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

Abstract

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.

Bifurcation of Limit Cycles and Center in 3D Cubic Systems with Z3-Equivariant Symmetry

“…Surprisingly, unlike the planar systems, where the number of limit cycles is finite [23], a simple three-dimensional system may have an infinite number of small amplitude limit cycles [24]. At present Huang, Gu and Wang [25] introduced some results on the bifurcation of limit cycles for three-dimensional smooth systems, such as Lotka-Volterra systems, Chen systems, Lorenz systems, Lü systems, and three-dimensional piecewise smooth systems. Sanchez and Torregrosa [26] studied several classes of three-dimensional polynomial differential systems by using the parallel computing method, and determined that there are at most 11, 31, 54, and 92 limit cycles at the origin for three-dimensional quadratic, cubic, quartic, and quintic systems, respectively.…”