lecnotes17 - 17 Intermittency(and quasiperiodicity In this...

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17 Intermittency (and quasiperiodicity) In this lecture we discuss the other two generic routes to chaos, intermittency and quasiperiodicity. Almost all our remarks will be on intermittency; we close with a brief de- scription of quasiperiodicity. Definition: Intermittency is the occurrence of a signal that alternates ran- domly between regular (laminar) phases and relatively short irregular bursts. In the exercises we have already seen examples, particulary in the Lorenz model (where it was discovered, by Manneville and Pomeau). Examples: The Lorenz model, near r = 166. Figure 1a,b Manneville and Pomeau (1980) Rayleigh-Benard convection. BPV, Figure IX.9 17.1 General characteristics of intermittency Let r = control parameter. The following summarizes the behavior with respect to r : For r < r i , system displays stable oscillations (e.g., a limit cycle). For r > r i ( r r i small), system in in the intermittent regime: stable oscillations are interrupted by fluctuations. As r r i from above, the fluctuations become increasingly rare, and disappear for r < r i . Only the average intermission time between fluctuations varies, not their amplitude nor their duration. 173
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We seek theories for Linear stability of the limit cycle and “relaminarization.” (i.e. return to stability after irregular bursts). Scaling law for intermission times. Scaling law for Lyaponov exponents. 17.2 One-dimensional map We consider the instability of a Poincar´ e map due to the crossing of the unit circle at (+1) by an eigenvalue of the Floquet matrix. This corresponds to the specific case of Type I intermittency . Let u be the coordinate in the plane of the Poincar´ e section that points in the direction of the eigenvector whose eigenvalue crosses +1. The lowest-order approximation of the 1-D map constructed along this line is u = ( r ) u. (39) Taking ( r i ) = 1 at the intermittency threshold, we have u = ( r i ) u = u. (40) We consider this to be the leading term of a Taylor series expansion of u ( u, r ) in the neighborhood of u = 0 and r = r i .
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