文档介绍：8 Poincar´e sections
The dynamical systems we study are of the form
d
ψx(t) = F (ψx, t)
dt
Systems of such equations describe a flow in phase space.
The solution is often studied by considering the trajectories of such flows.
But the phase trajectory is itself often difficult to determine, if for no other
reason than that the dimensionality of the phase space is too large.
Thus we seek a geometric depiction of the trajectories in a lower-dimensional
space—in essence, a view of phase space without all the detail.
Construction of Poincar´e sections
Suppose we have a 3-D flow . Instead of directly studying the flow in 3-D,
consider, ., its intersection with a plane (x3 = h):
Γ
x3
S
P
P0 1 P2
h
x2
x1
Points of intersection correspond (in this case) to x˙< 0 on .
• 3
Height h of plane S is chosen so that continually crosses S.
•
The points P , P , P form the 2-D Poincar´e section.
• 0 1 2
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The Poincar´e section is a continuous mapping T of the plane S into itself:
2
Pk+1 = T (Pk) = T [T (Pk 1)] = T (Pk 1) = . . .
−−
Since the flow is deterministic, P0 determines P1, P1 determines P2, etc.
The Poincar´e section reduces a continuous flow to a discrete-time map­
ping. However the time interval from point to point is not necessarily con­
stant.
We expect some geometric properties of the flow and the Poincar´e section to
be the same:
Dissipation areas in the Poincar´e section should contract.
•∞
If the flow has an attractor, we should see it in the Poincar´e section.
•
Essentially the Poincar´e section provides a means to visualize an otherwise
messy, possibly aperiodic, attractor.
Types of Poincar´e sections
As we did with power spectra, we classify three types of flows: periodic,
quasiperiodic, and aperiodic.
Periodic
The flow is a closed orbit (., a limit cycle):
P0
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P0 is a fixed point of the Poincar´e map:
2
P0 = T (P0) = T (P0) = . . . .