Methods of Bifurcation Theory
Methods of Bifurcation Theory
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DOI：
10.1007/978-1-4613-8159-4
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发表时间：
1996-04
期刊：
影响因子：
--
通讯作者：
S. Chow;J. Hale
中科院分区：
文献类型：
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作者：
S. Chow;J. Hale
这本书的另一个标题可能是非线性分析，分歧理论和微分方程。我们的主要目标是讨论分歧理论中对微分方程特别有意义的那些方面。为了实现这一目标，并使这本书获得更广泛的，我们已经详细介绍了许多相关的背景观众，材料从非线性泛函分析和定性理论的微分方程。由于有些材料没有很好的参考资料，因此将其包括在内似乎是必要的。分叉理论的两个不同的方面进行了讨论静态和动态。静态分叉理论关注的是当函数中的参数变化时，函数的零点集的结构发生的变化。如果函数是一个梯度，那么变分技术起着重要的作用，甚至可以有效地用于全局问题。如果函数不是梯度，或者需要更详细的信息，一般理论通常是局部的。同时，当函数中出现多个独立参数时，该理论是构造性的，是有效的。在微分方程中，平衡解是向量场的零点。因此，静态分叉理论中的方法是直接适用的。
An alternative title for this book would perhaps be Nonlinear Analysis, Bifurcation Theory and Differential Equations. Our primary objective is to discuss those aspects of bifurcation theory which are particularly meaningful to differential equations. To accomplish this objective and to make the book accessible to a wider we have presented in detail much of the relevant background audience, material from nonlinear functional analysis and the qualitative theory of differential equations. Since there is no good reference for some of the mate rial, its inclusion seemed necessary. Two distinct aspects of bifurcation theory are discussed-static and dynamic. Static bifurcation theory is concerned with the changes that occur in the structure of the set of zeros of a function as parameters in the function are varied. If the function is a gradient, then variational techniques play an important role and can be employed effectively even for global problems. If the function is not a gradient or if more detailed information is desired, the general theory is usually local. At the same time, the theory is constructive and valid when several independent parameters appear in the function. In differential equations, the equilibrium solutions are the zeros of the vector field. Therefore, methods in static bifurcation theory are directly applicable.