# Hauptseminar "Numerik Dynamischer Systeme"

*Prof. Junge*Das Seminar schließt an die gleichnamige Vorlesung im WS 06/07 an. Es sollen neuere Arbeiten gelesen werden, die sich mit der numerischen Approximation der Langzeitdynamik von dynamischen Systemen beschäftigen: Approximation invarianter Mengen, fast invarianter Mengen, aber auch von Kenngrößen aus der Ergodentheorie. Interessenten schicken bitte eine email mit dem gewünschten Vortragsthema aus der untigen Liste und einem Wunschtermin aus der folgenden Liste an jungema.tum.de. Die Vorträge werden dann nach Eingangsdatum der email verteilt.

### Termine

Di, 14:15-15:45, MI 02.08.011 noch verfügbar: 26.6., 17.7. vergeben:Termin | Thema | Vortragende(r) | ||
---|---|---|---|---|

17.04.07 | Grüne, L.: Attraction rates, robustness and discretization of attractors | Harald Schmid | ||

08.05.07 | Siegmund, S.; Taraba, P.: Approximation of box dimension of attractors using the subdivision algorithm | Alexander Bernhard | ||

05.06.07 | Murray, R.: Optimal partition choice for invariant measure approximation for one-dimensional maps | Martin Ronis | ||

12.06.07 | Dellnitz, M.; Hessel-von Molo, M.: Topological Entropy and Almost Invariant Sets | Frederic Moresmau | ||

19.06.07 | Beyn, W.-J.; Rieger, J.: Numerical Fixed Grid Methods for Differential Inclusions | Manuel Villegas Caballero | ||

10.07.07 | I. Horenko, E. Dittmer, A. Fischer, and Ch. Schütte: Automated Model Reduction for Complex Systems exhibiting Metastability | Thomas Wegner |

### Literatur

- Dellnitz, M.; Hessel-von Molo, M.:

**Topological Entropy and Almost Invariant Sets**,

*Submitted to Physica D, 2006.*

Over the last years, several numerical techniques have been proposed for the approximation of almost invariant sets. In this article we show that in this context, also the computation of the topological entropy with respect to certain partitions of state space can be useful. In fact, as a main result we prove that typically a large entropy can be expected when the boundaries of sets of the partition are near boundaries of exit sets from almost invarioent sets. We illustrate our results by numerical examples. - Murray, R.:

**Optimal partition choice for invariant measure approximation for one-dimensional maps**,

*Nonlinearity 17(5):1623-1644, 2004.*

The problem of finding absolutely continuous invariant measures (ACIMs) for a dynamical system can be formulated as a fixed point problem for a Markov operator (the Perron–Frobenius operator). This is an infinite-dimensional problem. Ulam's method replaces the Perron–Frobenius operator by a sequence of finite rank approximations whose fixed points are relatively easy to compute numerically. This paper concerns the optimal choice of Ulam approximations for one-dimensional maps; an adaptive partition selection is used to tailor the approximations to the structure of the invariant measure. The main idea is to select a partition which equally distributes the square root of the derivative of the invariant density amongst the bins of the partition. The results are illustrated for the logistic map where the ACIMs may have inverse square root singularities in their density functions. O(log n/n) convergence rates can be expected, whereas a non-adaptive algorithm yields O(n-1/2) at best. Studying the convergence of the adaptive algorithm allows an estimate to be made of the measure of the Jakobson parameter set (those logistic maps which admit an ACIM). - Bose, Ch.J., Murray, R.:

**The exact rate of approximation in Ulam's method**,

*Discrete and Continuous Dynamical Systems, 7(1):219-235, 2001.*

This paper investigates the exact rate of convergence in Ulam’s method: a well-known discretization scheme for approximating the invariant density of an absolutely continuous invariant probability measure for piecewise expanding interval maps. It is shown by example that the rate is no better than O( log n / n ), where n is the number of cells in the discretization. The result is in agreement with upper estimates previously established in a number of general settings, and shows that the conjectured rate of O( 1/n ) cannot be obtained, even for extremely regular maps. - Siegmund, S.; Taraba, P.:

**Approximation of box dimension of attractors using the subdivision algorithm**,

*Dyn. Syst. 21, 1-24, 2006.*

In this paper we use the subdivision algorithm to approximate the box dimension of attractors of dynamical systems. Although in theory the subdivision algorithm provides a covering of the attractor with boxes of arbitrarily small diameter, in practice we have to overcome two obstructions: 1. ensure that the covering is (almost) minimal and 2. enhance the speed of convergence to the box dimension. We solve both problems and apply our results to the Hénon, Lorenz, Rössler und Chua attractors. The method suggested in this paper uses information from several subdivision steps and converges to the box dimension much faster than the expression in the definition of the box dimension which uses only one covering of the attractor with boxes of a prescribed diameter. - Grüne, L.:

**Attraction rates, robustness and discretization of attractors**,

*SIAM Journal on Numerical Analysis, 41, 2096-2113, 2003.*

We investigate necessary and sufficient conditions for the convergence of attractors of discrete time dynamical systems induced by numerical one-step approximations of ordinary differential equations (ODEs) to an attractor for the approximated ODE. We show that both the existence of uniform attraction rates (i.e., uniform speed of convergence towards the attractors) and uniform robustness with respect to perturbations of the numerical attractors are necessary and sufficient for this convergence property. In addition, we can conclude estimates for the rate of convergence in the Hausdorff metric. - Froyland, G.:

**On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps**,

*Discrete and Continuous Dynamical Systems, Series A. To appear, 17(3), 2007.*

Perron-Frobenius operators and their eigendecompositions are increasingly being used as tools of global analysis for higher dimensional systems. The numerical computation of large, isolated eigenvalues and their corresponding eigenfunctions can reveal important persistent structures such as almost-invariant sets, however, often little can be said rigorously about such calculations. We attempt to explain some of the numerically observed behaviour by constructing a hyperbolic map with a Perron-Frobenius operator whose eigendecomposition is representative of numerical calculations for hyperbolic systems. We explicitly construct an eigenfunction associated with an isolated eigenvalue and prove that a special form of Ulam's method well approximates the isolated spectrum and eigenfunctions of this map. - Ban, H.; Kalies, W.D.:

**A Computational Approach to Conley’s Decomposition Theorem**,

*Journal of Computational and Nonlinear Dynamics 1, pp. 312-319, 2006.*

The discrete dynamics generated by a continuous map can be represented combinatorially by an appropriate multivalued map on a discretization of the phase space such as a cubical grid or triangulation. In this paper we provide explicit algorithms and computational complexity bounds for computing dynamical structures for the resulting combinatorial multivalued maps. Specifically we focus on the computation attractor-repeller pairs and Lyapunov functions for Morse decompositions. These discrete Lyapunov functions are weak Lyapunov functions and well-approximate a continuous Lyapunov function for the underlying map. - Henderson, M.E.:

**Computing Invariant Manifolds by Integrating Fat Trajectories**,

*SIAM J. Appl. Dyn. Sys. 4(4), 832--882, 2005.*

We present a new method of computing a well distributed set of points on k-dimensional manifolds which are invariant under a flow. The method uses on chains of local approximations along trajectories (fat trajectories) to cover the manifold with well spaced points. Points between two diverging fat trajectories are interpolated by either interpolating over a certain dual simplex, or by solving a two point boundary value problem. We derive formulae for the evolution of a second local approximation of the invariant manifold along a trajectory, show that interpolation points in the cleft where k (the dimension of the manifold) trajectories diverge will exist, and apply the method to the stable manifold of the origin in the "standard" Lorenz system. - I. Horenko, E. Dittmer, A. Fischer, and Ch. Schütte:

**Automated Model Reduction for Complex Systems exhibiting Metastability**,

*Mult. Mod. Sim. 5 (3), pp. 802-827, 2006.*

We present a novel method for the identification of the most important metastable states of a system with complicated dynamical behavior from time series information. The novel approach represents the effective dynamics of the full system by a Markov jump process between metastable states, and the dynamics within each of these metastable states by rather simple stochastic differential equations (SDEs). Its algorithmic realization exploits the concept of Hidden Markov Models (HMMs) with output behavior given by SDEs. The numerical effort of the method is linear in the length of the given time series, and quadratic in terms of the number of metastable states. The performance of the resulting method is illustrated by numerical tests and by application to molecular dynamics time series of a trialanine molecule. - Beyn, W.J; Kolezhuk, V.S.; Pilyugin, S.:

**Convergence of discretized attractors for parabolic equations on the line**,

*Preprint, Universität Bielefeld, 2004.*

We show that, for a semilinear parabolic equation on the real line satisfying a dissipativity condition, global attractors of time-space discretizations converge (with respect to the Hausdorff semi-distance) to the attractor of the continuous system as the discretization steps tend to zero. The attractors considered correspond to pairs of function spaces (in the sense of Babin-Vishik) with weighted and locally uniform norms (taken from Mielke-Schneider) used for both the continuous and the discrete system. - Beyn, W.-J.; Rieger, J.:

**Numerical Fixed Grid Methods for Differential Inclusions**,

*Preprint Universität Bielefeld, 2006.*

Numerical methods for initial value problems for differential inclusions usually require a discretization of time as well as of the set valued right hand side. In this paper, two numerical fixed grid methods for the approximation of the full solution set are proposed and analyzed. Convergence results are proved which show the combined influence of time and (phase) space discretization.