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Research on the conditions of reaction-diffusion equation occurring Hopf bifurcation under prescribed boundary condition

机译：在规定边界条件下发生Hopf分支的反应扩散方程的条件研究

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摘要

Reaction diffusion equation, which is well practically applied at present, is a good model to describe the natural phenomenon. Although Hopf bifurcation is easy, it is vitally important as a dynamic bifurcation. Hopf bifurcation theory has been the essential method to analyze the appearance and disappearance of the small amplitude periodic solution of the differential equation, mainly because its research theory and numerical research method are significant for the dynamic bifurcation and limit cycle. Moreover, there is inseparable relationship between Hopf bifurcation and the theory of self-excited vibration, so Hopf bifurcation is widely used in the modern engineering. First of all, this research elaborates the research status and significance of reaction diffusion equation. Then, the paper explains reaction diffusion equation and discusses its equilibrium, distribution and linear stability. In the end, the research discusses the conditions of one dimension reaction equation occurring Hopf bifurcation and analyzes the stability and conditions. Through the study of the whole process, it can be seen that reaction diffusion equation can be widely applied. Therefore, it is important for describing the natural movements and involves a number of disciplines. Furthermore, many mathematical models can be switched into reaction diffusion equation so that it is more beneficial for research. The research on one dimension reaction diffusion equation will be useful for the analysis and awareness of several natural phenomena.
机译：反应扩散方程是目前描述自然现象的良好模型，目前在实践中已得到很好的应用。尽管Hopf分叉很容易，但作为动态分叉至关重要。 Hopf分岔理论一直是分析微分方程小振幅周期解的出现和消失的基本方法，主要是因为其研究理论和数值研究方法对于动态分岔和极限环具有重要意义。而且，霍普夫分岔与自激振动理论之间有着密不可分的关系，因此霍普夫分岔在现代工程中得到了广泛的应用。首先，本文阐述了反应扩散方程的研究现状和意义。然后，本文解释了反应扩散方程，并讨论了它的平衡，分布和线性稳定性。最后，研究了一维反应方程发生霍普夫分支的条件，并分析了稳定性和条件。通过对整个过程的研究，可以看出反应扩散方程可以得到广泛的应用。因此，描述自然运动很重要，涉及许多学科。此外，许多数学模型可以转换成反应扩散方程，因此对研究更有利。一维反应扩散方程的研究将有助于分析和认识几种自然现象。

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