Bifurcation theory and applications

在数学方程系统的研究中，分岔理论解释了方程参数的微小变化如何导致方程解的不成比例的变化。分岔理论告诉我们，这种“因果”关系不是任意的，而很可能只以几种典型的（或一般的）方式发生，称为分岔，数学家们开始相当好地理解这一点。在本工作中，所研究的方程是微分方程的非线性系统；也就是说，决定系统如何随时间演化的方程。这类方程在科学中被广泛使用，并有助于理解许多种类的实际系统，例如在物理学、工程学和生物学中。

In the study of systems of mathematical equations, Bifurcation Theory explains how small changes in the parameters of the equations can lead to disproportionate changes in the solutions of the equations. Bifurcation Theory tells us that this "cause and effect" relationship is not arbitrary, but is most likely to occur only in a few typical (or generic) ways, called bifurcations, which mathematicians are beginning to understand fairly well. In the present work, the equations under study are nonlinear systems of differential equations; that is, equations that determine how a system evolves with time. Equations of this kind are widely used in science, and help in the understanding of many kinds of real systems, for example in physics, engineering and biology.