Codimension

INTERACTION OF SHALLOW CELLS CELLULAR DYNAMICS Evolution of Shallow Cells The Role of Codimension Two Bifurcations. The importance of nonlinear interactions between spatially resonant structures is... 

The stability of the (lAe)-family is lost at a Hopf bifurcation point denoted by the open circle (o) on Fig. 7, where the real parts of a complex conjugate pair of eigenvalues change sign. No stable time-periodic solutions were found near this point, indicating that the time-periodic states evolve sub-critically in P and are unstable. Haug (1986) predicted Hopf bifurcations for codimension two bifurcations of the form shown in Fig. 7. but did not compute the stability of the time-periodic states. 

Expanding the sample size to 2Xc admits the other shape families shown on Fig. 6 into the analysis and leads to additional codimension-two interactions between the shapes is the (1A<.)- family and shapes with other numbers of cells in the sample. The bifurcation diagram computed for this sample size with System I and k = 0.865 is shown as Fig. 11. The (lAc)- and (Ac/2)-families are exactly as computed in the smaller sample size, but the stability of the cell shapes is altered by perturbations that are admissible is the larger sample. The secondary bifurcation between the (lAc)- and (2Ae/3)-families is also a result of a codimension two interaction of these families at a slightly different wavelength. Two other secondary bifurcation points are located along the (lAc)-family and may be intersections with the (4Ac and (4A<./7) families, as is expected because of the nearly multiple eigenvalues for these families. 

Y is smooth of codimension r. In particular we see in this case for the top Chern classes... 

Proposition 3.2.17. Let IcP n be a smooth closed subvariety of dimension m. The locus where X has nth order contact with l-codimensional linear subvarieties of Pn has at most codimension... 

Obviously (l)-(4) only hold in the case that the locus where the contact occurs has the right codimension in X. 

Proposition 3.2.18. Let X be a smooth projective variety of dimension d in Pjv-If the locus where X has second order contact with m-planes has codimension at least... 

Proposition 3.2.19, Let X C P n be a smooth variety of dimension d. If the codimension of the locus, where X has third order contact with lines, has codimension 2N — 3d + 1, then its class is... 

Once the parametric representation of the Jacobian is obtained, the possible dynamics of the system can be evaluated. As detailed in Sections VILA and VII.B, the Jacobian matrix and its associated eigenvalues define the response of the system to (small) perturbations, possible transitions to instability, as well as the existence of (at least transient) oscillatory dynamics. Moreover, by taking bifurcations of higher codimension into account, the existence of complex dynamics can be predicted. See Refs. [293, 299] for a more detailed discussion. 

Proposition 1 1.1. Soit f X — S un morphisme fidelement plat de schemas localement noethdriens qui est quasi-compact ou localement de type fini. On suppose que S est normal et que, pour tout point s de S de codimension 1, la fibre X est lntegre. Alors, si D est un diviseur sur X, dont le support ne rencontre pas les fibres maximales de X au-dessus de S, il existe un unique diviseur sur S tel que D = f ( ). 

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