By Freddy Dumortier, Robert Roussarie, Jorge Sotomayor, Henryk Zoladek

ISBN-10: 3540384332

ISBN-13: 9783540384335

ISBN-10: 3540545212

ISBN-13: 9783540545217

The ebook stories on contemporary paintings by way of the authors at the bifurcation constitution of singular issues of planar vector fields whose linear components are nilpotent. The bifurcation diagrams of an important codimension-three instances are studied intimately. the consequences awarded achieve the bounds of what's presently identified at the bifurcation thought of planar vector fields. whereas the remedy is geometric, particular analytical instruments utilizing abelian integrals are wanted, and are explicitly built. The rescaling and normalization tools are superior for program right here. The reader is believed to be accustomed to the weather of Bifurcation and Dynamical platforms concept. The ebook is addressed to researchers and graduate scholars operating in usual Differential Equations and Dynamical structures, in addition to a person modelling advanced multiparametric phenomena.

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**Additional info for Bifurcations of Planar Vector Fields: Nilpotent Singularities and Abelian Integrals**

**Example text**

A. The critical Location points and nature of critical points of XA are given by y = 0 and ~x 3 + #2 x + #I = 0 = (I) 58 Let SN be the cuspidal surface defined by the zeroes of the discriminant of the first equation {(#i,#2,~) 2 3 I 27#1 + 4E#2 = 0}. The intersection 2 2 the sphere S = {~21 + #2 + ~ = ~} gives the lips shaped curve in of SN w i t h figures : SN = 2, 3, and 4). We verify now that the critical points are non degenerate outside SN. Let m ° we calculate j2XA(mo) = Y = (Xo,0) be any critical point the 2-jet of Xi at m ~a o Taking x .

DCST ' centre are tangent going DCST b (DCST) : c separatrix to the boundary. C and some We have They are illustrated of the 3 distinct in figure 28 53 C DCST. DCSTb DCSTc Figure 28 In the of non-isolated there DCST (8 c exists , the in the define near ~(1), case DCST central just one separatrix picture). C the a and an orbit versal parameter separatrix ones. In the case tangency point (~ is also In each a shift f u n c t i o n u(i) C isolated tangent DCST b in the in the bundle case, taking a it the boundary.

Let m ° we calculate j2XA(mo) = Y = (Xo,0) be any critical point the 2-jet of Xi at m ~a o Taking x . = x° +X, y = Y , : + (-Det(Xo,A)X + Tr(Xo,A)Y + 3c x ° x 2 + (b(k)+2Xo+3X~h(Xo,X) + x 3 ah o ~x (Xo ' ~)) XY + Q(x ° ' O,X)Y 2) aaY (2) Where I -Det(Xo'%) = 3~x2 + >2 o (3) Tr(Xo'~) = ~' + b(3")Xo + X2o + x~h(Xo')~) In particular, we see that : (0 -Det(Xo,~) The determinant, and saddle or focus/node Tr(xo,X) Det(Xo,~), nature of the ) (:) 1-jet (4) is non zero w h e n of the singular point % ~ SN is given by the sign of Det(Xo,A), equation points point in in : so the problem reduces to the study of the roots of the cubic 3 ~x + ~2 x + #I = O.

### Bifurcations of Planar Vector Fields: Nilpotent Singularities and Abelian Integrals by Freddy Dumortier, Robert Roussarie, Jorge Sotomayor, Henryk Zoladek

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