Jiazhong Yang

Linearization of germs of hyperbolic vector fields

We develop a normal form to express asymptotically a conjugacy between a germ of resonant vector field and its linear part. We show that such an asymptotic expression can be written in terms of functions of the Logarithmic Mourtada type. To cite this article: P Bonckaert et al., C. R. Acad. Sci. Paris, Ser. I336 (2003)

Hopf-zero bifurcations of reversible vector fields

We study the dynamics of a class of reversible vector fields having eigenvalues (0 ,α i, −αi) around their symmetric equilibria. We give a complete list of all normal forms for such vector fields, their versal unfoldings, and the corresponding bifurcation diagrams of the codimensional-one case. We also obtain some important conclusions on the existence of homoclinic and heteroclinic orbits, invariant tori and symmetric periodic orbits. AMS classification scheme numbers: 34K18, 37C29, 34K17, 37G10

On the hyperelliptic limit cycles of Liénard systems

In this paper we study hyperelliptic limit cycles of the Lienard systems where, respectively, fm(x) and gn(x) are polynomials of degree m and n, gn(0) = 0. We prove that, if m ≥ 5 and m + 1 < n < 2m, then there always exist Lienard systems of the above form such that they have a hyperelliptic limit cycle. This gives a positive answer to the open problem posed in the paper by Yu and Zhang (2011 J. Math. Anal. Appl. 376 535–9). By combining all the results obtained up to now, we in fact give a complete classification of the hyperelliptic limit cycles of the Lienard systems: Lienard systems of the above form have hyperelliptic limit cycles only in the following cases: (i) m = 2, 3 and m + 3 ≤ n; (ii) 4 ≤ m and m + 2 ≤ n.

Linearization of families of germs of hyperbolic vector fields

In this article, we develop some techniques to linearize families of smooth vector fields in a neighbourhood of a hyperbolic equilibrium point. In particular, we present the linearizing conjugacy in an explicit way and describe the smoothness of the conjugacy in terms of the eigenvalues of the vector fields.

Unique normal forms for Hopf-zero vector fields

Abstract We consider normal forms of Hopf-zero vector fields in R 3 . Unique normal forms under conjugacy and orbital equivalence for the generic case are given. To cite this article: G.xa0Chen etxa0al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Linearization of hyperbolic resonant germs

In this paper we develop an explicit normal form conjugacy procedure, called an LMT normal form, to study linearization of a smooth vector field in the neighbourhood of a hyperbolic equilibrium point with resonant eigenvalues. We give an asymptotic expression for such a linearization in terms of functions of Logarithmic Mourtada type.

On the C1 normal forms for hyperbolic vector fields

Abstract Given two germs of hyperbolic vector fields associated to autonomous ordinary differential equations x =Ax+⋯ and y =By+⋯ , where x,y∈ R n , and A and B are n×n matrices, we prove that under some algebraic conditions on the eigenvalues of the matrices and genericity condition on the nonlinear terms, they are at least C1 conjugate if and only if A and B are similar. To cite this article: Z. Ren, J. Yang, C. R. Acad. Sci. Paris, Ser. I 336 (2003).