive limit set.

Specifically, this includes the point $x = 0$. Thus, while the point belongs to the positive limit set of other trajectories, it does not constitute the entire limit set for any trajectory other than itself [FIGURE:4]. This system possesses properties similar to the system constructed by Zaremba in his article, yet it maintains a simpler structure.

References

Myshskis, A. D., and Reizin, L. E. Izvestiya Akademii Nauk Latviyskoy SSR, Seriya Fizicheskikh i Tekhnicheskikh Nauk, 80–86, S. K. Ann. Soc. Pol.

Math., (1937) 1938, Received by the editorial office

February 16th

1966, Physicotechnical Institute of Low Temperatures of the Ukrainian Academy of Sciences; Institute of Physics of the Latvian Academy of Sciences