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Use of Approximate Inertial Manifolds in Bifurcation Calculations

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Continuation and Bifurcations: Numerical Techniques and Applications

Part of the book series: NATO ASI Series ((ASIC,volume 313))

Abstract

The theory of inertial manifolds provides a rigorous connection between certain evolution PDEs and low-dimensional dynamical systems. The restriction of the flow of the PDE to the inertial manifold is given by a finite set of ODEs called an inertial form, which captures all the long time dynamic behavior of the PDE. We give a brief survey of several numerical schemes to approximate inertial manifolds and hence inertial forms. We then implement these approximate inertial forms and use them to construct bifurcation diagrams of two model PDEs (the Kuramoto-Sivashinsky equation and a coupled reaction diffusion system). Some numerical issues arising in these schemes are discussed.

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References

  • Armbruster, D., J. Guckenheimer, and P. J. Holmes (1988a), Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry, Physica, 29D, pp. 257–282.

    MathSciNet  Google Scholar 

  • Armbruster, D., J. Guckenheimer, and P. J. Holmes (1988b), Kuramoto-Sivashinsky dynamics on the center-unstable manifold, SIAM J. Appl. Math., (to appear).

    Google Scholar 

  • Billotti, J. E. and J. P. LaSalle (1971), Dissipative periodic processes, Bull. Amer. Math. Soc., 77, pp. 1082–1088.

    Article  MathSciNet  MATH  Google Scholar 

  • Brown, H. S., M. S. Jolly, I. G. Kevrekidis and E. S. Titi (1990), Implementation of approximate inertial manifolds for systems of coupled reaction-diffusion equations, in preparation,.

    Google Scholar 

  • Chang, H.-C. (1986), Travelling waves on fluid interfaces: normal form analysis of the Kuramoto-Sivashinsky equation, Phys. Fluids, 29, pp. 3142–3147.

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, L.-H. and H.-C. Chang (1986), Nonlinear waves on thin film surfaces- II. Bifurcation analyses of the long-wave equation, Chem. Eng. Sci., 41, pp. 2477–2486.

    Article  Google Scholar 

  • Constantin, P. (1989), Private communication.

    Google Scholar 

  • Constantin, P. (1987), in: Proceedings of the AMS/SIAM Summer Conference on the Connection between Finite and Infinite Dimensional Systems, Boulder, Col., 1987, C. Foias, B. Nicolaenko, and R. Team eds. Contemporary Mathematics (to appear).

    Google Scholar 

  • Constantin, P. and C. Foias (1985), Global Lyapunov exponents, Kapian Yorke formulas and the dimension of the attractor for 2D Navier-Stokes equations, Comm. Pure Appl. Math., 38, pp. 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  • Constantin, P., C. Foias, B. Nicolaenko, R. Témam (1988), Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Appl. Math. Sciences, No. 70 Springer Verlag, New York.

    Google Scholar 

  • Constantin, P., C. Foias, B. Nicolaenko, R. Témam (1989), Spectral barriers and inertial manifolds for dissipative partial differential equations, J. Dynamics and Differential Equations, 1, pp. 45–73.

    Article  MATH  Google Scholar 

  • Constantin, P., C. Foias, R. Témam (1984), On the large time Galerkin Approximation of the Navier-Stokes equations, SIAM J. Numerical Analysis, 21, pp. 615–634.

    Article  MATH  Google Scholar 

  • Dhabolkar, V. R., V. Balakotaiah and D. Luss, Stationary concentration profiles on an isothermal catalytic wire, Chem. Eng. Sci., 44, pp. 1915–1928.

    Google Scholar 

  • Doedel, E. J. (1981), AUTO: a program for the bifurcation analysis of autonomous systems, Cong. Num., 30, pp. 265–285.

    MathSciNet  Google Scholar 

  • Doering, C. R., J. D. Gibbon, D. D. Holm, and B. Nicolaenko (1988), Low dimensional behavior in the complex Ginzburg-Landau equation, Nonlinearity, 1, pp. 279–309.

    Article  MathSciNet  MATH  Google Scholar 

  • Fabes, E., M. Luskin and G. E. Sell (1988), Construction of inertial manifolds by elliptic regularization, IMA Preprint No. 459.

    Google Scholar 

  • Foias, C., M. S. Jolly, I. G. Kevrekidis, G. R. Sell, and E. S. Titi (1988a), On the computation of inertial manifolds, Physics Letters A, 131, pp. 433–436.

    Article  MathSciNet  Google Scholar 

  • Foias, C., O. Manley, and R. Témam (1988b), Modelling of the interaction of small and large eddies in two dimensional turbulent flows, Math. Modelling Numerical Anal., 22, pp. 93– 118.

    Google Scholar 

  • Foias, C., O. Manley, and R. Témam (1989b), Approximate inertial manifolds and effective viscosity in turbulent flows, (in preparation).

    Google Scholar 

  • Foias, C., B. Nicolaenko, G. R. Sell, R. Témam (1988c), Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimensions, J. Math. Pures Appl., 67, pp. 197–226.

    MathSciNet  MATH  Google Scholar 

  • Foias, C., G.R. Sell and R. Témam (1988d), Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73, pp. 309–353.

    Google Scholar 

  • Foias, C. and J.C. Saut (1983), Remarques sur les équations de Navier-Stokes stationnaires, Annali Scuola Norm. Sup.-Pisa, Ser. IV, 10, pp. 169–177.

    MathSciNet  MATH  Google Scholar 

  • Foias, C., G.R. Sell and E. S. Titi (1989a), Exponential tracking and approximation of inertial manifolds for dissipative equations, J. Dynamics and Differential Equations, 1, pp. 199–224.

    Article  MathSciNet  MATH  Google Scholar 

  • Foias, C. and R. Témam (1978), Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation, Annali Scuola Norm. Sup.-Pisa, Ser. IV, 5, pp. 29–63.

    Google Scholar 

  • Foias, C. and R. Témam (1979), Some analytic and geometric properties of the solutions of the Navier-Stokes equations, J. Math. Pures Appl., 58, pp. 339–368.

    MathSciNet  MATH  Google Scholar 

  • Foias, C. and R. Témam (1988a), The algebraic approximation of attractors: The finite dimensional case, Physica D, 32, pp. 163–182.

    Article  MathSciNet  MATH  Google Scholar 

  • Foias, C. and R. Témam (1988b), Approximation algébrique des attracteurs. I. Le cas de la dimension finie, C. R. Acad. Sci., Paris, 307, pp. Série I. 5–8.

    Google Scholar 

  • Foias, C. and R. Témam (1988c), Approximation algébrique des attracteurs. II. Le cas de la dimension infinie, C. R. Acad. Sci., Paris, 307, pp. Série I. 67–70.

    Google Scholar 

  • Foias, C., R. Témam and E. S. Titi (1989c), Inertial manifold interpretation of the finite difference method., (in preparation).

    Google Scholar 

  • Ghidaglia, J. M. and B. Héron (1987), Dimension of the attractor associated to the Ginzburg-Landau equation, Physica, 28D, pp. 282–304.

    MathSciNet  Google Scholar 

  • Glendinning, P. and C. Sparrow (1984), Local and global behavior near homoclinic orbits, J. Stat. Phys., 35, pp. 645–696.

    MathSciNet  Google Scholar 

  • Green, J. M. and J.-S. Kim (1988), The steady states of the Kuramoto-Sivashinsky equation, Physica, 33D, pp. 99–120.

    MathSciNet  Google Scholar 

  • Hale, J. K. (1988), Asymptotic behavior of dissipative systems, Math. Surveys and Monographs, 25, AMS, Providence, R.I..

    Google Scholar 

  • Hyman, J. M. and B. Nicolaenko (1986), The Kuramoto-Sivashinsky equation: A bridge between PDEs and dynamical systems, Physica, 18D, pp. 113–126.

    MathSciNet  Google Scholar 

  • Hyman, J.M., B. Nicolaenko, and S. Zaleski (1986), Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interfaces, Physica, 23D, pp. 265–292.

    MathSciNet  Google Scholar 

  • Jauberteau, F., C. Rosier, and R. Temam (1989a), A nonlinear Galerkin method for the Navier-Stokes equations, Proc. Conf. on “Spectral and High Order Methods for Partial Differential Equations” ICOSAHOM’ 89, Como, Italie, (to appear).

    Google Scholar 

  • Jauberteau, F., C. Rosier, and R. Temam (1989b), The nonlinear Galerkin method in computational fluid dynamics, Applied Numerical Mathematics, (to appear).

    Google Scholar 

  • Jolly, M.S. (1989), Explicit construction of an inertial manifold for a reaction diffusion equation, J. Differential Equations, 78, pp. 220–261.

    Article  MathSciNet  MATH  Google Scholar 

  • Jolly, M.S. (1989), Explicit construction of an inertial manifold for a reaction diffusion equation, J. Differential Equations, 78, pp. 220–261.

    Article  MathSciNet  MATH  Google Scholar 

  • Jolly, M. S., I. G. Kevrekidis, and E. S. Titi (1989b), Preserving dissipation in approximate inertial forms, J. Dynamics and Differential Equations, (to appear).

    Google Scholar 

  • Kevrekidis, I. G. and M. Jolly (1987), On the use of interactive graphics in the numerical study of chemical dynamics, paper no. 22c, presented to the 1987 Annual AIChE Meeting, New York, Nov. 1987.

    Google Scholar 

  • Kevrekidis, I. G. and M. Jolly (1987), On the use of interactive graphics in the numerical study of chemical dynamics, paper no. 22c, presented to the 1987 Annual AIChE Meeting, New York, Nov. 1987.

    Google Scholar 

  • LaQuey, R. E., S. M. Mahajan, P. H. Rutherford, and W. M. Tang (1975), Nonlinear saturation of the trapped-ion mode, Phys. Rev. Lett., 34, pp. 391–394.

    Article  Google Scholar 

  • Mallet-Paret, J.(1976), Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, J. Differential Equations, 22, pp. 331–348.

    Article  MathSciNet  MATH  Google Scholar 

  • Mallet-Paret, J. and G. R. Sell (1988), Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Amer. Math. Soc., 1, pp. 805–866.

    Article  MathSciNet  Google Scholar 

  • Marion, M. (1989a), Approximate inertial manifolds for the pattern formation Cahn Hilliard equations, Mathematical Modelling and Numer. Anal., 23, pp. 463–488.

    MathSciNet  Google Scholar 

  • Marion, M. (1989b), Approximate inertial manifolds for reaction diffusion equations in high space dimension, J. Dynamics and Differential Equations, 1, pp. 245–267.

    Article  MathSciNet  MATH  Google Scholar 

  • Marion, M. and R. Temam (1989), Nonlinear Galerkin methods, SIAM J. Numer. Anal., 26, pp. 1139–1157.

    Article  MathSciNet  MATH  Google Scholar 

  • Michelson, D. (1986), Steady solutions of the Kuramoto-Sivashinsky equation, Physica, 19D, pp. 89–111.

    MathSciNet  Google Scholar 

  • Nicolaenko, B. (1987), in: Proceedings of the A MS/SIAM Summer Conference on the Connection between Finite and Infinite Dimensional Systems, Boulder, Col., 1987, C. Foias, B. Nicolaenko, and R. Temam eds. Contemporary Mathematics (to appear).

    Google Scholar 

  • Nicolaenko, B., B. Scheurer, R. Temam (1989), Some global dynamical properties of a class of pattern formation equations, Commun. in Partial Differential Equations, 14, pp. 245–297.

    Article  MathSciNet  MATH  Google Scholar 

  • Orszag, S. A. (1971), Numerical simulation of incompressible flows within simple boundaries: Accuracy, J. Fluid Mech., 49, pp. 75–112.

    Article  MathSciNet  MATH  Google Scholar 

  • Orszag, S. A. (1972), Comparison of pseudospectral and spectral approximation, Studies in Appl. Math., 51, pp. 253–259.

    MATH  Google Scholar 

  • Scovel, C., I. G. Kevrekidis, and B. Nicolaenko (1988), Scaling laws and the prediction of bifurcations in systems modeling pattern formation, Phys. Lett. A, 130, pp. 73–80.

    Article  Google Scholar 

  • Sell, G. R., (1989), Approximation dynamics: hyperbolic sets and inertial manifolds, University of Minnesota Supercomputer Institute, Preprint No. 89/139.

    Google Scholar 

  • Smoller, J. (1983), Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York

    MATH  Google Scholar 

  • Taboada, M. (1989), Finite dimensional asymptotic behavior for the Swift-Hohenberg model of convection, Nonlinear Analysis, TMA, (to appear).

    Google Scholar 

  • Témam, R. (1988a), Variétés inertiélles approximatives pour les équations de Navier-Stokes bidimensionnelles, C. R. Acad. Sci. Paris, Serie II, 306, pp. 399–402.

    Google Scholar 

  • Témam, R. (1988b), Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer Verlag, New York.

    MATH  Google Scholar 

  • Témam, R. (1989), Induced trajectories and approximate inertial manifolds, Mathematical Modelling and Num. Anal. 23, pp. 541–561.

    MATH  Google Scholar 

  • Titi, E. S. (1987), On a criterion for locating stable stationary solutions to the Navier-Stokes equations, Nonlinear Anal., Theory, Methods and Appl., 11, pp. 1085–1102.

    Article  MathSciNet  MATH  Google Scholar 

  • Titi, E. S. (1988), Une vaHété approximante de Vattracteur universel des équations de Navier-Stokes, non linéaire, de dimension finie, C. R. Acad. Sci., Paris, 307, pp. Série I. 383–385.

    MathSciNet  Google Scholar 

  • Titi, E. S. (1989), On approximate inertial manifolds to the Navier Stokes equations, MSI Preprint 88–119, also in J. Math. Anal. and Appl., (to appear).

    Google Scholar 

  • Wiggins, S. (1988), Global Bifurcations and Chaos, Appl. Math. Sciences, No. 73, Springer Verlag, New York.

    Book  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Chemical Engineering, Princeton University Princeton, NJ, 08544, USA

    H. S. Brown & I. G. Kevrekidis

  2. Department of Mathematics, Indiana University Bloomington, IN, 47405, USA

    M. S. Jolly

  3. Department of Mathematics, University of California - Irvine, Irvine, CA, 92717, USA

    E. S. Titi

Authors
  1. H. S. Brown
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  2. M. S. Jolly
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  3. I. G. Kevrekidis
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  4. E. S. Titi
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Katholieke Universiteit Leuven, Leuven, Belgium

    Dirk Roose

  2. Avery STDE, Turnhout, Belgium

    Bart De Dier

  3. School of Mathematical Sciences, University of Bath, Bath, UK

    Alastair Spence

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© 1990 Kluwer Academic Publishers

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Brown, H.S., Jolly, M.S., Kevrekidis, I.G., Titi, E.S. (1990). Use of Approximate Inertial Manifolds in Bifurcation Calculations. In: Roose, D., Dier, B.D., Spence, A. (eds) Continuation and Bifurcations: Numerical Techniques and Applications. NATO ASI Series, vol 313. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0659-4_2

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  • DOI: https://doi.org/10.1007/978-94-009-0659-4_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6781-2

  • Online ISBN: 978-94-009-0659-4

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