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| Existence and Uniqueness of Limiting Cycle of Equations $\[\dot x = \varphi (y) - F(x),\dot y = - g(x)\]$ and its Existence of Two and only Two Limiting Cycle |
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Citation： |
Zhou Yurong.Existence and Uniqueness of Limiting Cycle of Equations $\[\dot x = \varphi (y) - F(x),\dot y = - g(x)\]$ and its Existence of Two and only Two Limiting Cycle[J].Chinese Annals of Mathematics B,1982,3(1):89~102 |
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Net amount： 895 |
Authors： |
Zhou Yurong; |
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| Abstract： |
In this paper, we consider the existence of limiting cycle of the system of equations
$\[\dot x = \varphi (y) - F(x),\dot y = - g(x)\]$(E)
its existence and uniqueness, and its existence of two and only two limiting cycle.
The theorems of existence and of existence and uniquence include following conditions:
1° All orbits of system (E) rotate round origin and not all orbits rotate round origin;
2° All or some of integral $\[\int_0^{ \pm \infty } {g(x)}dx \]$ and $\[\int_0^{ \pm \infty } {F'(x)} dx\]$ diverge or converge;
3° System (E) has one or two (one of them is saddle point) singular points.
These theorems include following results:
1° All orbits of system (E), if not zero, tend to the unique cycle as $\[t \to + \infty \]$.
2° The result allow us to decide the place and the number of cycle etc.
In the theorem of existence of two and only two limiting cycle, F(x) and g (x) needn't odd functions; number of zero point of F(jx) may be five or over five; F (x) may ascend or descend repeatedly in certain finite interval.
Combining § 2 with § 3, in fact, we can give a result of the existence of n and only n limiting cycle of system (E). |
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