A Quartic System with Twenty-Six Limit Cycles

A quartic system with 26 limit cycles

Limit Cycles Bifurcating from a Perturbed Quartic Center

We consider the quartic center ẋ = −yf(x, y), ẏ = xf(x, y), with f(x, y) = (x+a)(y+b)(x+c) and abc 6= 0. Here we study the maximum number σ of limit cycles which can bifurcate from the periodic orbits of this quartic center when we perturb it inside the class of polynomial vector fields of degree n, using the averaging theory of first order. We prove that 4[(n− 1)/2] + 4 ≤ σ ≤ 5[(n− 1)/2 + 14, ...

Bifurcation of Limit Cycles from Quartic Isochronous Systems

This article concerns the bifurcation of limit cycles for a quartic system with an isochronous center. By using the averaging theory, it shows that under any small quartic homogeneous perturbations, at most two limit cycles bifurcate from the period annulus of the considered system, and this upper bound can be reached. In addition, we study a family of perturbed isochronous systems and prove th...

Bifurcation of limit cycles from a quadratic reversible center with the unbounded elliptic separatrix

The paper is concerned with the bifurcation of limit cycles in general quadratic perturbations of a quadratic reversible and non-Hamiltonian system, whose period annulus is bounded by an elliptic separatrix related to a singularity at infinity in the poincar'{e} disk. Attention goes to the number of limit cycles produced by the period annulus under perturbations. By using the appropriate Picard...

Twenty-six compelling conversations.

I continue to believe I have the best job in science. Since May 2012, I’ve been the Editor at Large for the JCI, and in this capacity, I’ve been the chief interviewer of those chosen for the series Conversations with Giants in Medicine. We’ve aired twentysix interviews with twenty-eight notable scientists in the 2.5 years that this series has been part of the JCI (Figure 1). In that time, I’ve ...