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Spatio-temporal numerical solutions of the coupled real and complex Ginzburg-Landau amplitude equations for one-dimensional systems near the Turing-Hopf bifurcation. (English) Zbl 1450.35245
Summary: In this paper, the solutions of the coupled Ginzburg-Landau equations are numerically studied with the aim to describe the dynamics of systems close to the Turing-Hopf bifurcation. We found that the spatial modulations of the Turing and Hopf amplitudes increase with the domain size due to inhomogeneous perturbations. By measuring the growth of spatial Fourier modes in systems initialized with random initial conditions, the effective size of the domain where the generalized Eckhaus and Benjamin-Feir-Newell instabilities occur was determined. Besides, we have numerically corroborated that such instabilities can be quantified by recent theoretical results on secondary instabilities of the Mixed mode solution. Furthermore, this study exemplifies diverse spatiotemporal patterns related to intermittency and chaos previously studied for the uncoupled complex Ginzburg-Landau equation as well a new features related to the interaction of Turing and Hopf modes, as the appearance of backbones patterns, foliar figures and wormlike structures in the space-time maps.
MSC:
35Q56 Ginzburg-Landau equations
35B32 Bifurcations in context of PDEs
35B36 Pattern formations in context of PDEs
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
65T50 Numerical methods for discrete and fast Fourier transforms
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