Computation of nonautonomous invariant and inertial manifolds. (English) Zbl 1169.65116
The authors consider a numerical procedure in order to compute the invariant manifolds for semi-linear nonautonomous difference equations as well as for semi-linear ordinary differential equations and partial differential equations, which by discretization methods reduce to such discrete dynamical systems. The procedure uses a truncated Lyapunov-Perron operator and reduces the computation to systems of nonlinear algebraic equations.
The authors provide some details on Newton-like methods used to solve both locally and globally such systems. They illustrate the effectiveness of the method by analyzing some clasical examples (a population dynamical model, Henon map and scalar Chafee-Infante equation with time-dependent coefficients).
The authors provide some details on Newton-like methods used to solve both locally and globally such systems. They illustrate the effectiveness of the method by analyzing some clasical examples (a population dynamical model, Henon map and scalar Chafee-Infante equation with time-dependent coefficients).
Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca)
MSC:
| 65P40 | Numerical nonlinear stabilities in dynamical systems |
| 39A11 | Stability of difference equations (MSC2000) |
| 65L12 | Finite difference and finite volume methods for ordinary differential equations |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 37D10 | Invariant manifold theory for dynamical systems |
| 65H10 | Numerical computation of solutions to systems of equations |
| 37M25 | Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.) |
Keywords:
semi-linear nonautonomous difference equations; invariant manifolds; fiber bundles; Lyapunov-Perron operator; systems of nonlinear algebraic equations; inexact Newton methods; discrete dynamical systems; population dynamical model; Henon map; Chafee-Infante equationReferences:
| [1] | Allgower, E., Georg, K.: Numerical continuation methods. An Introduction. Springer Series. In: Computational Mathematics 13. Springer, Berlin (1990) · Zbl 0717.65030 |
| [2] | Aulbach, B., Rasmussen, M., Siegmund, S.: Invariant manifolds as pullback attractors of nonautonomous difference equations. In: Proceedings of the Eighth International Conference of Difference Equations and Application, Brno, Czech Republic, 2003, pp 23–37. Chapman & Hall/CRC, Boca Raton (2005) · Zbl 1125.37009 |
| [3] | Beyn W.-J.: On the numerical approximation of phase portraits near stationary points. SIAM J. Numer. Anal. 24(5), 1095–1112 (1987) · Zbl 0632.65083 · doi:10.1137/0724072 |
| [4] | Beyn W.-J., Lorenz J.: Center manifolds of dynamical systems under discretization. Numer. Funct. Anal. Optim. 9, 381–414 (1987) · Zbl 0597.34047 · doi:10.1080/01630568708816239 |
| [5] | Beyn W.-J., Kleß W.: Numerical Taylor expansion of invariant manifolds in large dynamical systems. Numer. Math. 80, 1–38 (1998) · Zbl 0909.65044 · doi:10.1007/s002110050357 |
| [6] | Broer H., Osinga H., Vegter G.: Algorithms of computing normally hyperbolic invariant manifolds. Zeitschrift für angewandte Mathematik und Physik 48(3), 480–534 (1997) · Zbl 0872.34030 · doi:10.1007/s000330050044 |
| [7] | Broer H., Hagen A., Vegter G.: Numerical continuation of normally hyperbolic invariant manifolds. Nonlinearity 20, 1499–1534 (2007) · Zbl 1123.34036 · doi:10.1088/0951-7715/20/6/011 |
| [8] | Clarke, F.H.: Optimization and nonsmooth analysis. Classics in Applied Mathematics 5. SIAM, Philadelphia (1990) · Zbl 0696.49002 |
| [9] | Chepyzhov, V., Vishik, M.: Attractors for Equations of Mathematical Physics. Colloquium Publications 49. American Mathematical Society, Providence (2001) · Zbl 1011.35104 |
| [10] | Dellnitz, M., Froyland, G., Junge, O.: The Algorithms behind GAIO–Set Oriented Numerical Methods for Dynamical Systems. In: Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pp 145–174. Springer, Heidelberg (2001) · Zbl 0998.65126 |
| [11] | Dellnitz M., Hohmann A.: A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math. 75, 293–317 (1997) · Zbl 0883.65060 · doi:10.1007/s002110050240 |
| [12] | Deuflhard, P.: Newton Methods For Nonlinear Problems. Affine Invariance And Adaptive Algorithms. Springer Series in Computational Mathematics 35. Springer, Berlin (2004) · Zbl 1056.65051 |
| [13] | Dorsselaer J., Lubich C.: Inertial manifolds of parabolic differential equations under higher-order discretization. IMA J. Numer. Anal. 18, 1–17 (1998) · Zbl 0908.65007 · doi:10.1093/imanum/18.1.1 |
| [14] | Eirola T., Von Pfaler J.: Taylor expansions for invariant manifolds numerically. Numer. Math. 99(1), 25–46 (2004) · Zbl 1063.65139 · doi:10.1007/s00211-004-0537-6 |
| [15] | Fuming M., Küpper T.: Numerical calculation of invariant manifolds for maps. Numer. Linear Algebra Appl. 1(2), 141–150 (1994) · Zbl 0837.65054 · doi:10.1002/nla.1680010205 |
| [16] | Garay, B.: Discretization and some qualitative properties of ordinary differential equations about equilibria. Acta Mathematica Universitatis Comenianae, LXII, pp. 249–275 (1993) · Zbl 0822.34042 |
| [17] | Guckenheimer J., Vladimirsky A.: A fast method for approximating invariant manifolds. SIAM J. Appl. Dyn. Syst. 3(3), 232–260 (2004) · Zbl 1059.37019 · doi:10.1137/030600179 |
| [18] | Henderson M.E.: Computing invariant manifolds by integrating fat trajectories. SIAM J. Appl. Dyn. Syst. 4(4), 832–882 (2005) · Zbl 1090.37012 · doi:10.1137/040602894 |
| [19] | Homburg A.J., Osinga H.M., Vegter G.: On the computation of invariant manifolds of fixed points. Zeitschrift für angewandte Mathematik und Physik 46(2), 171–187 (1995) · Zbl 0827.58043 · doi:10.1007/BF00944751 |
| [20] | Jolly M.S., Rosa R.: Computation of non-smooth local centre manifolds. IMA J. Numer. Anal. 25, 698–725 (2005) · Zbl 1076.76062 · doi:10.1093/imanum/dri013 |
| [21] | Jones D., Stuart A.: Attractive invariant manifolds under approximation. Inertial manifolds. J. Diff. Equ. 123, 588–637 (1995) · Zbl 0840.35041 · doi:10.1006/jdeq.1995.1174 |
| [22] | Kanat Camlibel M., Pang J.-S., Shen J.: Conewise linear systems: non-zeroness and observabiliy. SIAM J. Control Optim. 45(5), 1769–1800 (2006) · Zbl 1126.93030 · doi:10.1137/050645166 |
| [23] | Keller S., Pötzsche C.: Integral manifolds under explicit variable time-step discretization. J. Diff. Equ. Appl. 12(3–4), 321–342 (2005) · Zbl 1104.34037 · doi:10.1080/10236190500489533 |
| [24] | Kelley, C.T.: Solving nonlinear equations with Newton’s method, Fundamentals of Algorithms 1. SIAM, Philadelphia (2003) · Zbl 1031.65069 |
| [25] | Koksch N., Siegmund S.: Pullback attracting inertial manifolds for nonautonomous dynamical systems. J. Dyn. Diff. Equ. 14, 889–941 (2002) · Zbl 1025.34042 · doi:10.1023/A:1020768711975 |
| [26] | Krauskopf B., Osinga H.M., Doedel E.J., Henderson M.E., Guckenheimer J., Vladimirsky A., Dellnitz M., Junge O.: A survey of methods for computing (un)stable manifolds of vector fields. Int. J. Bifurcation Chaos 15(3), 763–791 (2005) · Zbl 1086.34002 · doi:10.1142/S0218127405012533 |
| [27] | Kuang Y., Cushing J.: Global stability in a nonlinear difference-delay equation model of flour beetle population growth. J. Diff. Equ. Appl. 2(1), 31–37 (1996) · Zbl 0862.39005 · doi:10.1080/10236199608808040 |
| [28] | Kummer, B.: Newton’s method for non-differentiable functions. In: Guddad, e.a.J. (ed.), Advances in Mathematical Optimization. Mathematical Research, pp. 114–125. Akademie, Berlin (1988) · Zbl 0662.65050 |
| [29] | Mifflin R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 959–972 (1977) · Zbl 0376.90081 · doi:10.1137/0315061 |
| [30] | Moore G., Hubert E.: Algorithms for constructing stable manifolds of stationary solutions. IMA J. Numer. Anal. 19, 375–424 (1999) · Zbl 0947.65135 · doi:10.1093/imanum/19.3.375 |
| [31] | Ombach, J.: Computation of the local stable and unstable manifolds. Universitatis Iagellonicae Acta Mathematica, XXXII, pp. 129–136 (1995) · Zbl 0831.58044 |
| [32] | Pötzsche C., Siegmund S.: C m -smoothness of invariant fiber bundles. Topol. Methods Nonlinear Anal. 24, 107–146 (2004) · Zbl 1075.39014 |
| [33] | Pötzsche C.: Attractive invariant fiber bundles. Appl. Anal. 86, 687–722 (2007) · Zbl 1127.37023 · doi:10.1080/00036810701354987 |
| [34] | Pötzsche C.: Discrete inertial manifolds. Mathematische Nachrichten 281(6), 847–878 (2008) · Zbl 1146.39030 · doi:10.1002/mana.200710645 |
| [35] | Pötzsche C., Rasmussen M.: Taylor approximation of invariant fiber bundles. Nonlinear Anal. (TMA) 60(7), 1303–1330 (2005) · Zbl 1070.39023 · doi:10.1016/j.na.2004.10.019 |
| [36] | Pötzsche, C., Rasmussen, M.: Computation of integral manifolds for Carathéodory differential equations. IMA J. Numer. Math. doi: 10.1093/imanum/drn059 · Zbl 1201.65218 |
| [37] | Qi, L., Sun, D.: A survey of some nonsmooth equations and smoothing Newton methods. In: Eberhard, e.a. A. (ed.), Progress in Optimization. Applied Optimization 30, pp. 121–146. Kluwer, Dordrecht (1999) · Zbl 0957.65042 |
| [38] | Robinson J.C.: Computing inertial manifolds. Discrete Contin. Dyn. Syst. 8(4), 815–833 (2002) · Zbl 1032.34060 · doi:10.3934/dcds.2002.8.815 |
| [39] | Sell, G.R., You, Y.: Dynamics of evolutionary equations, Applied Mathematical Sciences, vol. 143. Springer, Berlin (2002) · Zbl 1254.37002 |
| [40] | Simó, C.: On the numerical and analytical approximation of invariant manifolds. In: Benest, D., Froeschlé, C. (eds.) Les Methodes Modernes de la Mechanique Céleste, pp. 285–329. Coutelas (1989) |
| [41] | Xu H., Chang X.: Approximate Newton methods for nonsmooth equations. J. Optim. Theor. Appl. 93(2), 373–394 (1997) · Zbl 0899.90150 · doi:10.1023/A:1022606224224 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
