Computation of nonautonomous invariant and inertial manifolds
Tóm tắt
We derive a numerical scheme to compute invariant manifolds for time-variant discrete dynamical systems, i.e., nonautonomous difference equations. Our universally applicable method is based on a truncated Lyapunov–Perron operator and computes invariant manifolds using a system of nonlinear algebraic equations which can be solved both locally using (nonsmooth) inexact Newton, and globally using continuation algorithms. Compared to other algorithms, our approach is quite flexible, since it captures time-dependent, nonsmooth, noninvertible or implicit equations and enables us to tackle the full hierarchy of strongly stable, stable and center-stable manifolds, as well as their unstable counterparts. Our results are illustrated using a test example and are applied to a population dynamical model and the Hénon map. Finally, we discuss a linearly implicit Euler–Bubnov–Galerkin discretization of a reaction diffusion equation in order to approximate its inertial manifold.
Tài liệu tham khảo
Allgower, E., Georg, K.: Numerical continuation methods. An Introduction. Springer Series. In: Computational Mathematics 13. Springer, Berlin (1990)
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)
Beyn W.-J.: On the numerical approximation of phase portraits near stationary points. SIAM J. Numer. Anal. 24(5), 1095–1112 (1987)
Beyn W.-J., Lorenz J.: Center manifolds of dynamical systems under discretization. Numer. Funct. Anal. Optim. 9, 381–414 (1987)
Beyn W.-J., Kleß W.: Numerical Taylor expansion of invariant manifolds in large dynamical systems. Numer. Math. 80, 1–38 (1998)
Broer H., Osinga H., Vegter G.: Algorithms of computing normally hyperbolic invariant manifolds. Zeitschrift für angewandte Mathematik und Physik 48(3), 480–534 (1997)
Broer H., Hagen A., Vegter G.: Numerical continuation of normally hyperbolic invariant manifolds. Nonlinearity 20, 1499–1534 (2007)
Clarke, F.H.: Optimization and nonsmooth analysis. Classics in Applied Mathematics 5. SIAM, Philadelphia (1990)
Chepyzhov, V., Vishik, M.: Attractors for Equations of Mathematical Physics. Colloquium Publications 49. American Mathematical Society, Providence (2001)
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)
Dellnitz M., Hohmann A.: A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math. 75, 293–317 (1997)
Deuflhard, P.: Newton Methods For Nonlinear Problems. Affine Invariance And Adaptive Algorithms. Springer Series in Computational Mathematics 35. Springer, Berlin (2004)
Dorsselaer J., Lubich C.: Inertial manifolds of parabolic differential equations under higher-order discretization. IMA J. Numer. Anal. 18, 1–17 (1998)
Eirola T., Von Pfaler J.: Taylor expansions for invariant manifolds numerically. Numer. Math. 99(1), 25–46 (2004)
Fuming M., Küpper T.: Numerical calculation of invariant manifolds for maps. Numer. Linear Algebra Appl. 1(2), 141–150 (1994)
Garay, B.: Discretization and some qualitative properties of ordinary differential equations about equilibria. Acta Mathematica Universitatis Comenianae, LXII, pp. 249–275 (1993)
Guckenheimer J., Vladimirsky A.: A fast method for approximating invariant manifolds. SIAM J. Appl. Dyn. Syst. 3(3), 232–260 (2004)
Henderson M.E.: Computing invariant manifolds by integrating fat trajectories. SIAM J. Appl. Dyn. Syst. 4(4), 832–882 (2005)
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)
Jolly M.S., Rosa R.: Computation of non-smooth local centre manifolds. IMA J. Numer. Anal. 25, 698–725 (2005)
Jones D., Stuart A.: Attractive invariant manifolds under approximation. Inertial manifolds. J. Diff. Equ. 123, 588–637 (1995)
Kanat Camlibel M., Pang J.-S., Shen J.: Conewise linear systems: non-zeroness and observabiliy. SIAM J. Control Optim. 45(5), 1769–1800 (2006)
Keller S., Pötzsche C.: Integral manifolds under explicit variable time-step discretization. J. Diff. Equ. Appl. 12(3–4), 321–342 (2005)
Kelley, C.T.: Solving nonlinear equations with Newton’s method, Fundamentals of Algorithms 1. SIAM, Philadelphia (2003)
Koksch N., Siegmund S.: Pullback attracting inertial manifolds for nonautonomous dynamical systems. J. Dyn. Diff. Equ. 14, 889–941 (2002)
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)
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)
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)
Mifflin R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 959–972 (1977)
Moore G., Hubert E.: Algorithms for constructing stable manifolds of stationary solutions. IMA J. Numer. Anal. 19, 375–424 (1999)
Ombach, J.: Computation of the local stable and unstable manifolds. Universitatis Iagellonicae Acta Mathematica, XXXII, pp. 129–136 (1995)
Pötzsche C., Siegmund S.: C m-smoothness of invariant fiber bundles. Topol. Methods Nonlinear Anal. 24, 107–146 (2004)
Pötzsche C.: Attractive invariant fiber bundles. Appl. Anal. 86, 687–722 (2007)
Pötzsche C.: Discrete inertial manifolds. Mathematische Nachrichten 281(6), 847–878 (2008)
Pötzsche C., Rasmussen M.: Taylor approximation of invariant fiber bundles. Nonlinear Anal. (TMA) 60(7), 1303–1330 (2005)
Pötzsche, C., Rasmussen, M.: Computation of integral manifolds for Carathéodory differential equations. IMA J. Numer. Math. doi:10.1093/imanum/drn059
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)
Robinson J.C.: Computing inertial manifolds. Discrete Contin. Dyn. Syst. 8(4), 815–833 (2002)
Sell, G.R., You, Y.: Dynamics of evolutionary equations, Applied Mathematical Sciences, vol. 143. Springer, Berlin (2002)
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)
Xu H., Chang X.: Approximate Newton methods for nonsmooth equations. J. Optim. Theor. Appl. 93(2), 373–394 (1997)