Bifurcation analysis on a class of three-dimensional quadratic systems with twelve limit cycles

Abstract

This paper concerns bifurcation of limit cycles in a class of 3-dimensional quadratic systems with a special type of symmetry. Normal form theory is applied to prove that at least 12 limit cycles exist with 6–6 distribution in the vicinity of two singular points, yielding a new lower bound on the number of limit cycles in 3-dimensional quadratic systems. A set of center conditions and isochronous center conditions are obtained for such systems. Moreover, some simulations are performed to support the theoretical results.