论文标题
关于高阶扰动微分方程的周期性解决方案平滑分支的稳定性
On the stability of smooth branches of periodic solutions for higher order perturbed differential equations
论文作者
论文摘要
与Lyapunov-Schmidt还原结合使用的平均方法为存在以下类扰动$ t $ t $ t $ t $ t $ t $ t $ t $ t $ t $ t $ x'= f_0（t，t，x）+\ varepsilon f（t，t，x，x，x，\ varepsilon）$的定期解决方案提供了足够的条件。这种定期解决方案从不易处理的系统的定期解决方案$ x'= f_0（t，x）$的歧管$ \ mathcal {z} $分叉。确定这种周期性解决方案的稳定性涉及计算矩阵值函数$ m（\ varepsilon）$的特征值，可以使用$ k $ hymyperbolic矩阵来完成。通常，在这个理论中，必须采用$ m（\ varepsilon）$的$ K $ -JET的对角线化过程，并且不存在一般算法。在本文中，我们制定了一种替代策略来确定周期性解决方案的稳定性，而无需进行这种对角度的过程，即使在无法对角线化的情况下，也可以起作用。还提出了我们的结果对两个$ 4 $ D的矢量场的申请。
The averaging method combined with the Lyapunov-Schmidt reduction provides sufficient conditions for the existence of periodic solutions of the following class of perturbative $T$-periodic nonautonomous differential equations $x'=F_0(t,x)+\varepsilon F(t,x,\varepsilon)$. Such periodic solutions bifurcate from a manifold $\mathcal{Z}$ of periodic solutions of the unperturbed system $x'=F_0(t,x)$. Determining the stability of this kind of periodic solutions involves the computation of eigenvalues of matrix-valued functions $M(\varepsilon)$, which can done using the theory of $k$-hyperbolic matrices. Usually, in this theory, a diagonalizing process of $k$-jets of $M(\varepsilon)$ must be employed and no general algorithm exists for doing that. In this paper, we develop an alternative strategy for determining the stability of the periodic solutions without the need of such a diagonalization process, which can work even when the diagonalization is not possible. Applications of our result for two families of $4$D vector fields are also presented.
