  Tracking the solution of time-varying variational inequalities is an
important problem with applications in game theory, optimization, and machine
learning. Existing work considers time-varying games or time-varying
optimization problems. For strongly convex optimization problems or strongly
monotone games, these results provide tracking guarantees under the assumption
that the variation of the time-varying problem is restrained, that is, problems
with a sublinear solution path. In this work we extend existing results in two
ways: In our first result, we provide tracking bounds for (1) variational
inequalities with a sublinear solution path but not necessarily monotone
functions, and (2) for periodic time-varying variational inequalities that do
not necessarily have a sublinear solution path-length. Our second main
contribution is an extensive study of the convergence behavior and trajectory
of discrete dynamical systems of periodic time-varying VI. We show that these
systems can exhibit provably chaotic behavior or can converge to the solution.
Finally, we illustrate our theoretical results with experiments.
