Title: QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS ON THE PROJECTIVE PLANE
Abstract: Ⅰ.Basic concepts and theoremsConsider the differential system where P(x, y), Q(x, y) are defined on the square S: [0, a] × [0, a], continuous and have continuous first partial derivative there, and satisfy the following relationsA 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 projective plane.On the projective plane, in addition to closed curves in the usual sense (we call it O-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 Γ on the projective plane an n-closed curve, if Γ consists of n arcs which do not meet each other in S, and each arc intersects the sides of S at its two end points only. If n is even (odd), then we also call Γ an evenclosed 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, upperlower and lower-upper oriented families. Lemma 2. Let Γ be a closed orbit of system (1) on the projective plane consisting of only oriented arcs of the same kind, then Γ contains two arcs at most.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] × [-a, a] by defining first in [-a, a] × [0, a]: and then in S~*: It is easily seen that is a C~1 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 ∪ m_2 and n=n_1 ∪ n_2 be two 2-closed curves in the projective plane, and n is in the interior of m. Suppose the domain Ω 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 interlor to exterior. Then Ω contains at least two 2-closed orbits Γ and L, where Γ is outer-stable, L is innerstable limit cycle. Here Γ may coincide with L, if this takes place, then Γ=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 S. 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 Γ 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 Γ contains no stationary points, then in the interior of Γ there is a 1-closed orbit. Theorem 3. Let G be a domain in the projective plane, and B(x, y) be asinglevalued continuous function in G which has continuous first partial derivatives. Suppose 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 singlevalued continuous function and has conti
Publication Year: 1981
Publication Date: 1981-01-01
Language: en
Type: article
Access and Citation
AI Researcher Chatbot
Get quick answers to your questions about the article from our AI researcher chatbot