| CHEN YIYUAN.[J].数学年刊A辑,1981,2(1):101~114 |
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| QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONSON THE PROJECTIVE PLANE |
| Received:January 09, 1980 |
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| Consider the differential system
\[\frac{{dx}}{{dt}} = P(x,y),\frac{{dy}}{{dt}} = Q(x,y)\](1)
where P(x, y), Q(x,y) are defined on the square S: [0, a]X[0, a], continuous and have continuous first partial derivative there, and satisfy the following relations
P(0, y)=P(a,a-y), Q(0, y) = -Q(a, a-y),
P(x, 0) = —P(a-x,a), Q(x,0)=Q(a-x, a),
\[(x,y) \in [0,a] \times [0,a]\],
A projective plane will be viewed as the square S in the (x,y) -plane, in which the points (0, y), (a, a -y) or (x, 0), (a—x, a) on opposite sides of the square are identified. Thus, under condition (2), (1) is a differential system defined on the pro?tective plane.
On the projective plane, in addition to closed curves in the usual sense (we call it 0-closed curve), there are also closed curves consisting of several arcs in the square, such closed curves are illustrated in Fig. 1 (in which we use arrows and numbers to show that a closed curve can be constructed according to this direction and order). Hereafter, we will call the closed curve JT on the projective plane an closed curve, if .T consists of n arcs which do not meet each other in S, and each arc intersects the sides of 8 at its two end points only. If n is even (odd), then we also call P an even- closed curve (odd-closed curve) on the projective plane.
Lemma 1. An even-closed curve on the projective plane divides the projective plane into two parts, but an odd-closed curve does not.
So we can define in a certain sense the interior and exterior of an even-closed curve, while for an odd-closed curve, we can not define its interior and exterior.
We call L left-right orientead family of directed arcs in S, if the origin of every arc in L is at the left hand side of its end. Similarly, we can define right-left, upper- lower and lower-upper oriented families.?
Lemma 2. Let \[\Gamma \] be a closed orbit of system (1) on the projective plane consisting of only oriented ares of the same kind, then \[\Gamma \] contains two arcs atmost.
On the projective plane, we can define limit cycle of the differential system (1) as in [1]. In particular, we can define stable and unstable cycles as well as semi-stable cycle for an even-closed orbit, but if an odd-closed orbit is a limit cycle, it must be a stable or unstable limit cycle.
Let us extend system (1) to the square S*: [—a, a] x [—a, a] by defining first in
[-a, a] X [0, a]:
\[\begin{gathered}
{P_1}(x,y) = \left\{ {\begin{array}{*{20}{c}}
{p(x,y),(x,y) \in [0,a] \times [0,a]} \\
{p(a + x,a - y),(x,y) \in [ - a,0] \times [0,a];}
\end{array}} \right. \hfill \ {Q_1}(x,y) = \left\{ {\begin{array}{*{20}{c}}
{Q(x,y),(x,y) \in [0,a] \times [0,a]} \\
{ - Q(a + x,a - y),(x,y) \in [ - a,0] \times [0,a];}
\end{array}} \right. \hfill \ \hfill \\
\end{gathered} \]
and then in s*:
\[\begin{gathered}
{P_2}(x,y) = \left\{ {\begin{array}{*{20}{c}}
{{p_1}(x,y),(x,y) \in [ - a,a] \times [0,a]} \\
{ - {p_1}( - x, - y),(x,y) \in [ - a,a] \times [ - a,0];}
\end{array}} \right. \hfill \ {Q_1}(x,y) = \left\{ {\begin{array}{*{20}{c}}
{{Q_1}(x,y),(x,y) \in [ - a,a] \times [0,a]} \\
{ - {Q_2}( - x, - y),(x,y) \in [ - a,a] \times [ - a,0];}
\end{array}} \right. \hfill \\
\end{gathered} \]
It is easily seen that
\[\frac{{dx}}{{dt}} = {P_2}(x,y),\frac{{dy}}{{dt}} = {Q_2}(x,y)\](6)
is a C1 differential system on the torus formed by identifying opposite sides of S*. A closed orbit of (1) on the projective plane must correspond to some closed orbits of (6) onthe torus. We can prove now the following theorems.
Theorem 1. Let \[m = {m_1} \cup {m_2}\] and \[n = {n_1} \cup {n_2}\] be two 2-closed curves in the projective plane, and n is in the interior of m. Suppose the domain Q bounded by m and n contains no stationary points., and trajectories of (1) crossing m all run from exterior to interior, while trajectories crossing n all run from interior to exterior. Then Q contains at least two 2-closed orbits F and L, where F is outer-stable, L is inner- stable limit cycle. Here r may coincide with L, if this takes place, then F=L is a stable limit cycle.
Theorem 1.1. Let m be a 2-closed curve in the projective plane which consists of arcs joining opposite sides of the square 8. The interior of m contains no stationary points, and trajectories crossing m all run into the interior of m, then in the interior of m there is at least a closed orbit of (1) which is an outer stable 2-limit cycle or a stabe 1-limit cycle.
Theorem 1.2. Let \[\Gamma \] be a 2-closed orbit of (1) in the projective plane which consists of arcs joining opposite sides of the square S. The interior of \[\Gamma \] contains no stationary points, then in the interior of \[\Gamma \] there is a 1-closed orbit.
Theorem 3. Let Q be a domain in the projective plane, and B(x, y) be a single- valued continuous function in G which has continuous first partial derivatives. Suppose
\[\frac{\partial }{{\partial x}}(BP) + \frac{\partial }{{\partial y}}(BQ)\] does not change its sign in G, and the set \[\frac{\partial }{{\partial x}}(BP) + \frac{\partial }{{\partial y}}(BQ) = 0\]contains no 2-dimensionaL domain, then the system (1) has no even-closed orbit whose interior is in G.
In particular, if G is the whole projective plane, then the system (1) has no closed orbit at all.
Theorem 4. Suppose F(x, y) = C is a family of curves, where F(x, y) is a single valued continuous function and has continuous first partial derivatives in the projective plane. \[P\frac{{\partial F}}{{\partial x}} + Q\frac{{\partial F}}{{\partial y}} does not change its sign in a domain G, and the subset of G in which \[P\frac{{\partial F}}{{\partial x}} + Q\frac{{\partial F}}{{\partial y}} = 0\] contains no closed orbits of (1), then system (1)
II. Examples
The systems in the following examples are all systems of differential equations (defined in the projective plane, and the projective plane is formed by the square [0, pi] x [0, pi]. A, B, C, D are constants other than zero.
\[Ex.1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{dx}}{{dt}} = A\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2x + B\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} y,\frac{{dy}}{{dt}} = C\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 4y\]
Using theorem 4 and theorem 1, we can prove that (8) has a 2-olosed orbit and an 1-closed orbit, which are stable and unstable limit cycles respectively.
\[Ex.2{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{dx}}{{dt}} = B\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} y,\frac{{dy}}{{dt}} = C\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x + \frac{D}{2}\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2y\]
system (9) has always a 1-closed orbit, if \[\left| {\frac{D}{{2C}}} \right| > 1\], then we can prove that (9) has no 2-closed orbit.
\[Ex.3{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{dx}}{{dt}} = \frac{A}{2}\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2x,\frac{{dy}}{{dt}} = C\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x + \frac{D}{2}\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2y\]
where D>-2C>0
This system has ho closed orbit, although the projective plane is a one-sided surface. |
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