Limit cycles for quadratic and cubic planar differential equations under polynomial perturbations of small degree (Q515261)

The authors consider the two-dimensional systems \[ \dot{x}= -y + \delta (x^{2} -y^{2}) +\varepsilon p_{n}(x,y),\,\dot{y}=x+2\delta xy + \varepsilon q_{n} (x,y), \] \[ \dot{x} = -y +\delta x^{2} +\varepsilon p_{n}(x,y),\,\dot{y}= x+ \delta xy + \varepsilon q_{n}(x,y), \] where \(p_{n}\) and \(q_{n}\) are polynomial functions in \(x\) and \(y\) of degree \(n\), \(\delta\) is a nonzero real number and \(\varepsilon\) is a positive parameter. It is well-known that the systems have isochronous centers at the origin when \(\varepsilon =0\). It is shown that the maximum number of limit cycles bifurcating from the origin is at most \(n-2\) when \(n \leq 7\) for the first system, and it is \(n\) when \(n \leq 6\) for the second system. These upper bounds are sharp. The method of first order averaging is used to prove the results. It is conjectured that the results remain true for all values of \(n\). Other systems are also considered with numerical methods.