Equation-Free Computations as DDDAS Protocols for Bifurcation Studies: A Granular Chain Example

This chapter discusses the development and implementation of algorithms based on Equation-Free/Dynamic Data-Driven Applications Systems (EF/DDDAS) protocols for the computer-assisted study of the bifurcation structure of complex dynamical systems, such as those that arise in biology (neuronal networks, cell populations), multiscale systems in physics, chemistry, and engineering, and system modeling in the social sciences. An illustrative example demonstrates the experimental realization of a chain of granular particles (a so-called engineered granular chain). In particular, the focus is on the detection/stability analysis of time-periodic, spatially localized structures referred to as “dark breathers.” Results in this chapter highlight, both experimentally and numerically, that the number of breathers can be controlled by varying the frequency as well as the amplitude of an “out-of-phase” actuation, and that a “snaking” structure in the bifurcation diagram (computed through standard, model-based numerical methods for dynamical systems) is also recovered through the EF/DDDAS methods operating on a black-box simulator. The EF/DDDAS protocols presented here are, therefore, a step toward general-purpose protocols for performing detailed bifurcation analyses directly on laboratory experiments, not only on their mathematical models, but also on measured data.