庞加莱-本迪克松定理 (Poincaré-Bendixson theorem)是平面定性理论的经典成果并是后续研究的重要基础。 给定系统dx/dt=X (x)， (1)，或平面系统：dx/dt=P (x,y),dy/dt=Q (x,y)， (2)，庞加莱-本迪克松定理断言：若系统 (2)的一条正半轨保持在某一不含奇点的有界区域内，则它 ...

In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere.

其实这叫 庞加莱-本迪克松环域定理 Poincare-Bendixson Ring Domain Theorem. 假设 R 是一个被两条简单曲线 D_1,D_2 包围的环域 且 D_2 被完全包围在 D_1 中， F 是系统的速度场. 则系统在 R 中存在一个闭合轨迹. 直观理解： 一条轨迹一旦进入了 R 就出不去了 （抑或是从R里开始的轨迹）因为在边界的速度场方向都指向内部 那末它只能逼近一个临界点或是一个极限环 而根据假设 R 内部没有临界点 所以只能是闭合轨迹（不可能是不稳定极限环 只可能是另三种情况） for a …

在 数学 中， 庞加莱-本迪克松定理 是一个关于二维平面上的 连续动力系统 的 轨道 的变化趋势的定理。 庞加莱-本迪克松定理说明了：如果在二维的平面上的连续动力系统的某一个解的轨道被限制在一个紧区域内，那么在时间足够长之后，这个轨道要么逼近某一个 奇点，要么逼近某一个周期轨道（极限环）。 因此，一维或者二维平面上的连续动力系统是不可能出现 混沌 现象的。 混沌现象只可能出现在三维或以上维数空间上的连续动力系统中。 但是要注意的是：庞加莱-本迪克 …

NOTES ON THE POINCARE{BENDIXSON THEOREM JONATHAN LUK long-time behavior of solutions to ODEs. For this it will be very useful to introduce the notion of !-limit sets. A remarkable result - the Poincare{Bendixson theorem -is that for planar ODEs, one can have a r

y′ = g(x, y) Poincare-Bendixson Theorem Suppose R is the finite region of the plane lying between two simple closed curves D1 and D2, and Fis the velocity vector field for. th. lying inside R. The hypotheses of the theorem are illust. ated by fig. 1. We will not give the proof of the theorem, which requires a background in Mathem.

PROOF OF THE POINCARE-BENDIXON THEOREM. The aim is to show that if the omega limit set !+(x0) is nonempty, then it either an equilibrium point or a closed periodic orbit.

2020年2月14日 · The Poincaré–Bendixson theory borders on: the connection, discovered by Poincaré, between the rotation of a certain field on the boundary of a domain and the indices of the equilibrium positions inside it (see Singular point, index of a); results of Bendixson and L.E.J. Brouwer on the possible types of behaviour of trajectories near ...

In this lecture notes, we will introduce and give the proof of a celebrated result – Poincaré-Bendixson’s Theorem – which can show a periodic solution exists for the glycolysis system.

Example Using The Poincare-Bendixson Theorem We illustrate t. e Poincare-Bendixson The. rem with an example. Consider the system x0 = (6 x2 y2)x 3y3; y0 = (4 x. y2)y + 2x; where x and y are functions of t. First. there is clearly a critical point at (0; 0). The proof that it is the only critical point turned out to be rather con.