The language and ideas of dynamical systems theory that have been developed over the last century have become ubiquitous in the applied sciences. While the analytic language of differential equations and maps is still the basis for most quantitative descriptions of scientific ideas, current scientific results are often obtained based on models which are not derived from first principles, for which many of the essential parameters have not been measured, and which often involve stochastic terms. The key objective of this project is to develop scalable computational techniques to provide correct robust information about global dynamics over large ranges of parameter values. Bifurcation theory implies that the cost for robustness is a coarse description. However, the fact that scientists and engineers are using numerical simulations of phenomenologically derived models to further their understanding of dynamic processes indicates that the information these techniques provide must be both quantitative and qualitative. Since the study of systems over broad ranges of possible parameter values produces considerable information, to be of practical use these methods must organize this information in an efficient, queriable manner. We expect the work proposed in this project will produce (1) reliable computational tools for global decompositions of dynamical systems by constructing a database in which the global dynamics is encoded in combinatorial and algebraic structures and (2) efficient methods for querying the database to identify dynamical structures and bifurcations of interest.
This work will address the fundamental question of determining global decompositions of dynamical systems over varying parameters. The global dynamics is stored in the form of a database based on calculations for deterministic systems, but within the framework of these computations we will also explore how to predict the effects of noise on the observable dynamical behavior. These computational techniques will be tested on and applied to a variety of problems from mathematical biology. The biological models which will be considered are used to address central questions in biology including the role of the spatial environment in ecology and evolution and the robustness of the dynamics of signal transduction/gene regulatory networks. These activities will produce computational tools for global decompositions of dynamical systems, which will be made available to scientists and engineers for potential applications in a wide variety of disciplines.
We have developed a Database for Global Dynamics. This is a software tool that allows the user to choose a priori the level of resolution, both with respect to the variables and the parameters, at which the dynamics is to be measured. For low dimensional systems the software produces a database that provides a mathematically rigorous description of the dynamics over a large (or even complete) region of parameter space. This database can be queried for specific types of dynamics, e.g. fixed points, periodic orbits, or global dynamic behavior, e.g. bistability. The database can also be queried to establish the range of parameter space over which specific types of dynamics or global dynamic behavior occurs and how these dynamical structures continue or bifurcate as a function of the parameter values. The Database approach is particularly relevant for problems in which the nonlinearities for the models cannot be derived from first principles or for which the parameter are unknown. This is typical of multiscale problems such as those arising in the life sciences. Examples include population models and signal transduction/gene regulatory networks. The computational tools used to develop the Database for Global Dynamics are of interest in their own right. In particular, algebraic topological invariants are required to obtain mathematically rigorous results from the numerical computations. In general the computation of these invariants begins with a potentially extremely large complex (that may not even be able to be fit into internal memory). Using ideas from combinatorial Morse theory we developed a preprocessing tool for homological computations. In practice this preprocessing tool typically performs in nearly linear time. Furthermore, it can be applied to complexes incrementally which has allowed us to compute homology, induced maps on homology and persistent homology for extremely large complexes.
