Spatio-temporal intermittency

We recall from our earlier discussion of chaos in one-dimensional continuous systems (see section 4.1) that period-doubling is not the only mechanism by which chaos can be generated. Another frequently occurring route to chaos is intermittency. But while intermittency in low dimensional dynamical systems appears to be constrained to purely temporal behavior [pomeau80], CMLs exhibit a spatio-temporal intermittency in which laminar eddies are intermixed with turbulent regions in a complex pattern in space-time. 

In their study of CMLs exhibiting a Pomeau-Manneville intermittency, Crutchfield and Kaneko [crutch87] have observed the following general behavior  

Localized Kink Regime when the diffusive coupling is too small for kinks to move, the initial kinks separating domains remain locked in position. The behavior is analogous to that of class c2 elementary CA. 

Transition Regime the kinks become unstable and are able to propagate but remain localized global intermittent pattern cannot yet be formed. The patterns arc reniiniscent of class c4 behavior. 

Fully Developed Turbulence burst regions spread throughout the entire lattice so that it becomes very difficult to discern any laminar regions. 

---

We must also emphasize that it is presently unclear if spatio-temporal intermittent patterns analog to those shown in Figure 24 could be obtained from Equation 20 in a 2D system or if an additional instability or space dimensionality has to be invoked. So far, such pattern dynamics have not been obtained in 2D model simulations [74]. 

Fig. 9. Spatio-temporal pattern forming phenomena in the reaction-diffusion system (3) with symmetric Dirichlet boundary conditions and the slow manifold (8). The model parameters are e = 0.01, a = 0.01, uo = ui = —2. (a) D = 0.0322560 (7,7 oscillating pattern confined to the lower branch (b) D = 0.0322550 Cr crisis-induced intermittent bursting (c) D = 0.0322307 homoclinic intermittent bursting (d) D = 0.0322400 PJ periodic bursting,...

---