Abstract
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.
Highlights
Oscillatory patterns are recurrently observed in a variety of physical and chemical systems
Despite the importance of oscillatory patterns, only a few works use the amplitude formalism mainly because the study of the stability of the Mixed mode solution has been restricted to the homogeneous Turing and Hopf states, i.e. solutions where both modes have constant amplitudes, giving place to spatial pattern of fixed wave number oscillating in phase along the domain[10]
We will present a novel form of presenting the stability of the Mixed mode solutions in what we name stability diagrams, and explain how to use them for extract quantitative and qualitative conclusions of the possible solutions that the system can have as the size of the domain increases
Summary
Oscillatory patterns are recurrently observed in a variety of physical and chemical systems. We will prove that two of these roots are related to generalizations of the Eckhaus and BFN instabilities for the Mixed Mode solution These theoretical results are compared with the numerical solutions of the Turing-Hopf amplitude equations showing an excellent quantitative agreement. We will present a novel form of presenting the stability of the Mixed mode solutions in what we name stability diagrams, and explain how to use them for extract quantitative and qualitative conclusions of the possible solutions that the system can have as the size of the domain increases These results allow us to extend our understanding of the solution of dynamical systems near the CTHP to more realistic physical situations where larger domains and spatial perturbations give place to secondary instabilities, intermittency and chaos
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