Although ODEs are one very typical and powerful means for defining dynamical systems, note that the basic definition makes no reference to differential equations - and hence is an independent concept from ODEs. The system is just defined by a manifold and a function \$$\phi^t(x)\$$ - whether that function be defined as the solutions to ODEs, or some other way. For the discrete case, it can, for example, be directly defined by an iteration involving procedural logic.