Winning entries were selected by The Dynamical Systems Web based on ease of installation, ease of use, expected impact, genericity and complexity of the type of problem the software can tackle, and quality of documentation. Software packages submitted with major upgrades were allowed to compete and were evaluated regarding their upgrades only.
Dankowicz’s COCO is a MATLAB-based, object-oriented platform for constructing composite continuation problems, mapping out their solution manifolds, and locating special points on these manifolds. The COCO core provides general-purpose support for these tasks, making the platform extremely versatile and flexible, even at run time. COCO-compatible toolboxes exemplify common classes of problems that arise in dynamical systems theory, including bifurcation analysis and design optimization.
The package is under ongoing development. Earlier releases (November 2015 and before) were co-developed by Harry Dankowicz and Frank Schilder (with contributions also by Mike Henderson and Erika Fotsch), while subsequent development has been developed entirely within Harry's research group. A major new release was posted to Sourceforge in November 2017. The algorithmic foundation for the new functionality in this release was documented in a SIADS paper that was accepted for publication in December 2017 and appeared online in April 2018. This was written by Harry Dankowicz and co-authored with his graduate student Mingwu Li, who also helped with initial development of code and testing of tutorial demos.
The November 2017 release includes the following new functionality, without precedent in any existing packages:
- Fully documented support for general-purpose, staged construction of adjoint equations, consistent with COCO’s object-oriented construction paradigm and the decomposition of continuation problems into coupled instances of individual continuation objects.
- Full support for adaptive remeshing of adjoint equations, consistent with adaptive updates to the problem discretization along families of solutions to integro-differential boundary-value problems.
- Detailed core and toolbox tutorials and demos illustrating a method of successive continuation for constrained single-objective optimization along
- Solutions to arbitrary algebraic continuation problems;
- Families of equilibrium points in autonomous dynamical systems;
- Families of constrained trajectory segments, e.g., periodic orbits;
- Solutions to composite continuation problems, e.g., coupled periodic orbits.
In this release, the COCO core was augmented to support an expanded definition of COCO-compatible continuation problems, for example for simultaneous continuation of a zero problem and the associated adjoint conditions. See help/CORE-Tutorial.pdf for tutorial and reference documentation, including a fully documented example of single-objective optimization along a family of solutions to an algebraic continuation problem and related exercises. See the core/examples folder for the corresponding demo.
Furthermore, in this release the existing ep, coll, and po toolboxes were updated to include support for construction of the associated adjoint equations, as well as demos of constrained design optimization along families of
- equilibria in smooth, autonomous dynamical systems;
- collections of constrained trajectory segments with independent adaptive discretization in autonomous or non-autonomous dynamical systems, including single- and multi-segmentboundary-value problems; and
- single-segment periodic orbits in smooth, autonomous or non-autonomous dynamical systems, and multi-segment periodic orbits in hybrid, autonomous dynamical systems.
Dankowicz, who is also the Associate Dean for Graduate, Professional and Online Education for the College of Engineering, conducts dynamical systems research at the intersection of engineering, math and physics. This work involves studying a wide range of complex systems that are governed by differential equations and learning the behavior of those systems through theory and experiments. His research efforts further seek to make original and substantial contributions to the development and design of existing or novel devices that capitalize on system nonlinearities for improved system understanding and performance.
A significant emphasis of Dankowicz's work since 2006 is on the development of computational tools for intelligent exploration of the design space of nonlinear dynamical systems, including those modeling human cognition and behavior, complex societal network, and power systems. This collaborative effort has resulted in innovative advances in algorithm design and a powerful, general-purpose tool that is expected to greatly broaden the applicability of these methods to actual engineering design.