1. Intermittency route to chaos

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

4. Intermittency: Tangent bifurcation

5. Cause of Intermittency: Tangent Bifurcation

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

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

8. Intermittency Type-II
Hopf bifurcation

9. Intermittency Type-III Inverse period doubling bifurcation

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

11. Ref. H. G. Schuster

12. 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

13. 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)

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

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

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

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

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

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

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

21. 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”

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

23. 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)

25. Riddling

28. Direct Transition:Fixed point to chaos

29. !