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Intermittency route to chaos. Regular behavior (laminar flow) is Intermittently Interrupted by chaotic outbreaks (bursts). Intermittency: Tangent bifurcation. Cause of Intermittency: Tangent Bifurcation. Re-injection (Global features). Ref.: Hu. Intermittency Type-I - PowerPoint PPT Presentation
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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
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0.48 0.49 0.51 0.52
0.48
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0.2 0.4 0.6 0.8 1
0.2
0.4
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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
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0.2 0.4 0.6 0.8 1
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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
!
