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In this project, a generally applicable, automatic method for the efficient computation of a database of global dynamics of a multiparameter dynamical system is developed. 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. | |
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Phase separation processes in compound materials can produce intriguing and complicated patterns. Yet, characterizing the geometry of these patterns quantitatively can be quite challenging. Here we propose the use of computational homology to obtain such a characterization. Our method is illustrated for the complex microstructures observed during spinodal decomposition and early coarsening in both the deterministic Cahn-Hilliard theory, as well as in the stochastic Cahn-Hilliard-Cook model. While both models produce microstructures that are qualitatively similar to the ones observed experimentally, our topological characterization points to significant differences. One particular aspect of our method is its ability to quantify boundary effects in finite size systems. | |
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Within this project, quantitative characterizations for the thermal-elastic response of calcite-based polycrystals are developed. The characterization is based on topological measurements such as the number of components and the number of handles of a complex microstructure. These characterizations are then applied to characterize both the grain boundary misorientations in the polycrystal and the resulting elastic energy density and principal stress fields. It is demonstrated that the topological analysis can quantitatively distinguish between different types of grain boundary misorientations, as well as between the resulting differences in the response fields. | |
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This project is aimed at the development of efficient algorithms and software for the homology computation. The necessity of such algorithms appears evident as new applications of the homology computation arise in research. Currently available software, including the package that can be downloaded from the CHomP website, allows to compute homology of n-dimensional cubical sets and maps, including relative homology, but there is still a lot of room for improvement, and processing large data is often very time and memory consuming. This motivates the search for improved algorithms and for new ways of approach to the task of homology computation.
This is a joint project of the CHomP group at Rutgers (USA) and the CAPD group in Kraków (Poland). | |
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We use persistent Homology to extract geometrical and topological information from protein data available in the Protein Data Bank (PDB). Using the CGAL library we construct Alpha Complexes from the PDB data and then use the Perseus software to compute persistence diagrams. | |
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A method based on the Conley index theory for proving the existence of periodic trajectories in a smooth dynamical system in Rn is developed. This method can be applied in the cases where a hyperbolic periodic orbit is numerically observed. The method uses cubical index pairs and requires the computation of the map induced in homology by the cubical index map. | |
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It is well established both numerically and experimentally that nonlinear systems involving diffusion, chemotaxis, and/or convection mechanisms can generate complicated time-dependent patterns. Since this phenomenon is global in nature, obtaining a quantitative mathematical characterization that to some extent records or preserves the geometric structures of the complex patterns is difficult. Here we show that using algebraic topology, in particular homology, we can measure Lyapunov exponents that imply the existence of spatial-temporal chaos and suggest a tentative step towards the classification and/or identification of patterns within the FitzHugh-Nagumo system. | |
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Computational Homology is used as a measure of the complexity of geometric structures in the weakly turbulent state of spiral defect chaos in experiments on Rayleigh-Bénard convection. Different attractors of spiral defect chaos are distinguished by their homology. This technique reveals asymmetric patterns. In addition global stochastic ergodicity is observed for system parameter values where locally chaotic dynamics has been previously reported. | |
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Coarse, topological measurements of dynamical systems defined by maps can be used to uncover information about invariant sets for the system. These techniques are based on the Conley index and have been used to detect invariant structures from fixed points and periodic orbits, to connecting orbits and sets which exhibit chaotic symbolic dynamics. More notably, these techniques have been used to study both finite-dimensional and infinite-dimensional systems and to prove the existence of unstable invariant sets. The method relies on first building a pair of compact sets called an index pair and then computing the relative homology of the pair. This information is then used to make conclusions about the associated invariant structures. | |
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One of the most efficient methods for determining the equilibria of a continuous parameterized family of differential equations is to use predictor-corrector continuation techniques. In the case of partial differential equations this procedure must be applied to some finite dimensional approximation which of course raises the question of the validity of the output. We introduce a new technique that combines the information obtained from the predictor-corrector steps with ideas from rigorous computations and verifies that the numerically produced equilibrium for the finite dimensional system can be used to explicitly define a set which contains a unique equilibrium for the infinite dimensional partial differential equation. | |
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Homology is an important computable tool for quantifying complex structures. In many applications these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study, based on suitable discretizations. Such an approach immediately raises the question of how accurate the resulting homology computations are. In the papers cited below, we present a probabilistic approach to quantifying the validity of homology computations for nodal domains of random Fourier series in one and two space dimensions, which furnishes explicit probabilistic a-priori bounds for the suitability of certain discretization sizes. In addition, we introduce a numerical method for verifying the homology computation using interval arithmetic. | |