Computer algebra derives correct initial conditions for low-dimensional dynamical models

To ease analysis and simulation we make low-dimensional models of complicated dynamical systems. Centre manifold theory provides a systematic basis for the reduction of dimensionality from some detailed dynamical prescription down to a relatively simple model. An initial condition for the detailed dynamics also has to be projected onto the low-dimensional model, but except in meteorology this issue has received scant attention. Herein, based upon the algorithm in (Roberts, 1997), I develop a straightforward algorithm for the computer algebra derivation of this projection. The method is systematic and is based upon the geometric picture underlying centre manifold theory. The method is applied to examples of a pitchfork and a Hopf bifurcation. There is a close relationship between this projection of initial conditions and the correct projection of forcing onto a model. I reaffirm this connection and show how the effects of forcing, both interior and from the boundary, should be properly included in a dynamical model.

[1]  J. Carr.  Applications of Centre Manifold Theory , 1981 .

[2]  M. Lewenstein,et al.  Adiabatic drag and initial slip in random processes , 1983 .

[3]  A. J. Roberts.  Low-dimensional modelling of dynamical systems , 1997 .

[4]  Stephen M. Cox,et al.  Initial conditions for models of dynamical systems , 1995 .

[5]  Anthony J. Roberts,et al.  A complete model of shear dispersion in pipes , 1994 .

[6]  Anthony J. Roberts,et al.  Low-dimensional modelling of dynamics via computer algebra , 1996, chao-dyn/9604012.

[7]  Robert G. Muncaster,et al.  The theory of pseudo-rigid bodies , 1988 .

[8]  John Guckenheimer,et al.  Kuramoto-Sivashinsky dynamics on the center-unstable manifold , 1989 .

[9]  Harold Grad,et al.  Asymptotic Theory of the Boltzmann Equation , 1963 .

[10]  S. Cox,et al.  C D ] 7 M ar 2 00 3 Initialization and the quasi-geostrophic slow manifold , 1994 .

[11]  Anthony J. Roberts,et al.  The Accurate Dynamic Modelling of Contaminant Dispersion in Channels , 1995, SIAM J. Appl. Math..

[12]  Shui-Nee Chow,et al.  Ck centre unstable manifolds , 1988, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[13]  Anthony J. Roberts,et al.  The utility of an invariant manifold description of the evolution of a dynamical system , 1989 .

[14]  G. Sell,et al.  Approximation theories for inertial manifolds , 1989 .

[15]  Stephen M. Cox,et al.  Centre manifolds of forced dynamical systems , 1991, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[16]  Inertial manifolds , 1990 .

[17]  落合 萌,et al.  23.Elimination of Fast Variables in Stochastic Processes(パターン形成,運動と統計,研究会報告) , 1985 .

[18]  Chao Xu,et al.  On the low-dimensional modelling of Stratonovich stochastic differential equations , 1996, chao-dyn/9705002.

[19]  A. J. Roberts,et al.  A lubrication model of coating flows over a curved substrate in space , 1997, Journal of Fluid Mechanics.

[20]  B. U. Felderhof,et al.  Systematic elimination of fast variables in linear systems , 1983 .

[21]  A. Winfree.  Patterns of phase compromise in biological cycles , 1974 .

[22]  Robert G. Muncaster,et al.  Invariant manifolds in mechanics II: Zero-dimensional elastic bodies with directors , 1984 .

[23]  R. Temam,et al.  Nonlinear Galerkin methods , 1989 .

[24]  G. Sell,et al.  On the computation of inertial manifolds , 1988 .

[25]  Anthony J. Roberts,et al.  Appropriate initial conditions for asymptotic descriptions of the long term evolution of dynamical systems , 1989, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[26]  E. A. Spiegel,et al.  Amplitude Equations for Systems with Competing Instabilities , 1983 .

[27]  Edward N. Lorenz,et al.  On the Nonexistence of a Slow Manifold , 1986 .

[28]  J. Guckenheimer,et al.  Isochrons and phaseless sets , 1975, Journal of mathematical biology.

[29]  Anthony J. Roberts,et al.  The invariant manifold of beam deformations , 1993 .

[30]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[31]  Edriss S. Titi,et al.  On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation , 1994 .