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Intermittency & Crisis

Intermittency & Crisis. What ’ s intermittency? Cause of intermittency. Quantitaive theory of intermittency. Types of intermittency & experiments. Crises Conclusions. What ’ s Intermittency ?.

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Intermittency & Crisis

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  1. Intermittency & Crisis • What’s intermittency? • Cause of intermittency. • Quantitaive theory of intermittency. • Types of intermittency & experiments. • Crises • Conclusions

  2. What’s Intermittency ? Intermittency: sporadic switching between 2 qualitatively different behaviors while all control parmeters are kept constant. periodic  chaotic periodic  quasi-periodic (Apparently) Y.Pomeau, P.Manneville, Comm.Math.Phys 74, 189 (80) Reprinted: P.Cvitanovic, “Universality in Chaos” fully periodic Intermittency fully chaotic ___________ Ac ________________________ A∞ ___________ logistic map A = 3.74, period 5 A = 3.7375, Intermittency

  3. Lorenz Eq. r = 165, periodic r = 167, intermittent

  4. Cause of Intermittency: Tangent Bifurcation f(5) Iterates of f(5)(0.5) Saddle-node bifurcation A = 3.74 period 5 5 stable, 7 unstable f.p. A = 3.7375 intermittent 2 unstable f.p. ~ 4 cycles of period 5

  5. Re-injection (Global features) n = 10 n = 21 n = 91:96 Ref: Schuster

  6. (Reverse) Tangent Bifurcation Condition for birth of tangent bifurcation at period-n window: where at AC For A > AC, for the unstable f.p. → Type I intermittency for A < AC C.f., for period-doubling, bifurcation is at Sine-circle map, K < 1: intermittency is similar but between freq-lock & quasi-periodicity

  7. 1/f noise Power spectra  1/fδ 0.8 < δ < 1.4 Power spectra of systems with intermittency also exhibit 1/fδ dependence. Too sensitive to external noise. See Schuster

  8. Quantitative Theory of Intermittency Tangent bifurcation near stable n-period fixed point x* ( periodic for A > AC, intermittent / chaotic for A < AC ): Set: →

  9.  < 0 : periodic  = 0 : tangent bif  > 0 : intermittent

  10. Average Duration of Bursts: Renormalization Arguments L = average length of bursts of periodicity  < 0 : periodic  = 0 : tangent bif  > 0 : intermittent L → 0 for  >> 1 L → ∞ as  → 0+ L  n() = number of iterations required to pass thru gap Analogous number for h(2) is Scaling: →  h(2) → h  → δ = 4  

  11. h(2) → h  → 4    h(2m) → h  → 4m    Ansatz:     Experimental confirmation: diode circuit Renormalization theory version: there exists g such that with

  12. Ansatz → See Schuster, p.45 Extension to other univ classes: B.Hu, J.Rudnick, PRL 48,1645 (82)

  13. Types of Intermittency ε< 0 → ε > 0 M xn On-off intermittency = Type III with new freq ~ 0

  14. Crises Unstable fixed point / limit cycle collides with chaotic attractor → sudden changes in latter • Boundary crisis: chaotic attractor disappears • Interior crisis: chaotic attractor expands Sudden changes in fractal structure of basin boundary of chaotic attractor: metamorphosis

  15. Boundary Crisis • Logistic map: • A*3 < A < 4: chaotic attractor expands as A increases. • A = 4: chaotic attractor fills [0,1] and collides with unstable fixed point at x = 0. • A > 4: chaotic attractor disappears; new attractive fixed point at x = -∞. • A  4: escape region = [ x-, x+ ], i.e., • f(x) > 1 if x  [ x-, x+ ] → Average duration of chaotic transient  Universal for quadratic maps. for A  4

  16. 2-D Henon map: crisis near C = 1.08

  17. Interior Crisis • Logistic map: • Unstable period-3 fixed points created by tangent bifurcation at A = 1+√8 collide with chaotic attractor at A*3. • Chaotic attractor suddenly expands at A*3 ( trajectories scattered by the unstable fixed point into previously un-visited regions).

  18. Universality I • Logistic map: • Average time spent in pre-expansion-chaotic region is proportional to (A-A*3)-½. • → Loss regions = penetration of unstable x* into chaotic bands  (A-A*3)½. • (launches into previously forbidden region). • → Re-injection region Xr ( back into chaotic bands) • → Crisis-induced intermittency. xj = f(j)(1/2)

  19. Universality II Logistic map: fraction of time spent in pre-expansion-forbidden region is For tN << tO, we have tN time spent in previously forbidden region before landing in Xr. For x  ½, f(3×2)(x)  x6 near x*. x6 - x*  a Ex 7.6-3

  20. Let d be the distance from x* to Xr. Let M be the Floquet multiplier for F = f(3) at x* Let → Suppose when a = an, F(n)(x) reaches Xr but not F(n-1)(x) → tN n(an). As a increases beyond an, F(n)(x) may overshoots Xr while F(n-1)(x) hasn’t arrived → tN becomes longer Further increase of a brings F(n-1)(x) to Xr → tN n(an-1)-1.  tN is a periodic function of lna with period lnM. P = some function with period ln M.

  21. Noise–Induced Crisis Noise can bump a system in & out of crisis. Average time τbetween excursions into pre-crisis gaps is described by a scaling law: where σ strength of noise Ref: J.Sommerer, et al, PRL 66, 1947 (91) Double crises H.B.Steward, et al, PRL 75, 2478 (95)

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