庞加莱截面

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2.1 Poincare section
Using Poincare section [7], we can qualitatively describe the characteristic of the motion trajectory in phase space. With this method, we can compress a dimension of the phase space; hence, it brings convenience to the research. For the motion system of four-dimensional phase space, the energy is a constant, and it makes the motion trajectory in a torus of three-dimensional space. We can take a section, for example: the section of q2=0, the trajectory in the section is

(12)
Where, (q1, p1) is the coordinate of X, namely, while the trajectory go through the section, a dot will be left on it. We can get a series of this kind of dots (Xn). There are two values of p for one Hamiltonian because Hamiltonian is the function of p2; hence,
Xn can’t determine Xn+1
exclusively. So, in order to make Xn determine Xn+1
exclusively, we stipulate: only p2>0
(or p2<0), a dot will be left on the section, when the trajectory go through it. Then, the trace on the section faithfully reflects the essence characteristic of trajectory. The Poincare section is shown in figure.1.
1). It indicates the movement of the system is periodic when a stationary dot or a few discrete dots are shown on the section.
2). It means the movement of the system is quasi-periodic when a successive close curve is shown on the section. Generally, on Poincare section, periodic trajectories are in the center of the oval island consisted of quasi-period trajectories.
3).It means the system is chaotic when large numbers of dispersive dots take on the Poincare section.