Defining Functions for Multiple Hopf Bifurcations
Defining Functions for Multiple Hopf Bifurcations
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DOI：
10.1137/s0036142995282182
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发表时间：
1997-06
影响因子：
2.9
通讯作者：
W. Govaerts;J. Guckenheimer;A. Khibnik
中科院分区：
文献类型：
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作者：
W. Govaerts;J. Guckenheimer;A. Khibnik
设$A(u,\alpha)=F_u(u,\alpha)$ （$u \in R^n, \alpha \in R^k$）是一组实数$n \times n$矩阵，它们是动力系统$ \stackrel{.}{u}=F(u,\alpha)$平衡解的雅可比矩阵。平衡点称为Hopf点，如果$A$有一对纯虚特征值$\pm i\omega$, $\omega> 0$。如果有两个这样的对，则称为双Hopf点$\pm i\omega_1, \pm i\omega_2$；如果另外有$\omega_1=\omega_2$，则称为1:1共振双Hopf点。得到了Hopf、双Hopf和1:1共振双Hopf点的数值检测、计算和延拓的定义函数。它们是基于矩阵双积和有边矩阵方法的结合。在一个相当现实和复杂的神经模型问题$n=13$和$k=29$中进行了实例计算。然而，为了使方法适用于大规模问题（例如，离散边值问题），我们将状态空间简化为本质上包含实部最大的特征值的广义特征空间的子空间。
Let $A(u,\alpha)=F_u(u,\alpha)$ ($u \in R^n, \alpha \in R^k$) be a family of real $n \times n$ matrices arising as the Jacobian matrices of equilibrium solutions to the dynamical system $ \stackrel{.}{u}=F(u,\alpha)$. An equilibrium point is called a Hopf point if $A$ has a conjugate pair of pure imaginary eigenvalues $\pm i\omega$, $\omega> 0$. It is called a double Hopf point if there are two such pairs $\pm i\omega_1, \pm i\omega_2$ and a 1:1 resonant double Hopf point if, in addition, $\omega_1=\omega_2$. Defining functions are obtained for the numerical detection, computation, and continuation of Hopf, double Hopf, and 1:1 resonant double Hopf points. They are based on a combination of matrix biproduct and bordered matrix methods. Example computations are done in a fairly realistic and complicated neural model problem with $n=13$ and $k=29$. However, to make the methods applicable to large-scale problems (e.g., discretized boundary value problems) we reduce the state space to a subspace that essentially contains the generalized eigenspaces of the eigenvalues with largest real part.