# Lattice Structures for Attractors II

@article{Kalies2016LatticeSF, title={Lattice Structures for Attractors II}, author={William D. Kalies and Konstantin Mischaikow and Robert C. Vandervorst}, journal={Foundations of Computational Mathematics}, year={2016}, volume={16}, pages={1151-1191} }

The algebraic structure of the attractors in a dynamical system determines much of its global dynamics. The collection of all attractors has a natural lattice structure, and this structure can be detected through attracting neighborhoods, which can in principle be computed. Indeed, there has been much recent work on developing and implementing general computational algorithms for global dynamics, which are capable of computing attracting neighborhoods efficiently. Here we address the question…

## 26 Citations

### Lattice structures for attractors I

- Mathematics
- 2014

We describe the basic lattice structures of attractors and repellers in dynamical systems.
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### Lattice Structures for Attractors III

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### Global Dynamics for Steep Sigmoidal Nonlinearities in Two Dimensions

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It is proved that for 2-dimensional systems, the Morse graph defines a Morse decomposition for the dynamics of any smooth differential equation that is sufficiently close to the original piecewise affine ordinary differential equation.

### The Morse Equation in the Conley Index Theory for Discrete Multivalued Dynamical Systems

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A recent generalization of the Conley index to discrete multivalued dynamical systems without a continuous selector is motivated by applications to data–drive dynamics. In the present paper we…

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This paper casts the connection matrix theory into appropriate categorical, homotopy-theoretic language and identifies objects of the appropriate categories which correspond to connection matrices and may be computed within the computational Conley theory paradigm by using the technique of reductions.

### An Algorithmic Approach to Lattices and Order in Dynamics

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Recurrent versus gradient-like behavior in global dynamics can be characterized via a surjective lattice homomorphism between certain bounded, distributive lattices, that is, between attracting blo...

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My interests lie in the development of theories, algorithms and software for the analysis of nonlinear data and systems. My recent work is in Conley index theory, where I developed both a categorical…

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