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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. Re-injection (Global features). Ref.: Hu. Intermittency Type-I

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Intermittency route to chaos

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  1. Intermittency route to chaos

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

  3. Intermittency: Tangent bifurcation

  4. Cause of Intermittency: Tangent Bifurcation

  5. Re-injection (Global features) Ref.: Hu

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

  7. Intermittency Type-II Hopf bifurcation

  8. Intermittency Type-III Inverse period doubling bifurcation

  9. Types of Intermittency Ref.: H. G. Schuster

  10. Ref. H. G. Schuster

  11. 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

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

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

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

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

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

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

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

  19. Boundary Crisis due to “unstable-unstable pair bifurcation.

  20. 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”

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

  22. Other Crises Noise induced crisis: J.Sommerer, et al, PRL 66, 1947 (91) Double crises H.B.Steward, et al, PRL 75, 2478 (95)

  23. Riddling

  24. Direct Transition:Fixed point to chaos

  25. !

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