  Recent years have seen a strong interest in topological effects within
periodically driven systems. In this work, we explore topological effects in
two closely related 2-dimensional driven systems described by Floquet operators
possessing chiral symmetry (CS). Our numerical and analytical results suggest
the following. Firstly, the CS is associated with the existence of the
anomalous counter-propagating (ACP) modes reported recently. Specifically, we
show that a particular form of CS protects the ACP modes occurring at
quasienergies of $\pm \pi$. We also find that these modes are only present
along selected boundaries, suggesting that they are a weak topological effect.
Secondly, we find that CS can give rise to protected $0$ and $\pi$ quasienergy
modes, and that the number of these modes may increase without bound as we tune
up certain system parameters. Like the ACP modes, these $0$ and $\pi$ modes
also appear only along selected boundaries and thus appear to be a weak
topological effect. To our knowledge, this work represents the first detailed
study of weak topological effects in periodically driven systems. Our findings
add to the still-growing knowledge on driven topological systems.
