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.
This project aimed to develop computational techniques to analyze the local and global behavior of dynamical systems. Theoretical foundations, algorithms, and practical implementations were studied. The developed methods are aimed at providing applied scientists with a new way of understanding the global structure of dynamical systems as well as specific tools to explore these structures. The methods can identify specific types of dynamical behavior such as fixed points, periodic orbits, and chaotic dynamics. They also characterize global structure such as bi-stability including unstable objects as well as attractors. These tools also allow for the tracking of how the local dynamics as well as the global structure changes with respect to changing parameters in the system. The developed tools were applied to systems that are defined by an explicit mathematical model as well as systems that are observed through time series data. For explicitly defined systems the results can be obtained as rigorous statements, in the form of computer-assisted proofs, about the dynamics by incorporating round-off error. The model systems studied came from population ecology and other applications. Some of the tools are implemented in a publicly available software package CHomP-- http://chomp.rutgers.edu and the Database for Global Dynamics-- http://chomp.rutgers.edu/database. Four PhD students have received some training through involvement in this research. The PI participated as a principle lecturer in two summer schools for graduate students interested in computational dynamics.
