HOPF BIFURCATION IN HIGHER DIMENSIONAL DIFFERENTIAL SYSTEMS VIA THE AVERAGING METHOD
HOPF BIFURCATION IN HIGHER DIMENSIONAL DIFFERENTIAL SYSTEMS VIA THE AVERAGING METHOD
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DOI：
10.2140/pjm.2009.240.321
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发表时间：
2009-03
影响因子：
0.6
通讯作者：
J. Llibre;Xiang Zhang
中科院分区：
文献类型：
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作者：
J. Llibre;Xiang Zhang
我们研究 Rn 中 C3 微分系统的 Hopf 分岔，表明 l 极限环可以从特征值 ±bi 的一个奇点和 n - 2 个零（其中 l 在 {0,1,�,2n-3} 中）分岔。据我们所知，这是第一次证明Hopf分岔中可以分岔的极限环数量随着空间维数呈指数增长。为了证明这个结果，我们使用一阶平均理论。此外，在维度 4 中，我们描述了分叉极限环的形状和稳定性类型。我们将结果应用于某些四阶微分方程，然后应用于描述免疫反应的简化 Marchuk 模型。
We study the Hopf bifurcation of C3 differential systems in Rn showing that l limit cycles can bifurcate from one singularity with eigenvalues ±bi and n - 2 zeros with l in {0,1,�,2n-3}. As far as we know this is the first time that it is proved that the number of limit cycles that can bifurcate in a Hopf bifurcation increases exponentially with the dimension of the space. To prove this result, we use first-order averaging theory. Further, in dimension 4 we characterize the shape and the kind of stability of the bifurcated limit cycles. We apply our results to certain fourth-order differential equations and then to a simplified Marchuk model that describes immune response.