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This tutorial will constitute an introduction to a novel approach to computational dynamics that provides a framework in which one can rigorously describe the dynamics of systems for which the nonlinearities are imprecisely known or have weakly constrained parameters. The need for this approach arises from the increasing quest to understand dynamics of multiscale models of systems in which the phenomena are too complex for the nonlinearities to be derived from first principles; the life sciences is a typical example. Clearly, if the problem and hence model is imprecisely understood, then one cannot hope to attain a precise description of the dynamics. However, it is reasonable to expect that one can accurately capture dynamics at the same level of accuracy as the model or to accurately identify the dynamics exhibited in experiments.

The topics in this tutorial will range from the underlying theory to computational tools to concrete applications of the methods. In particular the following topics will be discussed:

  1. A coarse combinatorial representation of dynamics and its justification.
  2. How the combinatorial framework allows for efficient computations in both phase space and across parameter space.
  3. The Conley index and how it can be used to move from combinatorial to classical descriptions of dynamics.
  4. Computational methods for the Conley index.
  5. Applications to identifying regulatory networks.
  6. Time series data analysis in the context of regulatory networks.

The tutorial will include hands on descriptions of related software.