Studies of the Ginzburg-Landau equation
The turbulence problem is the motivation for the study of reduction of phase space dimension in the Ginzburg-Landau equation. Chaotic solutions to this equation provide a turbulence analog. A basis set for the chaotic attractor is derived using the orthogonal decomposition of the correlation matrix. This matrix is computed explicitly at the point of maximal Liapunov dimension in the parameter range under study. The basis set is shown to be optimal in a least squares sense. Galerdin projection is then used to obtain a small set of O.D.E.'s. The case of spatially periodic, even initial data is studied first. Three complex O.D.E.'s were sufficient to reproduce the solution of the full system as given by a 16 point pseudo-spectral Fourier method. The case of homogeneous boundary conditions was studied next. Ten complex O.D.E.'s were required versus 128 for the pseudo-spectral solution. Using power spectra and Poincare sections the reduced systems were shown to reproduce the exact behavior over a wide parameter range. Savings in C.P.U. time of an order of magnitude were attained over pseudo-spectral algorithms. New results on the asymptotic behavior of limit cycle solutions were also obtained. Singular solutions, zero almost everywhere, with strong boundary layer character were found in the limit of large domain size. An infinite hierarchy of subharmonic solutions was shown to exist for the spatially periodic case, and a countable number of fixed point solutions was found for both spatially periodic and homogeneous cases.
- Research Organization:
- Brown Univ., Providence, RI (USA)
- OSTI ID:
- 6091537
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
ALGORITHMS
BOUNDARY CONDITIONS
DIFFERENTIAL EQUATIONS
EQUATIONS
GALERKIN-PETROV METHOD
GINZBURG-LANDAU THEORY
ITERATIVE METHODS
LEAST SQUARE FIT
LYAPUNOV METHOD
MATHEMATICAL LOGIC
MATHEMATICAL SPACE
MATRICES
MAXIMUM-LIKELIHOOD FIT
NUMERICAL SOLUTION
PHASE SPACE
SPACE