Differential equations, in particular evolutionary differential equations, are used to model, often with uncanny effectiveness, a vast number of natural and social phenomena. The study of dynamical behaviors of the underlying models, e.g. their stabilities, robustness, bifurcations, and complexities, has become truly essential to understand the phenomena being modeled. It is precisely because of the modeling effectiveness of evolutionary differential equations that their dynamical study is among the fastest growing and most active multidisciplinary research fields today. Not only is the existing theory of dynamics of evolutionary equations closely intertwined with many areas of mathematics, but also the concepts, methods, and paradigms, introduced in the study have become an indispensible component of a multitude of works of more applied nature, in disciplines including chemistry, physics, material science, mechanics, electrical engineering, as well as in the emerging applications of social sciences, not to mention the explosion during the last several decades of applications related to the biological and life sciences. As we push the boundaries of applicability of the existing theory, we routinely discover that new mathematical and computational tools are needed, often symbiotically. To witness, it is widely accepted that an important factor contributing to the explosive growth in studying dynamics of evolutionary differential equations has been the availability of sophisticated and powerful computational tools. Yet, huge systems, systems with multiple time scales, multi-physics models, stochastic differential equations, non-smooth evolutionary problems, still transcend our collective understanding, and exhibit inherent complexities which are requiring the development of sophisticated new modeling and numerical techniques, and whose validity will need to be validated by rigorous mathematics.