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By Mitchell J. Feigenbaum (auth.), Roberto Artuso, Predrag Cvitanović, Giulio Casati (eds.)

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Thus the winding ratio will change on breaking the saddle-node, varying continuously in a devil's staircase, being incomplete or complete, respectively. If it has both flats and jumps, then the situation is much more complicated and remains to be fully analysed. There are intervals of parameter arbitrarily close to the boundary where the flow is fully mode-locked and intervals arbitrarily close where it is partially mode-locked.

Under reasonable conditions we can prove existence of at least two of these points on each curve of saddle-nodes [KMG). 2. Relation between mode locking and resonance How do the regions of full and partial mode-locking relate to the resonance regions? If F is (p,q) fully mode-locked, then it can be proved that it has a type (p,q) periodic orbit [BGKM]. So the (p,q)-fully mode-locked region is contained in the (p,q) resonance regjon. 7 near the (3,4)/5 resonance, also indicating a region of full mode-locking inside it.

Examples are shown in Figure 18. Definitions: We say that w E SI is rational if it is of the fonn (l,m)/I(I,m)l, for some (I,m) Z2\O. If cP E Dw u Ew' then w is rational [PM). We say that cp is partially mode-locked if cp E P = U (Bw u fully mode-locked if cp E F. e. Cp~t(x) = x, 'it. This is equivalent to saying that 0 E W11\1. 52 Figure 19. 5. Cherry flows Type B flows always have a cross-section. Hence their study reduces to that of diffeomorphisms of the circle, with a natural correspondence between winding ratio and rotation number.

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