Bifurcation of limit cycles and isochronous centers on center manifolds for four-dimensional systems

This paper is concerned with the bifurcation of limit cycles and isochronous centers on center manifolds for four-dimensional nonlinear dynamic systems, which have a pair of purely imaginary eigenvalues and a pair of eigenvalues with negative real part for the Jacobian matrix corresponding to the singularity situated at the origin. In order to investigate the isochronous centers for four-dimensional systems, a new algorithm to calculate so-called isochronous constants is given. Moreover, with the calculation of the isochronous constants, the conditions for the existence of an isochronous center are determined in a way which does not need first computing the center manifolds of the four-dimensional system. Finally, as an application, we investigate a class of quadratic polynomial systems for a model of airfoil and prove that the system can have five small limit cycles around the origin on the center manifold. Moreover, we apply our new methodology to find the sufficient and necessary conditions for the origin to be an isochronous center restricted to the center manifold, illustrating the reliability and availability of the new algorithm.