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We consider the Lotka-Volterra Equationṡ x = x(1 + ax + by),ẏ = y(−λ + cx + dy), with λ a non negative number. Our aim is to understand the mechanisms which lead to the origin being linearizable, integrable or normalizable. In the case of integrability and linearizability, there is a natural dichotomy. When the system has an invariant line other than the axes, then the system is integrable and we give necessary and sufficient conditions for linearizability in this case. When there is no such line, then the conditions for linearizability and integrability are the same. In this case we show that the monodromy groups of the separatrices play a key role. In particular for λ = p/q with p + q ≤ 12 and λ = n/2, 2/n with n ∈ N the origin is linearizable if and only if the monodromy groups can be shown to be linearizable by elementary arguments. We give 4 classes of these conditions, and their duals, in terms of the parameters of the system, and conjecture that these, together with two exceptional cases of Darboux linearizability, are the only integrability mechanisms for rational values of λ.The work on normalizability is more tentative. We give some sufficient conditions for this via monodromy groups, and give a complete classification when λ = 0. We also investigate in detail the case λ = 1, with a + c = 0. Much of our ideas here are based on recent work on the unfolding of the Ecalle-Voronin modulus of analytic classification [12]. In particular we give examples of "halfnormalizable" systems as well as an experimental example of a "transcritical bifurcation" of the functional moduli associated to the critical point.

In this paper we consider germs of k-parameter generic families of analytic 2-dimensional vector fields unfolding a saddle-node of codimension k and we give a complete modulus of analytic classification under orbital equivalence and a complete modulus of analytic classification under conjugacy. The modulus is an unfolding of the corresponding modulus for the germ of a vector field with a saddle-node. The point of view is to compare the family with a "model family" via an equivalence (conjugacy) over canonical sectors. This is done by studying the asymptotic homology of the leaves and its consequences for solutions of the cohomological equation. This paper is dedicated to the memory of Adrien Douady.

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We derive the global bifurcation diagram of a three-parameter family of cubic Li enard systems. This family seems to have a universal character in that its bifurcation diagram (or parts of it) appears in many models from applications. In fact, it is the 3-jet of a three-dimensional slice of the universal unfolding of a doubly degenerate Bogdanov-Takens point, that is, of the codimension-four singularity with nilpotent linear part and no quadratic terms in the normal form.

In this paper we present an extensive qualitative study of the phase portraits of Lidnard equations f = A ( x ) + B ( x ) i , with A cubic and B linear. We encounter limit cycles surrounding one singularity and limit cycles surrounding two or three singularities. We prove that the equations can have at most one limit cycle of the first kind which can never be surrounded by one of the second kind. We adduce strong evidence that there is at most one limit cycle in each case. After an affine coordinate change in phase space these equations reduce to the vector fields y a/& + [*x3 + p2x + p1 + y ( v + bx)] a / d y , with b > 0, whereby we recover the families of vector fields obtained previously after (principal) rescaling. The results obtained here can be used to complete some proofs missing in earlier work. In an appendix we present a complete study for the case A quadratic and B linear, and show that at most one limit cycle is present.