Spatio-temporal secondary instabilities near the Turing-Hopf bifurcation

Scientific Reports
Aldo Ledesma-Durán, José L Aragón

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.

References

Dec 1, 1993·Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics· De Wit AP Borckmans
Jul 1, 1996·Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics· De Wit AP Borckmans
Aug 23, 1993·Physical Review Letters·J PerraudP Borckmans
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Aug 11, 2001·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·W JustE Schöll
Sep 26, 2012·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·J L AragónP K Maini

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Citations

Apr 14, 2020·Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences·A P KolinichenkoL B Ryashko
Oct 21, 2020·Physical Review. E·Aldo Ledesma-DuránIván Santamaría-Holek

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