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首页> 外文期刊>Zeitschrift fur Angewandte Mathematik und Mechanik >Group Symmetry of Bifurcation Equation in Dynamic Branching
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Group Symmetry of Bifurcation Equation in Dynamic Branching

机译：动态分支中分歧方程的群对称性

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摘要

A variant of Lyapounov-Schmidt method for the finding of periodic solutions of nonlinear autonomous evolution equations with degenerate operator at the derivative is suggested. By virtue of shift invariance by time variable and supposed additional symmetry on spatial variables to this problem group theoretical and group analysis methods are applicable. We consider the most general case of dynamic (Andronov-Hopf) bifurcation when there are some multiple eigenvalues on imaginary axis. We decompose the set of imaginary eigenvalues in the upper half-plane into some disjoint subclasses. To definite subclass belong all eigenvalues iα_s, s = 1,...,m, of generalized eigenvalue problem,such that α_s = k_sα, where k_s are natural numbers without common nontrivial divisors. For each α on the base of the group analysis methods we construct its own branching equation of 2π/(α + μ) -periodical solutions. Some typical situations of construction and investigation of branching equation are considered. A variant of Lyapounov-Schmidt method in nonstationary branching was developed for the finding of periodic solutions of nonlinear autonomous evolution equations in Banach spaces. Invariance of this equation relative to shifts by the time gives the possibilities of application group theoretical and group analysis methods of differential equations. Here we consider the problem of construction of the general form of branching equation (BEq) on nonstationary branching by admitting group symmetry and finding of its solutions asymptotics. It is considered the most general case in comparison with when there are some multiple eigenvalues on imaginary axis.
机译：建议使用Lyapounov-Schmidt方法的一种变体，用于寻找在导数上具有简并算子的非线性自治演化方程的周期解。借助于时间变量的移位不变性以及假定在空间变量上的其他对称性，该问题组可以应用理论和组分析方法。当假想轴上存在多个本征值时，我们考虑动态（Andronov-Hopf）分支的最一般情况。我们将上半平面中的虚构特征值集合分解为一些不相交的子类。对于确定的子类，属于广义特征值问题的所有特征值iα_s，s = 1，...，m，使得α_s=k_sα，其中k_s是没有共同非平凡因数的自然数。对于每个基于组分析方法的α，我们构造其自己的2π/（α+μ）周期解的分支方程。考虑了分支方程的构造和研究的一些典型情况。开发了非平稳分支中Lyapounov-Schmidt方法的一种变体，用于发现Banach空间中非线性自治演化方程的周期解。该方程相对于时间的不变性为应用微分方程组理论和组分析方法提供了可能性。在这里，我们通过考虑群对称性及其解渐近性来考虑在非平稳分支上构造分支方程（BEq）的一般形式的问题。与假想轴上有多个特征值相比，这是最一般的情况。

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