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We investigate the simultaneous existence of two centers and their isochronicity for two families of planar Z 2 -equivariant differential systems. First, for a family studied by Liu and Li (2011) we present the necessary and sufficient conditions for the existence of an isochronous bi-center. Next, we give conditions for the existence of a bi-center and study its isochronicity for a planar Z 2 -equivariant quintic system having two weak foci or centers. We also give two examples of non-Hamiltonian cubic systems with three isochronous centers. [ABSTRACT FROM AUTHOR]

Abstract: For real planar autonomous analytic differential equations we introduce the notion of persistent center and show a list of equations with this property. We face the problem of whether our list is exhaustive or not and we prove that it is for several families of planar systems, like cubic or rigid systems. [Copyright &y& Elsevier]

This work concerns the generalized Hopf bifurcation analysis of a piecewise-smooth wheel system with higher order discontinuities. Center manifold reduction is used to reduce the towed caster wheel system to a planar dynamical system, in which the non-smoothness factor is considered. The analysis can be divided into two cases by the coefficient of quadratic term. If the coefficient of quadratic term is equal to zero, the type of generalized Hopf bifurcation can be determined by the stability of first-order fine focus. Otherwise, we have to construct a Poincaré map for illustrating the type of generalized Hopf bifurcation, which is the heart of our discussion. Besides, we put forward another convenient way to judge the bifurcation types due to the symmetrical characteristic of this wheel system. Numerical simulations show the feasibility of theoretical analysis. • This work concerns the generalized Hopf bifurcation analysis of a piecewise-smooth wheel system with higher order discontinuities. • Center manifold reduction is used to reduce the towed caster wheel system to a planar dynamical system. • The type of generalized Hopf bifurcation could be judged by the stability of first-order fine focus or Poincare map. [ABSTRACT FROM AUTHOR]

Abstract: By using an effective complex algorithm to calculate the Lyapunov constants of polynomial systems : , where is a homogeneous polynomial of degree n, in this note we construct two concrete examples, and , such that in both cases, the corresponding orders of fine focus can be as high as 18. The systems are given, respectively, by the following ordinary differential equations: where , and where c is the root between of the equation [Copyright &y& Elsevier]