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

The ebook reviews on contemporary paintings by means 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 situations are studied intimately. the consequences offered succeed in the bounds of what's at the moment identified at the bifurcation conception of planar vector fields. whereas the therapy is geometric, distinctive analytical instruments utilizing abelian integrals are wanted, and are explicitly built. The rescaling and normalization tools are enhanced for program the following. The reader is thought to be accustomed to the weather of Bifurcation and Dynamical platforms conception. The ebook is addressed to researchers and graduate scholars operating in usual Differential Equations and Dynamical platforms, in addition to somebody modelling advanced multiparametric phenomena.

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Andronov-Hopf I bifurcations bifurcation : (H) ~) m2 : j• 1 X l (0) = + (y ~ax x (o E is a non-degenerate o Oy linear part of X1 ). ] and also P-H-case : Suppose that integral Suppose also that : : bifurcation Lyapounov = 0). (h,l) point. Let us write = ~l(1)h + ~2(I)h2 is given by #i(I) = 0 and ~2(Io) (b02 + b20) ~ 0 (see the h-expansion + O(h3). (SN) and Saddle-node We may that jlxI a (0) = + y ~y set and that Ox is a center manifold of X 1 , invariant for each I. Then we may suppose that o a a xl = + y(l + 0(I m I )) ~y + (f(x,A) + O(y)) ~x with : set is given by The bifurcation loop (SNLI~.

Recovered of its in an at the end of Chapter a generic parts easier advantages of the b i f u r c a t i o n we consider Some way is to make responds or the also to the saddle elliptic case that b = b(0) case and (b > 2 ~ ) ; the in the clear set. family w r i t t e n in the normal III: ~ for ~I = + 1 and take Recall are of 8 a X~=y ~xx+ (ex3+~2x +#l+y(v+b(),)x+x 2 +x3h(x,~,))+y 2 Q(x,y,),))) ~y Here we write we consideration. saddle results critical bifurcations, bifurcations). families and to any rescaling.

1 and the that o a(l) > : x ~ b(1) is s O, rl(~) this other where given and : s = b(~) on {i=O} for x ~ 0, and gives are lines connections). (-i) r2(l) In s field and the X1 i are left by on for is linear defined. : x ~ the right, some b(l) to limit cycles is s + along > 0). : (s) (-i + x)i/r2 (I) for -i+x ~ 0). an equation This equation (i,s). lines of bifurcation. is given (-i+x)i/r2(1)(again is : 1 LQ and = b(1)(-i+x)I/r2 (~) equation, at Sl). [s=O} the vector coordinates to in the space inferior for the x ~ ~.