  The axiomatic theory of ordinary differential equations, owing to its
simplicity, can provide a useful framework to describe various generalizations
of dynamical systems. In this study, we consider how dynamical properties can
be generalized to this setting. First, we study the generalizations of
dynamical properties, such as invariance and limit sets and show that compact
weakly invariant sets can be characterized in terms of the corresponding
shift-invariant sets in the function space. This result enables us to consider
weakly invariant sets in terms of shift-invariant sets. Next, we compare the
axiomatic theory of ODE with other formalisms of generalized dynamical systems,
namely generalized semiflow and multivalued semigroup. Finally, we introduce
the notion of invariant measures and show that we can generalize classical
results, such as Poincar\'e recurrence theorem.
