Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Optimal low-dimensional dynamical approximations

Authors: L. Sirovich, B. W. Knight and J. D. Rodriguez
Journal: Quart. Appl. Math. 48 (1990), 535-548
MSC: Primary 58F39; Secondary 35C99, 35Q55, 58F12, 76E30
DOI: https://doi.org/10.1090/qam/1074969
MathSciNet review: MR1074969
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Abstract | References | Similar Articles | Additional Information

Abstract: We present a method for determining optimal coordinates for the representation of an inertial manifold of a dynamical system. The condition of optimality is precisely defined and is shown to lead to a unique basis system. The method is applied to the Neumann and Dirichlet problems for the Ginzburg-Landau equation. Substantial reduction in the size of the dynamical system, without loss of accuracy is obtained from the method.

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  • [1] J. Kevorkian and J. D. Cole, Perturbation Method in Applied Mathematics, Springer-Verlag, NY, 1981 MR 608029
  • [2] H. Haken, Synergetics, 3rd Edition, Springer-Verlag, NY, 1983 MR 2062548
  • [3] N. G. van Kampen, Elimination of fast variables, Phys. Rep. 124, 69-160 (1985) MR 795762
  • [4] S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases, Cambridge University Press, 1952 MR 1148892
  • [5] C. Cercignani, The Boltzmann Equation and its Application, Springer-Verlag, NY, 1988 MR 1313028
  • [6] E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 30, 130 (1963)
  • [7] I. Shimada and T. Nagaskima, A numerical approach to the ergodic problem of dissipative dynamical systems, Prog. Theor. Phys. 61, 1605 (1979) MR 539440
  • [8] V. L. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz. 20 (1950)
  • [9] L. Keefe, Dynamics of perturbed wavetrain solutions to the Ginzburg-Landau equation, Stud. Appl. Math. 73, 91-153 (1985) MR 804366
  • [10] L. Sirovich and J. D. Rodriguez, Coherent structures and chaos: A model problem, Physics Letter A 120, 211 (1987) MR 879949
  • [11] L. Sirovich, J. D. Rodriguez, and B. Knight, Two boundary value problems for the Ginzburg Landau equation, Physica D 43 (1990) MR 1060044
  • [12] J. D. Rodriguez and L. Sirovich, Low dimensional dynamics for the complex Ginzburg Landau equation, Physica D 43 (1990) MR 1060045
  • [13] R. B. Ash and M. F. Gardner, Topics in Stochastic Processes, Academic Press, NY, 1975 MR 0448463
  • [14] C. Foias, G. R. Sella, and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations 73, 309-353 (1988) MR 943945
  • [15] J. Mallet-Paret and G. R. Sell, Inertial manifolds for reaction-diffusion equations in higher space dimensions, J. Amer. Math. Soc., 805-866 (1988) MR 943276
  • [16] P. Constantin, C. Foias, B. Nicolenko, and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York-Berlin, 1989 MR 966192
  • [17] L. Sirovich, Turbulence and the dynamics of coherent structures, Pt. I: Coherent Structures, Quart. Appl. Math. 45 (3), 561-571 (1987) MR 910462
  • [18] L. Sirovich, Chaotic dynamics of coherent structures, Physica D 37, 126-145 (1989) MR 1024387
  • [19] D. Foias, O. P. Manley, and R. Temam, Sur l'interaction des petits et grands tourbillars dans les écoulements turbulents, C. R. Acad. Sci. Paris, Serie I, 305, 497-500 (1987)
  • [20] E. Titi, On approximate inertial manifolds to the Navier-Stokes equations, Math. Sci. Inst. Rep. (Cornell), 1989 MR 1057693
  • [21] P. Newton and L. Sirovich, Instabilities in the Ginzburgh-Landau equation: Periodic solutions, Quart. Appl. Math. 44 (49), (1986) MR 840442
  • [22] F. Riesz and B. Sz. Nagy, Functional Analysis, Ungar, N.Y., 1955 MR 0071727
  • [23] E. Titi, private communication
  • [24] J. M. Ghiadaglia and B. Héron, Dimension of the attractors associated to the Ginzburg-Landau partial differential equation, Physica 28D, 282-304 (1987)

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DOI: https://doi.org/10.1090/qam/1074969
Article copyright: © Copyright 1990 American Mathematical Society

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