Dimension of the attractors associated to the Ginzburg-Landau partial differential equation

Abstract 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.

[1]  John Whitehead,et al.  Finite bandwidth, finite amplitude convection , 1969, Journal of Fluid Mechanics.

[2]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[3]  L. Keefe.  Integrability and structural stability of solutions to the Ginzburg–Landau equation , 1986 .

[4]  H. Moon,et al.  Transitions to chaos in the Ginzburg-Landau equation , 1983 .

[5]  R. Temam.  Behaviour at Time t=0 of the Solutions of Semi-Linear Evolution Equations. , 1982 .

[6]  R. Temam,et al.  Determining modes and fractal dimension of turbulent flows , 1985, Journal of Fluid Mechanics.

[7]  Benoit B. Mandelbrot,et al.  Fractals: Form, Chance and Dimension , 1978 .

[8]  R. C. DiPrima,et al.  The Eckhaus and Benjamin-Feir resonance mechanisms , 1978, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[9]  H. Moon,et al.  Three-Frequency Motion and Chaos in the Ginzburg-Landau Equation , 1982 .

[10]  J. Yorke,et al.  The liapunov dimension of strange attractors , 1983 .

[11]  Jack K. Hale,et al.  Infinite dimensional dynamical systems , 1983 .

[12]  L. Keefe,et al.  Dynamics of Perturbed Wavetrain Solutions to the Ginzburg‐Landau Equation , 1985 .

[13]  N. Bourbaki.  Espaces vectoriels topologiques , 1955 .

[14]  L. Nirenberg,et al.  On elliptic partial differential equations , 1959 .

[15]  Roger Temam,et al.  Attractors for the Be´nard problem: existence and physical bounds on their fractal dimension , 1987 .

[16]  L. Sirovich,et al.  Periodic solutions of the Ginzburg-Landau equation , 1986 .

[17]  P. J. Blennerhassett,et al.  On the generation of waves by wind , 1980, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[18]  John C. Wells.  Invariant manifolds on non-linear operators. , 1976 .

[19]  J. Ghidaglia.  Some backward uniqueness results , 1986 .

[20]  R. Temam,et al.  Attractors Representing Turbulent Flows , 1985 .

[21]  M. Vishik,et al.  Attractors of partial differential evolution equations and estimates of their dimension , 1983 .

[22]  Lawrence Sirovich,et al.  Instabilities of the Ginzburg-Landau equation: periodic solutions , 1986 .