Zero singularities of codimension two and three in delay differential equations
Zero singularities of codimension two and three in delay differential equations
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DOI：
10.1088/0951-7715/21/11/010
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发表时间：
2008-11
期刊：
影响因子：
1.7
通讯作者：
S. A. Campbell;Yuan Yuan-Yuan
中科院分区：
文献类型：
--
作者：
S. A. Campbell;Yuan Yuan-Yuan
给出了一类时滞微分方程存在Bogdanov-Takens点或三零点分支的条件。我们展示了如何中心流形投影的延迟方程减少了动态的二维或三维系统的常微分方程。我们把这些方程的规范形式，并确定如何规范形式的系数依赖于模型中的原始参数。最后，我们将我们的结果应用到两个神经模型和比较的理论预测与数值分岔分析的完整方程。一个模型涉及跨临界分岔，因此，我们推导和分析这种情况下适当的展开。
We give conditions under which a general class of delay differential equations has a point of Bogdanov–Takens or a triple zero bifurcation. We show how a centre manifold projection of the delay equations reduces the dynamics to two- or three-dimensional systems of ordinary differential equations. We put these equations in normal form and determine how the coefficients of the normal forms depend on the original parameters in the model. Finally we apply our results to two neural models and compare the predictions of the theory with numerical bifurcation analysis of the full equations. One model involves a transcritical bifurcation, hence we derive and analyse the appropriate unfoldings for this case.