MATHEMATICS

Yu. S. BOGDANOV

SOME CRITERIA FOR THE ABSENCE OF CLOSED TRAJECTORIES

(Presented by Academician V. I. Smirnov on 13 XII 1957)

1°. Below a number of sufficient criteria are established for the absence, in a certain domain of the plane, of closed trajectories of a dynamical system described by the given differential system with continuous and twice continuously differentiable right-hand sides. The indicated criteria are not contained in Bendixson’s criterion ((1), p. 142). Nor do they follow from Poincaré’s criterion ((1), p. 144), if in the latter the topological system is assumed fixed.

2°. Basic notation. The symbol $\equiv$ should be understood as “is denoted below by”; $E^2 \equiv$ the two-dimensional real Euclidean space; $X \equiv$ an open subset of $E^2$; $x \equiv (x_1,x_2) \in X$; $|x| \equiv (x_1^2+x_2^2)^{1/2}$; $p(x) \equiv$ a real vector-function $(p_1(x),p_2(x))$, defined and twice differentiable for all $x \in X$; $D \equiv (dx/dt=p(x),\, x \in X,\, t \in (-\infty,\infty))$; $X_D \equiv$ the set of singular points of $D$; $V \equiv$ a bounded subset of $X$; $h \equiv$ the diameter of $V$; $\underline h \equiv$ the exact upper bound of the diameters of circles inscribed in $V$; $G \equiv$ a subset of $X$; $G' \equiv G \setminus X_D$;

[
k(x)\equiv p_1^2(x)\,\partial p_2(x)/\partial x_1
+p_1(x)p_2(x)\bigl(\partial p_2(x)/\partial x_2-\partial p_1(x)/\partial x_1\bigr)
-
]

[
{}-p_2^2(x)\,\partial p_1(x)/\partial x_2;
]

[
k(x;t)\equiv |k(x)|-t|p(x)|^3;
]

[
m(x)\equiv p(x)\,\partial\bigl(k(x)/|p(x)|^3\bigr)/\partial x
\quad \text{for } x\in G';
]

[
\Delta(x)=[0,|p(x)|^2/k(x)]
\quad \text{for } k(x)\ne 0;
]

[
\Delta(x)=(-\infty,\infty)
\quad \text{for } k(x)=0;
]

[
\xi(G)\equiv \bigcup_{x\in G'}\ \bigcup_{t\in \Delta(x)}
\bigl(x+t\bigr)\bigl(-p_2(x),\,p_1(x)\bigr);
]

[
\eta(G)\equiv \bigcup_{x\in G'}\ \bigcup_{t\in(-\infty,\infty)}
\bigl(x+t(-p_2(x),\,p_1(x))\bigr).
]

3°. Definitions. $\gamma \equiv$ a closed Jordan curve; $I_\gamma \equiv$ the interior of $\gamma$ ($\gamma$ encloses $V$) $\equiv (V\subset I_\gamma)$.

$(G$ is acyclic $(D)) \equiv$ (there exists no closed trajectory of $D$ having common points with $G$, in particular $(G\cap X_D=0)$); $(G$ is acyclic $(D)$ in itself$) \equiv$ (there exists no closed trajectory of $D$ situated entirely in $G$); $(G$ is acyclic $(D)$ with respect to $V) \equiv$ (there exists no closed trajectory of $D$ situated entirely in $G$ and enclosing $V$).

4°. Propositions $\mathfrak A$, $\mathfrak B$, $\mathfrak C$, $\mathfrak D$, $\mathfrak E$. $\mathfrak A \equiv$ (if $k(x,2/\underline h)>0$ for all $x\in G'$, then $G$ is acyclic $(D)$ with respect to $V$); $\mathfrak B \equiv$ (if $k(x)\ne 0$ for all $x\in G'$ and $\xi(G)\cap V\ne V$, then $G$ is acyclic $(D)$ with respect to $V$); $\mathfrak C \equiv$ (if $\eta(G)\cap V\ne V$, then $G$ is acyclic $(D)$ with respect to $V$); $\mathfrak D \equiv$ (if $k(x,2/h)<0$ for all $x\in V$, then $V$ is acyclic $(D)$ in itself); $\mathfrak E \equiv$ (if $m(x)\ne 0$ for all $x\in G'$, then $G'$ is acyclic $(D)$ in itself).

5°. Propositions (\mathfrak A_1, \mathfrak B_1, \mathfrak C_1, \mathfrak D_1, \mathfrak E_1). The propositions (4^\circ) are not difficult to translate into geometric language. Thus there arise the propositions (respectively) (\mathfrak A_1, \mathfrak B_1, \mathfrak C_1, \mathfrak D_1, \mathfrak E_1). (\mathfrak A_1 \equiv) (if at every point of (G') the curvature of the trajectory (D) is greater than (2/h), then (G) is acyclic ((L')) with respect to (V)); (\mathfrak B_1 \equiv) (if the trajectories (D) have no points of straightening in (G'), and if the set (\xi(G)), formed by the segments joining the points of (G') with the centers of curvature of the trajectories (D) passing through these points, does not cover (V), then (G) is acyclic ((D)) with respect to (V)); (\mathfrak C_1 \equiv) (if the set (\eta(G)), formed by the normals to the trajectories (D) drawn at all points of (G'), does not cover (V), then (G) is acyclic ((D)) with respect to (V)); (\mathfrak D_1 \equiv) (if (V \cap X_D = 0) and the curvature of the trajectory (D) passing through any point of (V) is less than (2/h), then (V) is acyclic ((D)) in itself); (\mathfrak E_1 \equiv) (if none of the trajectories (D) passing through (G') has stationary points of curvature in (G'), then (G') is acyclic ((D)) in itself).

6°. Propositions (\mathfrak A_2, \mathfrak B_2, \mathfrak C_2, \mathfrak D_2, \mathfrak E_2). The propositions (\mathfrak A_1, \mathfrak B_1, \mathfrak C_1, \mathfrak D_1, \mathfrak E_1), and consequently also the propositions (\mathfrak A, \mathfrak B, \mathfrak C, \mathfrak D, \mathfrak E), follow (respectively) from the following geometric facts ((l_1 \equiv) (a closed trajectory of the system (D)); (l_2 \equiv (l_1), enclosing (V)); (\varkappa l_i x =) (the curvature of (l_i) at the point (x \in l_i)), (i=1,2)); (\mathfrak A_2 \equiv) (if (G' \supset l_2), then (\varkappa l_2 x \leq 2/h) at least at one point (x \in G')); (\mathfrak B_2 \equiv) (if (G' \supset l_2) and (l_2) has no points of straightening, then every point of (V) lies on the segment joining some point of (l_2 \cap G') with the corresponding point of the evolute of (l_2)); (\mathfrak C_2 \equiv) (if (G' \supset l_2), then every point of (V) lies on some normal to (l_2)); (\mathfrak D_1 \equiv) (if (V \cap X_D = 0) and (V \supset l_1), then (\varkappa l_1 x \geq 2/h) for some (x \in V)); (\mathfrak E_2 \equiv) (if (G' \supset l_1), then (\varkappa l_1 x) attains an extremum for some (x \in G')).

7°. Remarks. 1) From the propositions (4^\circ) it is not difficult to derive criteria for acyclicity ((D)) of a domain (G), if, for example, one assumes that (G) is bounded by curves each of which consists either only of entry points or only of exit points. The latter will certainly be fulfilled if the boundary of (G) consists of smooth closed curves that are curves without contact. 2) The twice differentiability of (p(x)) is used only in the propositions (\mathfrak C, \mathfrak C_1, \mathfrak C_2); in the remaining propositions continuous differentiability of (p(x)) is sufficient. 3) If at some points (x \in G') (k(x)=0), then the condition (\xi(G) \cap V \ne V) is sufficient for the absence in (G') of convex trajectories (D) enclosing (V) (cf. (\mathfrak B)).

REFERENCES

1. V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, 2nd ed., Moscow–Leningrad, 1949.