Intermittency route to chaos

Intermittency route to chaos

Regular behavior (laminar flow) is Intermittently Interrupted by chaotic outbreaks (bursts)

Intermittency: Tangent bifurcation

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

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

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Ref.: Hu

Intermittency Type-I

Tangent/saddle-node bifurcation

HOT 21 nnn xxx

Laminar length?

Intermittency Type-II

Hopf bifurcation

n

nnn rrr

1n

31 HOT )1(

Intermittency Type-III

Inverse period doubling bifurcation

HOT )1( 31 nnn xxx

Types of Intermittency

Ref.: H. G. Schuster

Ref. H. G. Schuster

On-off intermittency

Stable/Unstable subspace

e.g. Synchronization:

n-D (n-m)-D

Collision of two repellers with a saddle Ref.:Y.-C. Lai

On-off intermittency

Existence of n-dimensional invariant manifolds(Synchronization)

Ott & Sommerer PLA 188, 39 (1994)Ding & Yang PRE 52, 207 (1995)

Crisis

Sudden change in chaotic attractorswith parameter variation

Ref.: E. Ott

Boundary Crisis

)( c

2

1

Ref.: E. Ott

1-D maps:

2

1n-D maps:

Boundary Crisis due to tangencies

Homoclinc

Ref. E. Ott

Hetroclinic

Boundary Crisis due to tangencies

Hmoclinc

Ref. E. Ott

Hetroclinic

Boundary Crisis due to tangencies

Homoclinc

Ref. E. Ott

Hetroclinic )( c

)||/(ln|)|(ln 2212 |||ln|/|)|(ln

2

121

eP )(

Ikeda Map

]||1

exp[21

nnn z

iikbzaz

-Transients: depend on ICs-Not an attractor-“leaky”

Ref. E. Ott

Boundary Crisis due to “unstable-unstable pair bifurcation.

])/(exp[ 2/1ck

Interior crisis: crisis induced intermittency

Unstable period-3 fixed points created by tangent bifurcation collide with chaotic attractor.

Chaotic attractor suddenly expands.

-No basin boundary-<> similar to basin boundary-Not “leaky”

Pomeau-Manneville intermittency:

Crisis induce intermittency:

Chaos Periodic

Chaos Chaos

Other Crises

Noise induced crisis:

J.Sommerer, et al, PRL 66, 1947 (91)

Double crises

H.B.Steward, et al, PRL 75, 2478 (95)

Riddling

Direct Transition:Fixed point to chaos

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