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Centre families in two-dimensional complex holomorphic dynamical systems. (English) Zbl 0920.34047
This paper is concerned with the two-dimensional complex holomorphic dynamical system $z_t=F(z,w),\quad w_t=G(z,w), \qquad (z,w,t)\in D\times I.$ Here, $$I\subset \mathbb{R}$$ is a connected open interval, $$D\subset \mathbb{C}^2$$ is a simply connected domain, and $$F,G:D\to \mathbb{C}$$ are holomorphic functions satisfying $$F(0,0)=G(0,0)=0.$$ Under the condition $\begin{pmatrix} F_z(0,0) & F_w(0,0) \\ G_z(0,0) & G_w(0,0) \\ \end{pmatrix} =\begin{pmatrix} \mu i & 0 \\ 0 & \nu i \\ \end{pmatrix}, \qquad \mu, \nu \in \mathbb{R}\setminus \{ 0\},$ it is proved that there exist isochronous centre families around the isolated equilibrium point $$(0,0).$$ These families lie on certain holomorphic invariant manifolds, which are constructed by using the singular point theory of Briot and Bouquet. The main result is applied to the complex Takens-Bogdanov system.

##### MSC:
 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems 34C30 Manifolds of solutions of ODE (MSC2000) 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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