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

Tangent/saddle-node bifurcation Intermittency Type-I Tangent/saddle-node bifurcation Laminar length?

Intermittency Type-II Hopf bifurcation

Intermittency Type-III Inverse period doubling bifurcation

Types of Intermittency Ref.: H. G. Schuster

Ref. H. G. Schuster

On-off intermittency Ref.:Y.-C. Lai Stable/Unstable subspace e.g. Synchronization: n-D  (n-m)-D Collision of two repellers with a saddle Ref.:Y.-C. Lai

Existence of n-dimensional invariant manifolds On-off intermittency Existence of n-dimensional invariant manifolds (Synchronization) Ott & Sommerer PLA 188, 39 (1994) Ding & Yang PRE 52, 207 (1995)

Sudden change in chaotic attractors with parameter variation Crisis Sudden change in chaotic attractors with parameter variation Ref.: E. Ott

Boundary Crisis 1-D maps: Ref.: E. Ott n-D maps:

Boundary Crisis due to tangencies Hetroclinic Homoclinc Ref. E. Ott

Boundary Crisis due to tangencies Hetroclinic Hmoclinc Ref. E. Ott

Boundary Crisis due to tangencies Hetroclinic Homoclinc Ref. E. Ott

Ikeda Map -Transients: depend on ICs -Not an attractor -“leaky” Ref. E. Ott

Boundary Crisis due to “unstable-unstable pair bifurcation.

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 -<t> similar to basin boundary -Not “leaky”

Pomeau-Manneville intermittency: Chaos  Periodic Crisis induce intermittency: Chaos  Chaos

Noise induced crisis: J.Sommerer, et al, PRL 66, 1947 (91) 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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