Twelve Limit Cycles in 3D Quadratic Vector Fields with Z3 Symmetry

Abstract

This paper is concerned with the number of limit cycles bifurcating in three-dimensional quadratic vector fields with [Formula: see text] symmetry. The system under consideration has three fine focus points which are symmetric about the [Formula: see text]-axis. Center manifold theory and normal form theory are applied to prove the existence of 12 limit cycles with [Formula: see text]–[Formula: see text]–[Formula: see text] distribution in the neighborhood of three singular points. This is a new lower bound on the number of limit cycles in three-dimensional quadratic systems.

Hopf bifurcation and the centers on center manifold for a class of three‐dimensional Circuit system

“…And the maximal number of limit cycles in the vicinity of a Hopf singular point is a very challenging problem. 5 At present, most of the results obtained are concentrated on small limit cycles bifurcated from a focus on the local center manifold, for the general polynomial systems, see, eg, previous studies, 5,8,9 for Lotka-Volterra systems and chaotic systems, see, eg, Gyllenberg and Yan and Wang et al, 10,11 besides, in Liu et al, 12 six limit cycles are proved in Maxwell-Bloch system. Recently, Garcia et al 13 derive the theorems that bound the maximum number of limit cycles within the center manifold from any center at the origin, and the upper bounds on the cyclicity of the centers on center manifolds for the Lorenz, Chen and Lü families were given in Garcia et al, 14 clearly, the center bifurcation is more difficult to study.…”

“…For cubic polynomial systems, so far the best result is H(3) ≥ 13, see Li et al, 1,2 and for other relevant results, one can refer to the previous studies. [3][4][5] As one of the most important topics about the above problem, the Hopf bifurcation is investigated just in the vicinity of isolated fixed points. Its main tasks will be computing the focus values and determining a focus or center.…”

“…For generic quadratic systems, center problem was solved completely in Dulac,6 the maximum number of small-amplitude limit cycles via Hopf bifurcation was obtained accordingly. For generic cubic systems, the center problem is still a puzzle, Hopf bifurcation of only some special cubic systems has been studied, and some good results were obtained, see for example, the one of 12 small-amplitude limit cycles given in Yu and Tian,7 one can also see Guo et al 5 If the studies of limit cycles bifurcation and center problem are extended from planar systems to three dimensional systems, these will become more difficult comparatively. And the maximal number of limit cycles in the vicinity of a Hopf singular point is a very challenging problem.…”

Multiple limit cycles bifurcation and Jacobi stability for a class of segmented disc dynamo system

“…[22,23,24]. For general threedimensional systems, only some low bounds were obtained, e.g., the examples of 12 smallamplitude limit cycles in quadratic vector fields were given recently [25,26], the readers can also refer to [21,27] for other results.…”

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

“…Lu et al [28] proved the existence of eight limit cycles in a class of Lorenz systems, which are Z 2 symmetric and quadratic three-dimensional systems. Guo, Yu and Chen [29] applied the theory of center manifold and normal forms to prove the existence of 12 limit cycles with 4 − 4 − 4 distribution in three-dimensional vector fields with Z 3 symmetry. Then, they analyzed a 3D quadratic system with two symmetric singular points by using the same approach in [30] and showed that there are twelve limit cycles with 6 − 6 distributed around the singular points.…”