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Dimension of the attractors associated to the Ginzburg-Landau partial differential equation. (English) Zbl 0623.58049
We study the long-time behavior of solutions to the Ginzburg-Landau partial differential equation. It is shown that a finite-dimensional attractor captures all the solutions. An upper bound on this dimension is given in terms of physical quantities, by estimating the Lyapunov exponents on the trajectories. Finally, using the well-known side-band instabilities of an exact, time-dependent solution (Stokes solution) we derive lower bounds on the dimension of the universal attractor. Moreover the lower and upper bounds agree.

58Z05 Applications of global analysis to the sciences
35Q99 Partial differential equations of mathematical physics and other areas of application
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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