文档介绍：16 Period doubling route to chaos
We now study the “routes” or “scenarios” towards chaos.
We ask: How does the transition from periodic to strange attractor occur?
The question is analogous to the study of phase transitions: How does a solid
e a melt; or a liquid e a gas?
We shall see that, just as in the study of phase transitions, there are universal
ways in which systems e chaotic.
There are three universal routes:
Period doubling
•
Intermittency
•
Quasiperiodicity
•
We shall focus the majority of our attention on period doubling.
Instability of a limit cycle
To analyze how a periodic regime may lose its stability, consider the Poincar´e
section:
x2
x1
x0
The periodic regime is linearly unstable if
ψx ψx < ψx ψx < . . .
| 1 − 0| | 2 − 1|
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or
νψx < νψx < . . .
| 1| | 2|
Recall that, to first order, a Poincar´e map T in the neighborhood of ψx0 is
described by the Floquet matrix
ωTi
Mij = .
ωXj
In a periodic regime,
ψx(t + φ) = ψx(t).
But the mapping T sends
ψx + νψx ψx + Mνψx.
0 ∗ 0
Thus stability depends on the 2 (plex) eigenvalues i of M.
If > 1, the fixed point is unstable.
| i|
There are three ways in which > 1:
| i|
Im λi
Re λi
−1 +1
1. = 1 + π, π real, π> 0. νψx is amplified is in the same direction:
x
x1 x2 x3 4
This transition is associated with Type 1 intermittency.
2. = (1 + π). νψx is amplified in alternating directions:
−
x3 x1 x0 x2
This transition is associated with period doubling.
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iε
3. = iλ= (1 + π)e± . νψx is amplified, νψx is rotated:
± | |
4 3
2
γγγ
1
x0
This transition is associated with quasiperiodicity.
In each of these cases, nonlinear effects eventually cause the instability to
saturate.
Let’s look more closely at the second case, 1.
◦−
Just before the transition, = (1 π), π> 0.
−−
Assume the Poincar´e section goes through x = 0. Then an initial pertur­
−
bation x0 is dam