  Our context is Filippov systems defined on two-dimensional manifolds having a
finite number of tangency points. We prove that topological transitivity is a
necessary and sufficient condition for the occurrence of non-deterministic
chaos when the Filippov system has non-empty sliding or escaping regions. A
fundamental result for continuous flows is the equivalence of topological
transitivity and existence of a dense orbit. We prove in our setting that
topological transitivity for Filippov systems is indeed equivalent to the
existence of a dense Filippov orbit, although, in contrast to the continuous
case, we are not able to garantee that the dense orbit implies the existence of
a residual set of dense orbits. Finally we prove that, in this context,
topological transitivity implies strictly positive topological entropy for the
Filippov system. This calculation is made using techniques similar to those
from symbolic dynamics.
