Global Phase Portraits of Kukles Differential Systems with Homogeneous Polynomial Nonlinearities of Degree 6 Having a Center and Their Small Limit Cycles
Global Phase Portraits of Kukles Differential Systems with Homogeneous Polynomial Nonlinearities of Degree 6 Having a Center and Their Small Limit Cycles
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DOI：
10.1142/s0218127416500449
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发表时间：
2016-04
期刊：
影响因子：
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通讯作者：
J. Llibre;Maurício Fronza da Silva
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文献类型：
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作者：
J. Llibre;Maurício Fronza da Silva
我们提供了形式为 ẋ = −y, ẏ = x + ax5y + bx3y3 + cxy5 的 Kukles 多项式微分系统中心族的庞加莱盘中的九个拓扑全局相图，其中 x,y ∈ ℝ 和 a,b,c 是满足 a2 + b2 + c2≠0 的实参数。使用六阶平均理论，我们确定当我们首先在所有 6 次齐次多项式微分系统类中扰动该系统，然后在所有 6 次多项式微分系统类中扰动该系统时，从原点分叉的极限环数。
We provide the nine topological global phase portraits in the Poincare disk of the family of the centers of Kukles polynomial differential systems of the form ẋ = −y, ẏ = x + ax5y + bx3y3 + cxy5, where x,y ∈ ℝ and a,b,c are real parameters satisfying a2 + b2 + c2≠0. Using averaging theory up to sixth order we determine the number of limit cycles which bifurcate from the origin when we perturb this system first inside the class of all homogeneous polynomial differential systems of degree 6, and second inside the class of all polynomial differential systems of degree 6.