Bi-center problem and Hopf cyclicity of a Cubic Liénard system

Abstract

In this paper we investigate the bi-center problem and the total Hopf cyclicity of two center-foci for the general cubic Liénard system which has three distinct equilibria and is equivalent to the general Liénard equation with cubic damping and restoring force. The location of these three equilibria is arbitrary, specially without any kind of symmetry. We find the necessary and sufficient condition for the existence of bi-centers and prove that there is no case of a unique center. On the Hopf cyclicity we prove that there are totally 9 possible styles of small amplitude limit cycles surrounding these two center-foci and 6 styles of them can occur, from which the total Hopf cyclicity is no more than 4 and no less than 2.

Isochronicity of bi-centers for symmetric quartic differential systems

“…Note that in Figure 1(D) and Figure 1(E) there exists a third center at the origin which is also isochronous (see [32] for a proof). Following we prove that in system (21) there are two isochronous centers in addiction of the ones in (1, 0) and (−1, 0), they are located at (1, 2) and (−1, −2), see Figure 1 Figure 1. Global phase portraits of systems ( 14), ( 16), ( 18), ( 19), ( 20), ( 21) and ( 23), respectively.…”

“…Later on, in [10] and [5] the authors provided conditions for the existence of two centers for cubic systems. Recently, a study on the bi-center problem for cubic Liénard systems was done in [21]. In [16], using the classification of the quadratic systems in four types: Hamiltonian, reversible Lotka-Volterra, Q 4 , the authors presented a complete description of the quadratic systems possessing two centers.…”