Topological and Computational Approaches

to Dynamical Systems and Applications

Date: February 8 (Thu) 13:00 - February 10 (Sat) 17:00

Place: Feb.8th,9th Bldg. No.3, Room #102 (Ground floor),

Feb.10th Bldg. No.21, Room #203 (2nd floor),

Fukakusa Campus, Ryukoku University

Access ( English / Japanese ); Campus Map ( English / Japanese )

Contact: Hiroe Oka ( oka@rins.ryukoku.ac.jp )

Department of Applied Mathematics and Informatics,

Ryukoku University

TEL 077-543-7518 / FAX 077-543-7524

This workshop is a part of the project “Analytical and Computational Methods in

Mathematical Sciences .” supported by Joint Research Center for Science and

Technology of Ryukoku University.

<Program>

February 8th (Thu)

1:00-2:00 Konstantin Mischaikow (Rutgers, USA)

Building a Database for the Global Dynamics of Multi-Parameter Systems

2:10-3:10 Toshiyuki Ogawa (Osaka, Japan)

Wave bifurcations in three-component RD systems

3:40-4:40 Pawel Pilarczyk (Kyoto, Japan)

Continuation of Morse Decompositions - an Algorithmic Approach

4:50-5:50 Yasuaki Hiraoka (Hiroshima, Japan)

Rigorous numerics for the existence of homoclinic orbits via exponential dichotomy

February 9th (Fri)

10:00-11:00 Tomas Gedeon (Montana, USA)

The Conley index for fast-slow systems

11:15-12:15 Atsushi Mochizuki (NIBB, Japan)

What is the origin of cell diversity?

1:30-2:30 Tom Wanner (George Mason, USA)

Homological analysis of complex transient patterns via discretization

2:40-3:40 Yasumasa Nishiura (Hokkaido, Japan)

Dynamics of traveling pulses in heterogeneous media

4:10-5:10 Marcio Gameiro (Rutgers, USA)

Rigorous Continuation of Equilibria of PDEs Over Long Parameter Ranges

5:20-6:20 Takashi Teramoto (Chitose, Japan)

Topological Computation of Triply Periodic Morphology in Polymer Mixtures

- Party -

February 10th (Sat)

10:00-11:00 Bill Kalies (Florida, USA)

Computing Global Decompositions of Dynamical Systems

11:15-12:15 Vidit Nanda (Rutgers, USA)

Computing the homology of maps with high confidence

2:00-3:00 Marian Mrozek (Krakow, Poland)

The coreduction homology algorithm

3:15-4:15 Zin Arai (Kyoto, Japan)

Hyperbolicity and Monodromy of Dynamical Systems

4:30-5:30 Rob Vandervorst (Amsterdam, The Netherlands)

TBA

<Abstract>

Pawel Pilarczyk (Kyoto, Japan)

Continuation of Morse Decompositions - an Algorithmic Approach

Abstract: We introduce an automated method for the analysis of continuation and bifurcations of Morse decompositions in discrete dynamical systems. In this method, the phase space is subdivided into boxes with respect to a uniform rectangular lattice, and the dynamical system is represented in terms of a multivalued map defined on the boxes. This map is computed in a rigorous way (with interval arithmetic). A rigorous outer approximation of the Morse decomposition is obtained using fast graph algorithms, and it consists of isolating neighborhoods of Morse sets. If one considers a parameter-dependent dynamical system, then by comparing the results obtained for different intervals of parameters, our method allows one to prove that some Morse decompositions can be continued into those computed for adjacent paramters. The effectiveness of the method is illustrated with a sample dynamical system, namely, a nonlinear density dependent Leslie population model. This is joint work with Hiroshi Kokubu, Hiroe Oka, and Zin Arai.

Yasuaki Hiraoka (Hiroshima, Japan)

Rigorous numerics for the existence of homoclinic orbits via exponential dichotomy

Abstract: In this talk, a new rigorous numerical approach to prove the existence of homoclinic orbits in vector fields is presented. The approach uses exponential dichotomy properties to rigorously calculate Melnikov integrals.

Tomas Gedeon (Montana, USA)

The Conley index for fast-slow systems

Abstract: We use Conley index theory to develop a general method used to prove existence of periodic and heteroclinic orbits in a singularly perturbed system of ODE's. The key new idea is the observation that the Conley index in fast-slow systems has a cohomological product structure. The factors in this product are slow index, which captures information about the flow in the slow direction transverse to the slow flow, and the fast index, which is analogous to the Conley index for fast-slow systems with one-dimensional slow flow. This is a joint work with H. Kokubu, H. Oka and K. Mischaikow.

Atsushi Mochizuki (NIBB, Japan)

What is the origin of cell diversity?

Abstract: The dynamical behavior of gene activities should strongly depend on structure of gene regulatory networks. I introduce a novel idea to determine diversity of possible gene activities from topological structure of gene regulatory networks. The basic idea is very simple: activity of each gene should be a function of controlling genes. Thus each gene should always show the same expression activity if the activities of the controlling genes are the same. Based on this idea, the condition of compatibility between different steady states of gene activity is derived, and then the possible maximum diversity of steady states was determined. By extending the analysis, some general properties were proved showing the relations between the topology of regulatory networks and the diversity of gene activities. For example, loop structures in regulatory networks are necessary for increasing the diversity of gene activity. On the other hand, connected loops may not contribute for the diversity if they share the same genes. I applied the method to the actual gene regulatory network responsible for early development of sea urchin. A set of important genes responsible for generating diversities of gene activities was derived based on the idea of steady states compatibility.

Tom Wanner (George Mason, USA)

Homological analysis of complex transient patterns via discretization

Abstract: Many partial differential equation models arising in applications generate complex time-evolving patterns which are hard to quantify due to the lack of any underlying regular structure. Such models may include some element of stochasticity which leads to variations in the detail structure of the patterns and forces one to concentrate on rougher common geometric features, such as their homology. In practice, the patterns are approximated using discretizations, which raises the question of the accuracy of the resulting homology computation. In this talk, I will present a probabilistic approach which gives insight into the suitability of this method in the context of random fields. We will obtain explicit probability estimates for the correctness of the homology computations, which in turn yield a-priori bounds for the suitability of certain grid sizes. In addition, we present a computational approach to homology validation in the above setting, and apply our results to certain stochastic partial differential equations arising in materials science.

Marcio Gameiro (Rutgers, USA)

Rigorous Continuation of Equilibria of PDEs Over Long Parameter Ranges

Abstract: A standard numerical method for determining the equilibria of a parameterized family of differential equations is the so called predictor-corrector continuation method. In the case of PDEs the computations must be performed on some finite dimensional approximation of the PDE. This raises the question of the validity of the results. We present a numerical method that uses the classical predictor-corrector continuation method and ideas from rigorous computations to produce validated results. We present examples for the Swift-Hohenberg PDE, and show how to use these techniques to compute validated equilibria for very large ranges of parameter values.

Bill Kalies (Florida, USA)

Computing Global Decompositions of Dynamical Systems

Abstract: Conley Theory has been an important tool for the topological analysis of dynamical systems. Recent results provide an algorithmic framework in which computational tools have been developed for the study of the dynamics of continuous maps. Using software to compute the homology of maps, the Conley index theory can be used not only to obtain coarse descriptions of global dynamics, but also to give rigorous proofs of the existence of certain localized dynamical structures such as periodic orbits, connecting orbits, and chaotic sets.

Vidit Nanda (Rutgers, USA)

Computing the homology of maps with high confidence

Abstract: Given two Riemannian submanifolds of Euclidean space X and Y with homology groups H_*(X) and H_*(Y ) respectively, and a lipschitz-continuous map f : X -> Y , we define the homology of f to be the homomorphism f_* : H_*(X) -> H_*(Y ) induced by f. Weinberger and Smale have provided bounds on the number of points we need to sample (iid, uniform measure) from X and Y to recover H_*(X) and H_*(Y) with high confidence. We use these bounds to provide an algorithm that computes f_* with high confidence.

Marian Mrozek (Krakow, Poland)

The coreduction homology algorithm

Abstract: Efficient homology computations constitute an important part in many approaches to rigorous numerics of dynamical systems. We present a new algorithm for computing homology of cubical sets. The algorithm is based on homology theory of representable sets and elementary coreductions, a concept dual to elementary reductions. The algorithm may be extended to compute homology of inclusions, projections and general continuous maps. The numerical experiments confirm that the algorithm is very fast.