Hopf bifurcation cyclicity and chaos for the Z_2-symmetric

摘要: In this paper, the Hopf cyclicity and hidden chaos for the three-dimensional (3D)  $Z_2$-symmetric R\"{o}ssler system are investigated. Applying the recursive formula of the singular point quantities, and by strict symbolic calculation, we determine the  highest order three of weak focus at the symmetric equilibria on center manifold. And under suitable perturbation, six and at most six small amplitude limit cycles can generate from symmetric equilibria via Hopf bifurcation. Furthermore, we study different cases that multiple Hopf bifurcation and chaos can simultaneously occur around the two symmetric equilibria, where one (2, 2) distribution of four limit cycles is accompanied by chaos. To our knowledge, this property is very rare in many chaotic systems.