半线性偏微分方程的分支理论及其应用
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摘要
在半线性偏微分方程的研究领域当中有一类非常重要的非线性现象,这类非线性现象就是分支现象,它反应的是流的拓扑结构随参数的变化而引起质的变异。分支问题主要包括局部分支问题、半局部分支问题和全局分支问题。偏微分方程分支问题的研究既要用到经典的动力系统理论,又要用到拓扑、代数、泛函等相关知识,其研究具有强烈的实际背景和重大的理论意义。
本文利用中心流形理论、规范型方法以及全局稳态分支定理等数学理论与数学方法,针对几类半线性偏微分方程的局部Hopf分支、全局稳态分支以及图灵分支进行了系统的研究。本文的主要工作如下:
1、利用中心流形理论和规范型方法,给出了一般的半线性偏微分方程的局部Hopf分支定理:给出了判定Hopf分支存在性的条件,以及判定分支方向和分支周期解稳定性的一般的计算公式。这一定理的提出,为半线性偏微分方程局部Hopf分支的研究提供了方法和途径;结合一类具有扩散项的捕食与被捕食模型和刻划CIMA化学反应的Lengyel-Epstein系统,我们证明了这两类半线性偏微分方程,不但具有空间齐次的周期解,而且还具有空间非齐次的周期解;同时,我们证明了,空间的“大小”影响着空间非齐次分支周期解的个数:当空间区域足够大时,空间越“大”,空间非齐次分支周期解的个数越多。
2、利用史峻平和王学峰的推广的全局稳态分支定理(该定理指出,在适当的条件下,局部稳态分支和全局稳态分支是等价的),给出了适用于一般半线性偏微分方程的简化的全局稳态分支定理。结合一类具有扩散项的捕食与被捕食模型,给出了这类半线性偏微分方程产生全局稳态分支的条件,并证明了在适当的条件下,系统的稳态分支曲线将形成一个“线圈”,该“线圈”连接系统的两个不同的稳态分支点。同时,研究了这类半线性偏微分方程中Hopf分支和稳态分支之间的相互作用与影响。
3、利用适用于一般的半线性偏微分方程处理图灵分支的方法,在一维的有界的空间区域上,考虑了刻划CIMA化学反应的Lengyel-Epstein系统的图灵分支,得到了易于验证的一般性的判别准则。同时,通过数值模拟对我们的理论分析加以验证。
In the field of nonlinear partial differential equations, there exists an importantnonlinear phenomenon, which is bifurcation phenomenon. So called bifurcation phe-nomenon means that, when the parameters cross through certain critical values, thephenomenon of the change of some structural properties in the system are studied.Bifurcation is made up of three parts: local bifurcation, semi-local bifurcation andglobal bifurcation. The study of bifurcation of partial differential equation is not onlyrelated to the theories of classical dynamical system, but also related to the otherknowledge such as topology, algebra and functional analysis. The study is of greattheoretical significance and practical background.
This paper mainly performs local Hopf bifurcation analysis, global steady statebifurcation analysis, and Turing bifurcation analysis to the semilinear partial differ-ential equations by using center manifold theory, normal form methods and globalsteady state bifurcation theorems. The main contents of the paper are as follows:
1、By using the center manifold theory and normal form methods, we obtainan abstract local Hopf bifurcation theorem for the semilinear partial differential equa-tions, which derives the conditions for the general semilinear reaction-diffusion equa-tions to undergo Hopf bifurcation, and obtains the general algorithm to determinethe bifurcation direction and the stability of the bifurcating periodic solutions. Thismakes it convenient for readers’future applications in the Hopf bifurcation analysisin semilinear partial differential equations. By applying our results to the diffusivepredator-prey system and the diffusive Lengyel-Epstein system modeling the CIMAchemical reaction, we prove that these two semilinear partial differential equationsnot only have the spatial homogenous periodic solutions but also have the spatial non-homogenous periodic solutions. Moreover, the“size”of the spatial domain affects thenumber of the spatial non-homogenous bifurcating periodic solutions in the followingway: whenever the domain is large enough, the larger the domain is, the more thespatial non-homogenous bifurcating periodic solutions.
2、By using Shi and Wang’s generalized global steady state bifurcation theorem,which says that under suitable conditions the local steady state bifurcation is equiv- alent to the global steady state bifurcation, we derive an abstract simplified globalsteady state bifurcation theorem for the semilinear partial differential equations. Byapplying our results to a diffusive homogenous predator-prey system, we prove that,under certain conditions, this system has a global steady state bifurcation, and partic-ularly we show that, under suitable conditions, the system may have loops consistingof the positive steady solutions, which connects with two different bifurcation points.Further, we consider the interaction between Hopf bifurcations and steady state bifur-cations.
3、By using the general results on Turing bifurcations of the general semilinearpartial differential equations, we consider the Turing bifurcation of a Lengyel-Epsteinsystem in CIMA chemical reaction, in a bounded interval in the one dimensionalspace, and obtain the criterion for the system to occur Turing bifurcation. Finally wepresent numerical simulations to support our theoretical analysis.
本文利用中心流形理论、规范型方法以及全局稳态分支定理等数学理论与数学方法,针对几类半线性偏微分方程的局部Hopf分支、全局稳态分支以及图灵分支进行了系统的研究。本文的主要工作如下:
1、利用中心流形理论和规范型方法,给出了一般的半线性偏微分方程的局部Hopf分支定理:给出了判定Hopf分支存在性的条件,以及判定分支方向和分支周期解稳定性的一般的计算公式。这一定理的提出,为半线性偏微分方程局部Hopf分支的研究提供了方法和途径;结合一类具有扩散项的捕食与被捕食模型和刻划CIMA化学反应的Lengyel-Epstein系统,我们证明了这两类半线性偏微分方程,不但具有空间齐次的周期解,而且还具有空间非齐次的周期解;同时,我们证明了,空间的“大小”影响着空间非齐次分支周期解的个数:当空间区域足够大时,空间越“大”,空间非齐次分支周期解的个数越多。
2、利用史峻平和王学峰的推广的全局稳态分支定理(该定理指出,在适当的条件下,局部稳态分支和全局稳态分支是等价的),给出了适用于一般半线性偏微分方程的简化的全局稳态分支定理。结合一类具有扩散项的捕食与被捕食模型,给出了这类半线性偏微分方程产生全局稳态分支的条件,并证明了在适当的条件下,系统的稳态分支曲线将形成一个“线圈”,该“线圈”连接系统的两个不同的稳态分支点。同时,研究了这类半线性偏微分方程中Hopf分支和稳态分支之间的相互作用与影响。
3、利用适用于一般的半线性偏微分方程处理图灵分支的方法,在一维的有界的空间区域上,考虑了刻划CIMA化学反应的Lengyel-Epstein系统的图灵分支,得到了易于验证的一般性的判别准则。同时,通过数值模拟对我们的理论分析加以验证。
In the field of nonlinear partial differential equations, there exists an importantnonlinear phenomenon, which is bifurcation phenomenon. So called bifurcation phe-nomenon means that, when the parameters cross through certain critical values, thephenomenon of the change of some structural properties in the system are studied.Bifurcation is made up of three parts: local bifurcation, semi-local bifurcation andglobal bifurcation. The study of bifurcation of partial differential equation is not onlyrelated to the theories of classical dynamical system, but also related to the otherknowledge such as topology, algebra and functional analysis. The study is of greattheoretical significance and practical background.
This paper mainly performs local Hopf bifurcation analysis, global steady statebifurcation analysis, and Turing bifurcation analysis to the semilinear partial differ-ential equations by using center manifold theory, normal form methods and globalsteady state bifurcation theorems. The main contents of the paper are as follows:
1、By using the center manifold theory and normal form methods, we obtainan abstract local Hopf bifurcation theorem for the semilinear partial differential equa-tions, which derives the conditions for the general semilinear reaction-diffusion equa-tions to undergo Hopf bifurcation, and obtains the general algorithm to determinethe bifurcation direction and the stability of the bifurcating periodic solutions. Thismakes it convenient for readers’future applications in the Hopf bifurcation analysisin semilinear partial differential equations. By applying our results to the diffusivepredator-prey system and the diffusive Lengyel-Epstein system modeling the CIMAchemical reaction, we prove that these two semilinear partial differential equationsnot only have the spatial homogenous periodic solutions but also have the spatial non-homogenous periodic solutions. Moreover, the“size”of the spatial domain affects thenumber of the spatial non-homogenous bifurcating periodic solutions in the followingway: whenever the domain is large enough, the larger the domain is, the more thespatial non-homogenous bifurcating periodic solutions.
2、By using Shi and Wang’s generalized global steady state bifurcation theorem,which says that under suitable conditions the local steady state bifurcation is equiv- alent to the global steady state bifurcation, we derive an abstract simplified globalsteady state bifurcation theorem for the semilinear partial differential equations. Byapplying our results to a diffusive homogenous predator-prey system, we prove that,under certain conditions, this system has a global steady state bifurcation, and partic-ularly we show that, under suitable conditions, the system may have loops consistingof the positive steady solutions, which connects with two different bifurcation points.Further, we consider the interaction between Hopf bifurcations and steady state bifur-cations.
3、By using the general results on Turing bifurcations of the general semilinearpartial differential equations, we consider the Turing bifurcation of a Lengyel-Epsteinsystem in CIMA chemical reaction, in a bounded interval in the one dimensionalspace, and obtain the criterion for the system to occur Turing bifurcation. Finally wepresent numerical simulations to support our theoretical analysis.
引文
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