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A survey of shadowing methods for numerical solutions of ordinary differential equations. (English) Zbl 1069.65075
Summary: A shadow is an exact solution to a set of equations that remains close to a numerical solution for a long time. Shadowing can thus be used as a form of backward error analysis for numerical solutions to ordinary differential equations. This survey introduces the reader to shadowing with a detailed tour of shadowing algorithms and practical results obtained over the last 15 years.

MSC:
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
Software:
pchip; UNCMND
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References:
[1] Anosov, D.V., Geodesic flows and closed Riemannian manifolds with negative curvature, Proc. Steklov inst. math., 90, 1, (1967) · Zbl 0163.43604
[2] Ascher, U.M.; Mattheij, R.M.M.; Russell, R.D., Numerical solution of boundary value problems for ordinary differential equations, Prentice-Hall series in computational mathematics, (1988), Prentice-Hall Englewood Cliffs, NJ · Zbl 0666.65056
[3] Beyn, W.-J., On invariant closed curves for one-step methods, Numer. math., 51, 103-122, (1987) · Zbl 0617.65082
[4] Bowen, R., ω-limit sets for axiom A diffeomorphisms, J. differential equations, 18, 333, (1975) · Zbl 0315.58019
[5] Braun, M., Differential equations and their applications, (1983), Springer Berlin · Zbl 0528.34001
[6] Channell, P.J.; Scovel, C., Symplectic integration of Hamiltonian systems, Nonlinearity, 3, 231-259, (1990) · Zbl 0704.65052
[7] Chow, S.N.; Lin, X.B.; Palmer, K.J., A shadowing lemma for maps in infinite dimensions, (), 127-136
[8] Chow, S.-N.; Palmer, K.J., On the numerical computation of orbits of dynamical systems: the one-dimensional case, Dynamics differential equations, 3, 361-380, (1991) · Zbl 0729.34010
[9] Chow, S.-N.; Palmer, K.J., On the numerical computation of orbits of dynamical systems: the higher dimensional case, J. complexity, 8, 398-423, (1992) · Zbl 0769.58040
[10] Chow, S.-N.; Van Vleck, E.S., A shadowing lemma for random diffeomorphisms, Random comput. dynamics, 1, 2, 197-218, (1992) · Zbl 0791.58057
[11] Chow, S.N.; Van Vleck, E.S., Shadowing of lattice maps, (), 97-113 · Zbl 0811.58041
[12] Chow, S.-N.; Van Vleck, E.S., A shadowing lemma approach to global error analysis for initial value odes, SIAM J. sci. comput., 15, 4, 959-976, (1994) · Zbl 0818.65068
[13] Chow, S.N.; Van Vleck, E.S., Shadowing of lattice maps, (), 97-113 · Zbl 0811.58041
[14] ()
[15] Coomes, B.A., Shadowing orbits of ordinary differential equations on invariant submanifolds, Trans. amer. math. soc., 349, 1, 203-216, (1997) · Zbl 0866.34010
[16] Coomes, B.A.; Koçak, H.; Palmer, K.J., Periodic shadowing, (), 115-130 · Zbl 0808.34059
[17] Coomes, B.A.; Koçak, H.; Palmer, K.J., Shadowing orbits of ordinary differential equations, J. comput. appl. math., 52, 35-43, (1994) · Zbl 0813.34042
[18] Coomes, B.A.; Koçak, H.; Palmer, K.J., Rigorous computational shadowing of orbits of ordinary differential equations, Numer. math., 69, 401-421, (1995) · Zbl 0822.65048
[19] Coomes, B.A.; Koçak, H.; Palmer, K.J., A shadowing theorem for ordinary differential equations, Z. angew. math. phys., 46, 85-106, (1995) · Zbl 0826.58027
[20] Coomes, B.A.; Koçak, H.; Palmer, K.J., Long periodic shadowing, Numer. algorithms, 14, 55-78, (1997) · Zbl 0890.65086
[21] Corless, R.M., Continued fractions and chaos, Amer. math. monthly, 99, 3, 203-215, (1992) · Zbl 0758.58020
[22] Corless, R.M., Defect-controlled numerical methods and shadowing for chaotic differential equations, Physica D, 60, 323-334, (1992) · Zbl 0779.34035
[23] Corless, R.M., Error backward, (), 31-62 · Zbl 0809.65090
[24] Corless, R.M., What good are numerical simulations of chaotic dynamical systems?, Comput. math. appl., 28, 10-12, 107-121, (1994) · Zbl 0813.65101
[25] Corless, R.M., Continued fractions and chaos, (), Amer. math. monthly, 99, 3, 203-215, (1992), Reprinted from · Zbl 0758.58020
[26] Corless, R.M.; Corliss, G.F., Rationale for guaranteed ODE defect control, () · Zbl 0838.65085
[27] Dahlquist, G.; Björck, Å., Numerical methods, Prentice-Hall series in automatic computation, (1974), Prentice-Hall Englewood Cliffs, NJ
[28] Dawson, S.; Grebogi, C.; Sauer, T.; Yorke, J.A., Obstructions to shadowing when a Lyapunov exponent fluctuates about zero, Phys. rev. lett., 73, 14, 1927-1930, (1994)
[29] W.H. Enright, W.B. Hayes, Robust defect control for continuous rk-methods with high-order interpolants, submitted for publication
[30] Farmer, J.D.; Sidorowich, J.J., Optimal shadowing and noise reduction, Physica D, 47, 373-392, (1991) · Zbl 0729.65501
[31] Fryska, S.T.; Zohdy, M.A., Computer dynamics and the shadowing of chaotic orbits, Phys. lett. A, 166, 340-346, (1992)
[32] Golub, G.H.; Van Loan, C.F., Matrix computations, (1991), Johns Hopkins University Press Baltimore, MD
[33] Gonzalez, O.; Higham, D.J.; Stuart, A.M., On the qualitative properties of modified equations, IMA J. numer. anal., 19, 169-190, (1999) · Zbl 0935.34003
[34] Góra, P.; Boyarsky, A., Why computers like Lebesgue measure, Comput. math. appl., 16, 4, 321-329, (1988) · Zbl 0668.28008
[35] Grebogi, C.; Hammel, S.M.; Yorke, J.A.; Sauer, T., Shadowing of physical trajectories in chaotic dynamics: containment and refinement, Phys. rev. lett., 65, 13, 1527-1530, (1990) · Zbl 1050.37521
[36] Hadeler, K.P., Shadowing orbits and Kantorovich’s theorem, Numer. math., 73, 65-73, (1996) · Zbl 0862.58017
[37] Hairer, E.; Nørsett, S.P.; Wanner, G., Solving ordinary differential equations, (1993), Springer Berlin, Two volumes · Zbl 0789.65048
[38] Hammel, S.M.; Yorke, J.A.; Grebogi, C., Do numerical orbits of chaotic dynamical processes represent true orbits?, J. complexity, 3, 136-145, (1987) · Zbl 0639.65037
[39] Hammel, S.M.; Yorke, J.A.; Grebogi, C., Numerical orbits of chaotic dynamical processes represent true orbits, Bull. amer. math. soc., 19, 465-470, (1988) · Zbl 0689.58026
[40] W. Hayes, Efficient shadowing of high dimensional chaotic systems with the large astrophysical n-body problem as an example, Master’s Thesis, Dept. of Computer Science, University of Toronto, 1995
[41] Hayes, W., Shadowing-based reliability decay in softened n-body simulations, Astrophys. J. lett., 587, 59-62, (2003)
[42] Hayes, W., Shadowing high-dimensional Hamiltonian systems: the gravitational n-body problem, Phys. rev. lett., 90, 5, (2003)
[43] W. Hayes, K.R. Jackson, A fast shadowing algorithm for high dimensional ODE systems, 1996, unpublished · Zbl 1146.37042
[44] W.B. Hayes, Rigorous Shadowing of Numerical Solutions of Ordinary Differential Equations by Containment, Ph.D. Thesis, Department of Computer Science, University of Toronto, 2001, Available on the web as http://www.cs.toronto.edu/NA/reports.html#hayes-01-phd
[45] Hayes, W.B.; Jackson, K.R., Rigorous shadowing of numerical solutions of ordinary differential equations by containment, SIAM J. numer. anal., 41, 5, 1948-1973, (2003) · Zbl 1059.37013
[46] Kahaner, D.; Moler, C.; Nash, S., Numerical methods and software, Prentice-Hall series in computational mathematics, (1989), Prentice-Hall Englewood Cliffs, NJ · Zbl 0744.65002
[47] Merritt, D., Elliptical galaxy dynamics, Publ. astronom. soc. Pacific, 111, 756, 129-168, (1999)
[48] ()
[49] Merritt, D.; Valluri, M., Chaos and mixing in triaxial stellar systems, Astrophys. J., 471, 82-105, (1996)
[50] N.S. Nedialkov, Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation, Ph.D. Thesis, Department of Computer Science, University of Toronto, 1999 · Zbl 1130.65312
[51] Palmer, K.J., Exponential dichotomies, the shadowing lemma and transversal homoclinic points, () · Zbl 0676.58025
[52] Palmer, K.J., Shadowing in dynamical systems—theory and applications, (2000), Kluwer Academic Dordrecht · Zbl 0997.37001
[53] Palmer, K.J.; Stoffer, D., Rigorous verification of chaotic behaviour of maps using validated shadowing, Nonlinearity, 12, 1683-1698, (1999) · Zbl 0988.37041
[54] Pilyugin, S.Y., Shadowing in dynamical systems, (1999) · Zbl 0954.37014
[55] Quinlan, G.D.; Tremaine, S., Shadow orbits and the gravitational N-body problem, (), 143-148
[56] Quinlan, G.D.; Tremaine, S., On the reliability of gravitational N-body integrations, Monthly notices roy. astronom. soc., 259, 505-518, (1992)
[57] Sanz-Serna, J.M., Symplectic integrators for Hamiltonian problems: an overview, (), 243-286 · Zbl 0762.65043
[58] Sauer, T.; Grebogi, C.; Yorke, J.A., How long do numerical chaotic solutions remain valid?, Phys. rev. lett., 79, 1, 59-62, (7 July 1997)
[59] Sauer, T.; Yorke, J.A., Rigorous verification of trajectories for the computer simulation of dynamical systems, Nonlinearity, 4, 961-979, (1991) · Zbl 0736.58032
[60] Shadwick, B.A.; Bowman, J.C.; Morrison, P.J., Exactly conservative integrators, SIAM J. appl. math., 59, 3, 1112-1133, (1999) · Zbl 0946.65130
[61] R. Skeel, The meaning of molecular dynamics, 1996, unpublished, e-mail: skeel@cs.uiuc.edu
[62] Skeel, R.D., Integration schemes for molecular dynamics and related applications, (), 119-176 · Zbl 0938.65150
[63] Smale, S., Diffeomorphisms with many periodic points, (), 63-80 · Zbl 0142.41103
[64] Smale, S., Differentiable dynamical systems, Bull. amer. math. soc., 73, 747-817, (1967) · Zbl 0202.55202
[65] Struck, C., Galaxy splashes: the effects of collisions between gas-rich galaxy disks, (), 225-230
[66] Van Vleck, E.S., Numerical shadowing near hyperbolic trajectories, SIAM J. sci. comput., 16, 5, 1172-1189, (1995) · Zbl 0837.65088
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