Third-order bifurcation of limit cycles for a perturbed quartic isochronous centre

In this article, we study how many limit cycles can bifurcate from the periodic orbits of a quartic uniform isochronous centre when it is perturbed inside a class of quartic polynomial differential systems. Using the first and second order averaging method, we provide the maximum number of limit cycles, 3 and 5 respectively, that can bifurcate from the periodic orbits around the centre. Using the third order averaging method, we show that at least five limit cycles can bifurcate from the periodic orbits around the centre. Our main theorem has improved and generalised some known results in published papers.