  Protecting coherent quantum dynamics from chaotic environment is key to
realizations of fragile many-body phenomena and their applications in quantum
technology. We present a general construction that embeds a desired periodic
orbit into a family of non-integrable many-body Hamiltonians, whose dynamics is
otherwise chaotic. Our construction is based on time dependent variational
principle that projects quantum dynamics onto a manifold of low-entangled
states, and it complements earlier approaches for embedding non-thermal
eigenstates, known as quantum many-body scars, into thermalizing spectra. By
designing terms that suppress "leakage" of the dynamics outside the variational
manifold, we engineer families of Floquet models that host exact scarred
dynamics, as we illustrate using a driven Affleck-Kennedy-Lieb-Tasaki model and
a recent experimental realization of scars in a dimerized superconducting qubit
chain.
