Ledesma-Durán Aldo, Aragón José L
Centro de Física Aplicada y Tecnología Avanzada, Universidad Nacional Autónoma de México, Boulevard Juriquilla 3001, 76230, Querétaro, Mexico.
Sci Rep. 2019 Aug 2;9(1):11287. doi: 10.1038/s41598-019-47584-9.
In this work, we provide a framework to understand and quantify the spatiotemporal structures near the codimension-two Turing-Hopf point, resulting from secondary instabilities of Mixed Mode solutions of the Turing-Hopf amplitude equations. These instabilities are responsible for solutions such as (1) patterns which change their effective wavenumber while they oscillate as well as (2) phase instability combined with a spatial pattern. The quantification of these instabilities is based on the solution of the fourth order polynomial for the dispersion relation, which is solved using perturbation techniques. With the proposed methodology, we were able to identify and numerically corroborate that these two kinds of solutions are generalizations of the well known Eckhaus and Benjamin-Feir-Newell instabilities, respectively. Numerical simulations of the coupled system of real and complex Ginzburg-Landau equations are presented in space-time maps, showing quantitative and qualitative agreement with the predicted stability of the solutions. The relation with spatiotemporal intermittency and chaos is also illustrated.
在这项工作中，我们提供了一个框架，用于理解和量化余维二的图灵-霍普夫点附近的时空结构，这些结构是由图灵-霍普夫振幅方程的混合模式解的二次不稳定性产生的。这些不稳定性导致了诸如（1）在振荡时改变其有效波数的模式，以及（2）与空间模式相结合的相位不稳定性等解。这些不稳定性的量化基于色散关系的四阶多项式的解，该解使用微扰技术求解。通过所提出的方法，我们能够识别并通过数值证实这两种解分别是著名的埃克豪斯和本杰明-费尔-纽厄尔不稳定性的推广。实部和复部金兹堡-朗道方程耦合系统的数值模拟以时空图的形式呈现，显示出与预测的解的稳定性在定量和定性上的一致性。同时也阐述了与时空间歇性和混沌的关系。
