The Fuller index and global Hopf bifurcation
The Fuller index and global Hopf bifurcation
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DOI：
10.1016/0022-0396(78)90041-4
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发表时间：
1978-07
影响因子：
2.4
通讯作者：
S. Chow;J. Mallet-Paret
中科院分区：
文献类型：
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作者：
S. Chow;J. Mallet-Paret
利用Fuller定义的自治方程周期解的指数，证明了Alexander和Yorke的全局Hopf分支定理。由于时滞泛函微分方程解的Fuller指数可以定义，因此在这种情况下也可以证明整体分支定理。这些结果表明具有多个有理相关时滞的时滞方程的周期解的存在性，例如ẋ(T)=−α[ax(t−1)+bx(t−2)]g(x(T))，其中a和b非负且α大于由线性化方程计算出的某个可计算量ξ(a，b).
Using an index for periodic solutions of an autonomous equation defined by Fuller, we prove Alexander and Yorke's global Hopf bifurcation theorem. As the Fuller index can be defined for retarded functional differential equations, the global bifurcation theorem can also be proved in this case. These results imply the existence of periodic solutions for delay equations with several rationally related delays, for example, x ̇ (t)=− α [ax (t− 1)+ bx (t− 2)] g (x (t)), with a and b non-negative and α greater than some computable quantity ξ (a, b) calculated from the linearized equation.