Intermittency & Crisis
• What’s intermittency?• Cause of intermittency.• Quantitaive theory of intermittency.• Types of intermittency & experiments.
• Crises• Conclusions
What’s Intermittency ?Intermittency: sporadic switching between 2 qualitatively different behaviors while all control parmeters are kept constant.
periodic chaotic periodic quasi-
periodic
fully periodic Intermittency fully chaotic
___________ Ac ________________________ A∞
___________
A = 3.74, period 5
A = 3.7375, Intermittency
logistic
map10 20 30 40
0.4
0.5
0.6
0.7
0.8
0.9
5 10 15 20 25 30 35
0.4
0.5
0.6
0.7
0.8
0.9
(Apparently)
Y.Pomeau, P.Manneville, Comm.Math.Phys 74, 189 (80) Reprinted: P.Cvitanovic, “Universality in Chaos”
15 20 25 30 35 40
-50
50
100
150
200
z
y
x
15 20 25 30 35 40-50
50
100
150
200
250
z
y
x
Lorenz Eq.
r = 165, periodic
r = 167, intermittent
Cause of Intermittency: Tangent Bifurcation
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0.48 0.49 0.51 0.52
0.48
0.49
0.51
0.52
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0.48 0.49 0.51 0.52
0.48
0.49
0.51
0.52
f(5
)Iterates of f(5)
(0.5)
A = 3.74 period 5
A = 3.7375 intermittent
~ 4 cycles of period 5
5 stable, 7 unstable f.p.
2 unstable f.p.
Saddle-node bifurcation
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
Re-injection (Global features)
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
n = 10 n = 21
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
n = 91:96
Ref: Schuster
(Reverse) Tangent Bifurcation
Condition for birth of tangent bifurcation at period-n window: *
1nd f xd x
* *nf x x at AC
For A > AC,
*1
nd f xd x
where
for the unstable f.p.
C.f., for period-doubling, bifurcation is at
*1
nd f xd x
→ Type I intermittency for A < AC
Sine-circle map, K < 1: intermittency is similar but between freq-lock & quasi-periodicity
1/f noise
Power spectra 1/fδ 0.8 < δ < 1.4
See Schuster
Power spectra of systems with intermittency also exhibit 1/fδ dependence.Too sensitive
to external noise.
Quantitative Theory of Intermittency
Tangent bifurcation near stable n-period fixed point x*
( periodic for A > AC, intermittent / chaotic for A < AC ): * *
C
nAf x x
2* * *nA Cf x x x x a x x b A A
Set:
1 *y x xb
c ab CA A
→ 2*nAf y x
y c yb
21k k k ky h y y c y
*
1C
nA
x
d fdx
-0.4 -0.2 0.2 0.4
-0.4
-0.2
0.2
0.4
0.6
0.8
c,2,.1c,2,0c,2,.1
2h y y c y
< 0 : periodic = 0 : tangent bif > 0 : intermittent
Average Duration of Bursts: Renormalization Arguments
L → 0 for >> 1 L → ∞ as → 0+
L = average length of bursts of periodicity
L n() = number of iterations required to pass thru gap Analogous number for h(2) is
22 2 2h y y c y c y c y
2h y y c y
22 2y c y , 0y
Scaling:
2
2 , 2 2y y yh c
2,h y y c y
→ 2, 4
h(2) → h → δ = 4
1 42n n
12n
< 0 : periodic = 0 : tangent bif > 0 : intermittent
h(2m) → h → 4m
1 42
mm n n
h(2) → h → 4
1 42n n
Ansatz:
kn C
1 42
kk mm C C
1 42
mkm
ln 2 2 ln 2m mk
12
k
Cn
Experimental confirmation: diode circuit
Renormalization theory version: there exists g such that
xg x g g
with
0 1g
0 0g
/ /g x g g x g x
0 0 0g g g g 0 0 0g g g
1xg xcx
/1 /
x xg g gcx
/1 /
/11 /
xcxcxcx
/1 2 /xcx
/1 2 /
xg g xcx
1xg xcx
→ 2
0 0g 2
11 1
cxg xcx cx
2
11 cx
0 1g
Extension to other univ classes: B.Hu, J.Rudnick, PRL 48,1645 (82)
0 1g g 4
Ansatz
See Schuster, p.45
Type I Type II Type IIITangent Hopf Period- doubling
xn+1 = ε+ xn + u xn
2
rn+1 = (1+ε) rn + u rn
3
Θn+1 = Θn + Ω
xn+1 = -(1+ε) xn - u xn
3
Types of Intermittency
On-off intermittency = Type III with new freq ~ 0
M
xn
ε< 0 → ε > 0
Crises
Unstable fixed point / limit cycle collides with chaotic attractor → sudden changes in latter
• Boundary crisis: chaotic attractor disappears• Interior crisis: chaotic attractor expands
Sudden changes in fractal structure of basin boundary of chaotic attractor: metamorphosis
Boundary CrisisLogistic map:
• A*3 < A < 4: chaotic attractor expands as A increases.
• A = 4: chaotic attractor fills [0,1] and collides with unstable fixed point at x = 0.
• A > 4: chaotic attractor disappears; new attractive fixed point at x = -∞.
• A 4: escape region = [ x-, x+ ], i.e.,
f(x) > 1 if x [ x-, x+ ]1 41 12
xA
Average duration of chaotic transient
1 141
x xA
Universal for quadratic maps.
1 1Ax x →
14A
for A 4
2-D Henon map: crisis near C = 1.08
Interior Crisis
Logistic map:
• Unstable period-3 fixed points created by tangent bifurcation at A = 1+√8 collide with chaotic attractor at A*3.
• Chaotic attractor suddenly expands at A*3 ( trajectories scattered by the unstable fixed point into previously un-visited regions).
Logistic map:
• Average time spent in pre-expansion-chaotic region is proportional to (A-A*3)-½. → Loss regions = penetration of unstable x* into chaotic bands (A-A*3)½.
(launches into previously forbidden region). → Re-injection region Xr ( back into chaotic bands) → Crisis-induced intermittency.
Universality I
xj = f(j)
(1/2)
Logistic map: fraction of time spent in pre-expansion-forbidden region is
lnlnlnaF a K a C P aM
*3a A A
Universality II
N N
O N O
t tFt t t
average time spent in new, previously forbidden, region
average time spent in old, chaotic band, region
For tN << tO, we have
1Ot a
tN time spent in previously forbidden region before landing in Xr.For x ½, f(3×2)(x) x6 near x*.
x6 - x* a
Ex 7.6-3
Let d be the distance from x* to Xr.Let M be the Floquet multiplier for F = f(3) at x*
12
nrF x x X
Let → 6 * n nd x x M kaM
ln lnd n Mka
ln / lnln lnd k anM M
lnlnaCM
tN is a periodic function of lna with period lnM.
Suppose when a = an, F(n)(x) reaches Xr but not F(n-1)(x) → tN n(an).As a increases beyond an, F(n)(x) may overshoots Xr while F(n-1)(x) hasn’t arrived → tN becomes longer
Further increase of a brings F(n-1)(x) to Xr → tN n(an-1)-1.
11
n nn na M a M
1ln ln lnn na a M
ln lnlnaF a K a C P aM
P = some function with period ln M.
Noise–Induced Crisis
Noise can bump a system in & out of crisis.
Average time τ between excursions into pre-crisis gaps is described by a scaling law:
*A AK g
where σ strength of noise
Ref: J.Sommerer, et al, PRL 66, 1947 (91)
Double crises
H.B.Steward, et al, PRL 75, 2478 (95)