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Bifurcation of limit cycles in a quintic system with ten parameters
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In this paper, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of quintic polynomial differential system are investigated. With the help of the computer algebra system MATHEMATICA, the first 11 quasi- Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there exist 11 small amplitude limit cycles created from the three-order nilpotent critical point is also proved. Henceforth, we give a lower bound of cyclicity of three-order nilpotent critical point for quintic Lyapunov systems. The results of Jiang et al. (Int. J. Bifurcation Chaos 19:2107–2113, 2009) are improved.
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Three-order nilpotent critical point, Center-focus problem, Bifurcation of limit cycles, Quasi-Lyapunov constant
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