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)