Abstract. A generally applicable, automatic method for the efficient computation of a database of global dynamics of a multiparameter dynamical system is introduced. An outer approximation of the dynamics for each subset of the parameter range is computed using rigorous numerical methods and is represented by means of a directed graph. The dynamics is then decomposed into the recurrent and gradient-like parts by fast combinatorial algorithms and is classified via Morse decompositions. These Morse decompositions are compared at adjacent parameter sets via continuation to detect possible changes in the dynamics. The Conley index is used to study the structure of isolated invariant sets associated with the computed Morse decompositions and to detect the existence of certain types of dynamics. The power of the developed method is illustrated with an application to the two-dimensional, density-dependent, Leslie population model. An interactive visualization of the results of computations discussed in the paper can be accessed at the website http://chomp.rutgers.edu/database/, and the source code of the software used to obtain these results has also been made freely available.

ABSTRACT. In this paper we give a new definition of the chain recurrent set of a continuous map using finite spatial discretizations. This approach allows for an algorithmic construction of isolating blocks for Morse sets of Morse decompositions which approximate the chain recurrent set arbitrarily closely as well as discrete approximations of Conley's Lyapunov function. This is a natural framework in which to develop computational techniques for the analysis of qualitative dynamics including rigorous computer-assisted proofs.KEYWORDS. chain recurrence, Lyapunov function, Conley's decomposition theorem, algorithms, computation. AMS SUBJECT CLASSIFICATION. 37M99, 37C70.Conley's Fundamental Decomposition Theorem and its extension to Morse decompositions is a powerful tool in dynamical systems theory. However, the framework on which the standard theory is built is not does not lead naturally to an algorithmic or computational approach for the approximation of the chain recurrent set, i.e. generation of Morse decompositions or the approximation of a Lyapunov function for the gradient-like part of the system. One can approximate the chain recurrent set by the -chain recurrent set for finite > 0, but there are no algorithmic or computational techniques for computing this set directly.In this paper, we present an alternative approach based on finite discretizations and combinatorial multivalued maps. This approach has several advantages. The basic elements of the theory can be proved in a straightforward manner. Moreover, the methods are inherently combinatorial and hence algorithmic. The framework leads naturally to computational techniques for analyzing qualitative dynamics including rigorous computer-assisted proofs, see e.g. [10, 14, 4, 5].

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ABSTRACT. We describe the basic lattice structures of attractors and repellers in dynamical systems. The structure of distributive lattices allows for an algebraic treatment of gradient-like dynamics in general dynamical systems, both invertible and noninvertible. We separate those properties which rely solely on algebraic structures from those that require some topological arguments, in order to lay a foundation for the development of algorithms to manipulate these structures computationally.

A method is described for quantitatively analyzing the level of interconnectivity of solid-oxide fuel cell electrode phases. The method was applied to the three-dimensional microstructure of a Ni-Y2O3-stabilized ZrO2 (Ni-YSZ) anode active layer measured by focused ion beam scanning electron microscopy. Each individual contiguous network of Ni, YSZ, and porosity was identified and labeled according to whether it was contiguous with the rest of the electrode. It was determined that the YSZ phase was 100% connected, whereas at least 86% of the Ni and 96% of the pores were connected. Triple-phase boundary (TPB) segments were identified and evaluated with respect to the contiguity of each of the three phases at their locations. It was found that 11.6% of the TPB length was on one or more isolated phases and hence was not electrochemically active.