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Numerical analysis of dynamical systems

Published online by Cambridge University Press:  07 November 2008

Andrew M. Stuart
Andrew M. Stuart
Affiliation:
Program in Scientific Computing and Computational MathematicsDivision of Applied MechanicsStanford UniversityCalifornia, CA94305-4040, USA E-mail: stuart@sccm.Stanford.edu
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Abstract

This article reviews the application of various notions from the theory of dynamical systems to the analysis of numerical approximation of initial value problems over long-time intervals. Standard error estimates comparing individual trajectories are of no direct use in this context since the error constant typically grows like the exponential of the time interval under consideration.

Instead of comparing trajectories, the effect of discretization on various sets which are invariant under the evolution of the underlying differential equation is studied. Such invariant sets are crucial in determining long-time dynamics. The particular invariant sets which are studied are equilibrium points, together with their unstable manifolds and local phase portraits, periodic solutions, quasi-periodic solutions and strange attractors.

Particular attention is paid to the development of a unified theory and to the development of an existence theory for invariant sets of the underlying differential equation which may be used directly to construct an analogous existence theory (and hence a simple approximation theory) for the numerical method.

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Type
Research Article
Information
Acta Numerica , Volume 3 , January 1994 , pp. 467 - 572
Copyright
Copyright © Cambridge University Press 1994

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References

REFERENCES

Alouges, F. and Debussche, A. (1991), ‘On the qualitative behaviour of the orbits of a parabolic partial differential equation and its discretization in the neighbourhood of a hyperbolic fixed point’, Numer. Funct. Anal. Opt. 12, 253–269.CrossRefGoogle Scholar
Alouges, F. and Debussche, A. (1993), ‘On the discretization of a partial differential equation in the neighbourhood of a periodic orbit’, Numer. Math. 65, 143–175.CrossRefGoogle Scholar
Aronson, D.G., Doedel, E.J. and Othmer, H.G. (1987), ‘An analytical and numerical study of the bifurcations in a system of linearly coupled oscillators’, Physica D, 25, 20–104.Google Scholar
Arrowsmith, D.K. and Place, C.M. (1990), An Introduction to Dynamical Systems, Cambridge Univeristy Press (Cambridge).Google Scholar
Babin, A. and Vishik, M.I. (1992), ‘Attractors of evolution equations’, in Studies in Mathematics and its Applications, North-Holland (Amsterdam).Google Scholar
Bai, F., Spence, A., Stuart, A.M. (1993), ‘The numerical computation of heteroclinic connections in systems of gradient partial differential equations’, SIAM J. Appl. Math. 53, 743–769.CrossRefGoogle Scholar
Beyn, W.-J. (1987a) ‘On invariant closed curves for one-step methods’, Numer. Math. 51, 103–122.CrossRefGoogle Scholar
Beyn, W.-J. (1987b) ‘On the numerical approximation of phase portraits near stationary points’, SIAM J. Numer. Anal. 24, 1095–1113.CrossRefGoogle Scholar
Beyn, W.-J. (1990), ‘The numerical computation of connecting orbits in dynamical systems’, IMA J. Numer. Anal. 9, 379–405.CrossRefGoogle Scholar
Beyn, W.-J. (1992), ‘Numerical methods for dynamical systems’, in Numerical Analyis; Proceedings of the SERC Summer School, Lancaster, 1990 (Light, W.A., ed.), Oxford University Press (Oxford).Google Scholar
Beyn, W.-J. and Dodel, E. (1981), ‘Stability and multiplicity of solutions to discretizations of nonlinear ordinary differential equations’, SIAM J. Sci. Stat. Comput. 2, 107–120.CrossRefGoogle Scholar
Beyn, W.-J. and Lorenz, J. (1987), ‘Center manifolds of dynamical systems under discretization’, Numer. Func. Anal. Opt. 9, 381–414.CrossRefGoogle Scholar
Bhatia, N.P. and Szego, G.P. (1970), Stability Theory of Dynamical Systems, Springer (New York).CrossRefGoogle Scholar
Braun, M. and Hershenov, J. (1977), ‘Periodic solution of finite difference equations’, Quart. Appl. Maths 35, 139–147.CrossRefGoogle Scholar
Brezzi, F., Ushiki, S. and Fujii, H. (1984), ‘Real and ghost bifurcation dynamics in difference schemes for ordinary differential equations’, in Numerical Methods for Bifurcation Problems (Kupper, T., Mittleman, H.D. and Weber, H., eds), Birkhauser (Boston).Google Scholar
Broomhead, D. and Iserles, A. (1992), Proceedings of the IMA Conference on the Dynamics of Numerics and the Numerics of Dynamics, 1990, Oxford University Press (Oxford).Google Scholar
Budd, C.J. (1990), ‘The dynamics of numerics and the numerics of dynamics’, Bristol Univerisity Applied Mathematics Report AM-90–13.Google Scholar
Budd, C.J. (1991), ‘IMA Conference on the dynamics of numerics and the numerics of dynamics. Report of the meeting’, Bull. IMA 27, 56–58.Google Scholar
Burrage, K. and Butcher, J. (1979), ‘Stability criteria for implicit Runge–Kutta processes’, SIAM J. Numer. Anal. 16, 46–57.CrossRefGoogle Scholar
Butcher, J.C. (1975), ‘A stability property of implicit Runge–Kutta methods’, BIT 15, 358–361.CrossRefGoogle Scholar
Butcher, J.C. (1987), The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta Methods and General Linear Methods, Wiley (Chichester).Google Scholar
Calvo, M.P. and Sanz-Serna, J.M. (1992), ‘Variable steps for symplectic integrators’, in Numerical Analysis, 1991 (Griffiths, D.F. and Watson, G.A., eds), Longman (London).Google Scholar
Calvo, M.P. and Sanz-Serna, J.M. (1993a), ‘Reasons for a failure. The integration of the two-body problem with a symplectic Runge–Kutta–Nystrom code with stepchanging facilities’, in Equadiff-91 (Perello, C., Simo, C. and de Sola-Marales, J., eds), World Scientific (Singapore).Google Scholar
Calvo, M.P. and Sanz-Serna, J.M. (1993b), ‘The development of variable-step symplectic integrators, with applications to the two-body problem’, SIAM J. Sci. Comput. 14, 936–952.CrossRefGoogle Scholar
Carr, J. (1982), Applications of Centre Manifold Theory, Springer (New York).Google Scholar
Chow, S.N. and Palmer, K.J. (1989), ‘The accuracy of numerically computed orbits of dynamical systems’, in Equadiff 1989, Prague.Google Scholar
Chow, S.N. and Palmer, K.J. (1990a), ‘The accuracy of numerically computed orbits of dynamical systems in ℝk’ Georgia Inst.Tech. Report CDSNS90–28.Google Scholar
Chow, S.N. and Palmer, K.J. (1990b), ‘On the numerical computation of orbits of dynamical systems: the higher dimensional case’, Georgia Inst. Tech. Report CDSNS90–32.Google Scholar
Chow, S.N. and Van Vleck, E. (1993), ‘A shadowing lemma approach to global error analysis for initial value ODEs’, to appear in SIAM J. Sci. Stat. Comput.Google Scholar
Constantin, P., Foias, C., Nicolaenko, B. and Temam, R. (1989), Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Applied Mathematical Sciences, Springer (New York).CrossRefGoogle Scholar
Corless, R.M. (1992), ‘Defect-controlled numerical methods and shadowing for chaotic differential equations’, Physica D 60, 323–334.Google Scholar
Corless, R.M. and Corliss, G.F. (1991), ‘Rationale for guaranteed ODE defect control’, Argonne Nat. Lab. Preprint MCS-P273–1191.Google Scholar
Corless, R.M. and Pilyugin, S.Y. (1993), ‘Approximate and real trajectories for generic dynamical systems’, to appear in J. Math. Anal. Appl.Google Scholar
Crouziex, M. and Rappaz, J. (1990), Numerical Approximation in Bifurcation Theory, Masson–Springer (Paris).Google Scholar
Dahlquist, G. (1975), ‘Error analysis for a class of methods for stiff non-linear initial value problems’, in Numerical Analysis, Dundee 1975 (Watson, G.A., ed.), Springer (New York), 60–74.Google Scholar
Dahlquist, G. (1978), ‘G-stability is equivalent to A-stability’ BIT 18, 384–401.CrossRefGoogle Scholar
Dekker, K. and Verwer, J.G. (1984), Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland (Amsterdam).Google Scholar
Demengel, F. and Ghidaglia, J.M. (1989), ‘Time-discretization and inertial manifolds’, Math. Mod. Numer. Anal. 23, 395–404.CrossRefGoogle Scholar
Devaney, R.L. (1989), An Introduction to Chaotic Dynamical Systems, Addison-Wesley (New York).Google Scholar
Dieci, L. and Lorenz, J. (1993), ‘Computation of invariant tori by the method of characteristics’, to be submitted.Google Scholar
Dieci, L., Lorenz, J. and Russell, R.D. (1991), ‘Numerical calculation of invariant tori’, SIAM J. Sci. Stat. Comput. 12, 607–647.CrossRefGoogle Scholar
Dieci, L., Russell, R.D. and Van Vleck, E. (1993), ‘On the computation of Lyapunov exponents for continuous dynamical systems’, submitted to SIAM J. Numer. Anal.Google Scholar
Doan, H.T. (1985), ‘Invariant curves for numerical methods’, Quart. Appl. Math. 3, 385–393.Google Scholar
Doedel, E.J. and Kervenez, J.P. (1986), ‘AUTO: Software for continuation of bifurcation problems in ordinary differential equations’, Appl. Maths Tech. Rep., California Institute of Technology.Google Scholar
Drazin, P.G. (1992), Nonlinear Dynamics, Cambridge University Press (Cambridge).Google Scholar
Eden, A., Foias, C., Nicolaenko, B. and Temam, R. (1990), ‘Inertial sets for dissipative evolution equations’, IMA Preprint Series, #694.Google Scholar
Eirola, T. (1988), ‘Invariant curves for one-step methods’, BIT 28, 113–122.CrossRefGoogle Scholar
Eirola, T. (1989), ‘Two concepts for numerical periodic solutions of ODEs’, Appl. Math. Comput. 31, 121–131.Google Scholar
Eirola, T. (1993), ‘Aspects of backward error analysis of numerical ODEs’, J. Comput. Appl. Math. 45, 65–73.CrossRefGoogle Scholar
Eirola, T. and Nevanlinna, O. (1988), ‘What do multistep methods approximate?’, Numer. Math. 53, 559–569.CrossRefGoogle Scholar
Elliott, C.M. and Larsson, S. (1992), ‘Error estimates with smooth and non smooth data for a finite element method for the Cahn-Hilliard equation’, Math. Comput. 58, 603–630.CrossRefGoogle Scholar
Elliott, C.M. and Stuart, A.M. (1993), ‘Global dynamics of discrete semilinear parabolic equations.’ to appear in SIAM J. Numer. Anal.CrossRefGoogle Scholar
Elliott, C.M. and Stuart, A.M. (1994), ‘Error analysis for gradient groups with applications’, in preparation.Google Scholar
Enquist, B. (1969), ‘On difference equations approximating linear ordinary differential equations’, Report 21, Department of Computer Science, Uppsala University.Google Scholar
Fenichel, N. (1971), ‘Persistence and smoothness of invariant manifolds for flows’, Indiana Univ. Math. J. 21, 193–226.CrossRefGoogle Scholar
Foias, C., Sell, G. and Temam, R. (1988), ‘Inertial manifolds for nonlinear evolutionary equations’, J. Diff. Eqns 73, 309–353.CrossRefGoogle Scholar
Gacay, B.M. (1993), ‘Discretization and some qualitative properties of ordinary differential equations about equilibria’, Budapest University of Technology, Technical Report.Google Scholar
Griffiths, D.F., Sweby, P.K. and Yee, H.C. (1992), ‘On spurious asymptotic numerical solutions of explicit Runge–Kutta methods’, IMA J. Numer. Anal. 12, 319–338.CrossRefGoogle Scholar
Guckenheimer, J. and Holmes, P. (1983), Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer (New York).CrossRefGoogle Scholar
Hairer, E., Iserles, A. and Sanz-Serna, J.M. (1990), ‘Equilibria of Runge–Kutta methods’, Numer. Math. 58, 243–254.CrossRefGoogle Scholar
Hairer, E., Norsett, S.P. and Wanner, G. (1987), Solving Ordinary Differential Equations, Parts I & II, Springer (New York).Google Scholar
Hale, J.K. (1969), Ordinary Differential Equations, Wiley (New York).Google Scholar
Hale, J.K. (1988), Asymptotic Behaviour of Dissipative Systems, AMS Mathematical Surveys and Monographs 25, American Mathematical Society (Providence, RI).Google Scholar
Hale, J.K. and Kŏcak, H. (1991), Dynamics and Bifurcations, Springer (New York).Google Scholar
Hale, J.K., Magalhaes, L. and Oliva, W. (1984), An Introdcution to Infinite Dimensional Dynamical Systems, Springer (New-York).Google Scholar
Hale, J.K. and Raugel, G. (1989), ‘Lower semicontinuity of attractors of gradient systems and applications’, Annali di Mat. Pura. Applic. CLIV, 281–326.Google Scholar
Hale, J.K., Lin, X.-B. and Raugel, G. (1988), ‘Upper semicontinuity of attractors for approximations of semigroups and partial differential equations’, Math. Comput. 50, 89–123.CrossRefGoogle Scholar
Hammel, S., Yorke, J.A. and Grebogi, C. (1987), ‘Do numerical orbits of chaotic dynamical processes represent true orbits?’, J. Complexity 3, 136–145.CrossRefGoogle Scholar
Hammel, S., Yorke, J.A. and Grebogi, C. (1988), ‘Numerical orbits of chaotic processes represent true orbits’, Bull. AMS 19, 465–470.CrossRefGoogle Scholar
Hartman, P. (1982), Ordinary Differential Equations, Birkhauser (Boston).Google Scholar
Henry, D. (1981), Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, Springer (New York).CrossRefGoogle Scholar
Heywood, J.G. and Rannacher, R. (1986), ‘Finite element approximations of the Navier–Stokes problem. Part II: Stability of solutions and error estimates uniform in time’, SIAM J. Numer. Anal. 23, 750–777.CrossRefGoogle Scholar
Homburg, A.J., Osinga, H.M. and Vegter, G. (1993), ‘On the numerical computation of invariant manifolds’, University of Groningen Technical Report.Google Scholar
Hill, A.T. (1994), ‘Global dissipativity for multistep methods’, in preparation.Google Scholar
Hill, A.T. and Suli, E. (1993), ‘Upper semicontinuity of attractors for linear multistep methods approximating sectorial evolution equations’, submitted to Math. Comput.Google Scholar
Hill, A.T. and Suli, E. (1994), ‘Set convergence for discretizations of the attractor’, submitted to Numer. Math.Google Scholar
Hirsch, M.W. and Smale, S. (1974), Differential Equations, Dynamical Systems and Linear Algebra, Academic Press (London).Google Scholar
Hubert, E. (1993), ‘Computation of stable manifolds’, MSc Thesis, Imperial College (London).Google Scholar
Humphries, A.R. (1993), ‘Spurious solutions of numerical methods for initial value problems’, IMA J. Numer. Anal. 13, 262–290.CrossRefGoogle Scholar
Humphries, A.R. (1994), ‘Approximation of attractors and invariant sets by Runge–Kutta methods’, in preparation, 1994.Google Scholar
Humphries, A.R. and Stuart, A.M. (1994), ‘Runge–Kutta methods for dissipative and gradient dynamical systems’, to appear in SIAM J. Numer. Anal.CrossRefGoogle Scholar
Humphries, A.R., Jones, D.A. and Stuart, A.M. (1994), ‘Approximation of dissipative partial differential equations over long-time intervals’, in Numerical Analysis, Dundee, 1993 (Griffiths, D.F. and Watson, G.A., eds), Longman (New York).Google Scholar
Iserles, A. (1990), ‘Stability and dynamics of numerical methods for nonlinear ordinary differential equations’, IMA J. Numer. Anal. 10, 1–30.CrossRefGoogle Scholar
Iserles, A., Peplow, A.T. and Stuart, A.M. (1991) ‘A unified approach to spurious soluions introduced by time discretization. Part I: basic theory’, SIAM J. Numer. Anal. 28, 1723–1751.Google Scholar
Jones, D.A. and Stuart, A.M. (1993), ‘Attractive invariant manifolds under approximation’, submitted to J. Diff. EqnsGoogle Scholar
Kirchgraber, U. (1986), ‘Multi-step methods are essentially one-step methods’, Numer. Math. 48, 85–90.CrossRefGoogle Scholar
Kloeden, P. and Lorenz, J. (1986), ‘Stable attracting sets in dynamical systems and their one-step discretizations’, SIAM J. Numer. Anal. 23, 986–995.CrossRefGoogle Scholar
Kloeden, P. and Lorenz, J. (1989), ‘Liapunov stability and attractors under discretization’, in Differential Equations, Proceedings of the Equadiff Conference (Dafermos, C.M., Ladas, G. and Papanicolaou, G., eds), Marcel-Dekker (New York).Google Scholar
Kloeden, P. and Lorenz, J. (1990), ‘A note on multistep methods and attracting sets of dynamical systems’, Num. Math. 56, 667–673.CrossRefGoogle Scholar
Kloeden, P. and Palmer, K.J. (1994), ‘Chaotic numerics: the approximation and computation of complicated dynamics’, Proceedings of the Conference on Chaotic Numerics,Geelong, Australia,July 12th–16th 1993.CrossRefGoogle Scholar
Ladyzhenskaya, O. (1991), Attractors for Semigroups and Evolution Equations, Cambridge University Press (Cambridge).CrossRefGoogle Scholar
Lambert, J. (1991), Numerical Methods for Ordinary Differential Equations, Wiley (Chichester).Google Scholar
Larsson, S. (1989), ‘The long-time behaviour of finite element approximations of solutions to semilinear parabolic problems’, SIAM J. Numer. Anal. 26, 348–365.CrossRefGoogle Scholar
Larsson, S. (1992), ‘Non-smooth data error estimates with applications to the study of long-time behaviour of finite element solutions of semilinear parabolic problems’, Preprint, Chalmers University (Sweden).Google Scholar
Larsson, S. and Sanz-Serna, J.M. (1993), ‘The behaviour of finite element solutions of semilinear parabolic problems near stationary points’, submitted to SIAM J. Numer. Anal.Google Scholar
Liu, L., Moore, G. and Russell, R.D. (1993), ‘Computing connecting orbits between steady solutions’, Technical Report, Mathematics Department, Simon Fraser University.Google Scholar
Lord, G.J. and Stuart, A.M. (1994), ‘Existence and convergence of attractors and inertial manifolds for a finite difference approximation of the Ginzburg–Landau equation’, in preparation.Google Scholar
Lorenz, E.N. (1963), ‘Deterministic noperiodic flow’, J. Atmos. Sci. 20, 130–141.2.0.CO;2>CrossRefGoogle Scholar
Lorenz, J. (1994), ‘Convergence of invariant tori under numerical approximation’, in Proceedings of the Conference on Chaotic Numerics,Geelong, Australia,July 12th–16th 1993.Google Scholar
Mallet-Paret, J. and Sell, G.R. (1988), ‘Inertial manifolds for reaction-diffusion equations in higher space dimensions’, J. Amer. Math. Soc. 1, 805–864.CrossRefGoogle Scholar
Medved, M. (1991), Fundamentals of Dynamical Systems and Bifurcation Theory, Adam-Hilger (Bristol).Google Scholar
Mitchell, A.R. and Griffiths, D.F. (1986), ‘Beyond the linearized stability limit in nonlinear problems’, in Numerical Analysis (Griffiths, D.F. and Watson, G.A., eds), Pitman (Boston).Google Scholar
Moore, G. (1993), ‘Computation and paramterization of connecting and periodic orbits’, submitted to IMA J. Numer. Anal.Google Scholar
Moore, G. (1993), ‘Computation and paramterization of invariant curves and tori’, in preparation.Google Scholar
Murdoch, T. and Budd, C.J. (1990), ‘Convergent and spurious solutions of nonlinear elliptic equations’, IMA J. Numer. Anal. 12, 365–386.CrossRefGoogle Scholar
Newell, A.C. (1977), ‘Finite amplitude instabilities or partial difference schemes’, SIAM J. Appl. Math. 33, 133–160.CrossRefGoogle Scholar
Pliss, V. (1966), Nonlocal Problems in the Theory of Oscillations, Academic Press (New York).Google Scholar
Pliss, V.A. and Sell, G.R. (1991), ‘Perturbations of attractors of differential equations’, J. Diff. Eqns 92, 100–124.CrossRefGoogle Scholar
Pugh, C. and Shub, M. (1988), ‘Cr stability of periodic solutions and solution schemes’, Appl. Math. Lett. 1, 281–285.CrossRefGoogle Scholar
Rheinboldt, W.C. (1986), Numerical Analysis of Parameterised Nonlinear Equations, Wiley (New York).Google Scholar
Sanz-Serna, J.M. (1992a), ‘Symplectic integrators for Hamiltonian problems: an overview’, Acta Numerica, Cambridge University Press (Cambridge), 244–286.Google Scholar
Sanz-Serna, J.M. (1992b), ‘Numerical ordinary differential equations vs. dynamical systems’, in Proceedings of the IMA Conference on the Dynamics of Numerics and the Numerics of Dynamics, 1990 (Broomhead, D. and Iserles, A., eds), Cambridge University Press (Cambridge).Google Scholar
Sanz-Serna, J.M. and Larsson, S. (1993), ‘Shadows, chaos and saddles’, to appear in Appl. Numer. Math.CrossRefGoogle Scholar
Sanz-Serna, J.M. and Stuart, A.M. (1992), ‘A note on uniform in time error estimates for approximations to reaction-diffusion equations’, IMA J. Numer. Anal. 12, 457–462.CrossRefGoogle Scholar
Sauer, T. and Yorke, J.A. (1991), ‘Rigorous verification of trajectories for the computer simulation of dynamical systems’, Nonlinearity 4, 961–979.CrossRefGoogle Scholar
Stephens, A.B. and Shubin, G.R. (1987), ‘Multiple solutions and bifurcations of finite difference approximations to some steady problems of fluid mechanics.’ SIAM J. Sci. Stat. Comput. 2, 404–415.CrossRefGoogle Scholar
Stetter, H. (1973), ‘Analysis of Discretization Methods for Ordinary Differential Equations, Springer (New York).CrossRefGoogle Scholar
Stoffer, D. (1993), ‘General linear methods: connection to one-step methods and invariant curves’, Numer. Math. 64.CrossRefGoogle Scholar
Stoffer, D. (1994) ‘Averaging for almost identical maps and weakly attractive tori’, submitted.Google Scholar
Stuart, A.M. (1989a) ‘Nonlinear instability in dissipative finite difference schemes’, SIAM Rev. 31, 191–220.CrossRefGoogle Scholar
Stuart, A.M. (1989b), ‘Linear instability implies spurious periodic solutions’, IMA J. Numer. Anal. 9, 465–486.CrossRefGoogle Scholar
Stuart, A.M. (1991), ‘The global attractor under discretization’, in Continuation and Bifurcations: Numerical Techniques and Applications (Roose, D., De Dier, B. and Spence, A., eds), Kluwer (Dordrecht).Google Scholar
Stuart, A.M. and Humphries, A.R. (1992a), ‘Model problems in numerical stability theory for initial value problems’, submitted to SIAM Rev.Google Scholar
Stuart, A.M. and Humphries, A.R. (1992b), ‘The essential stability of local error control for dynamical systems’, submitted to SIAM J. Numer. Anal.Google Scholar
Temam, R. (1988), Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer (New York).CrossRefGoogle Scholar
Titi, E.S. (1991) ‘Un critère pour l'approximation des solutions périodiques des équations de Navier–Stokes’, C.R. Acad. Sci. Paris, 312, 41–43.Google Scholar
Van Veldhuizen, M. (1988), ‘Convergence results for invariant curve algorithms’, Math. Comput. 51, 677–697.CrossRefGoogle Scholar
Wiggins, S. (1990), Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer (New York).CrossRefGoogle Scholar
Yee, H. C., Griffiths, D. F. and Sweby, P. K. (1991), ‘Dynamical approach to the study of spurious steady-state numerical solutions for nonlinear differential equations, I: The dynamics of time discretization and its implications for algorithm development in CFD’, J. Comput. Phys. 97, 249–310.CrossRefGoogle Scholar
Yin-Yan, (1993), ‘Attractors and error estimates for discretizations of incompressible Navier–Stokes equations’, submitted to SIAM J. Numer. Anal.Google Scholar
Yoshizawa, T. (1966), Stability Theory by Lyapunov's Second Method, Publ. Math. Soc. Japan (Tokyo).Google Scholar