Invariants and Liapunov functions for nonautonomous systems

If for a difference equation no stability theorem applies, it is necessary to examine this difference equation directly. For many equations, however, it is neither obvious whether solutions are bounded or stable, nor is it trivial to prove such behavior. A useful way to prove boundedness is to find the difference equationʹs invariant (e.g., see [1–5]). But what is an invariant? It is hard to find a definition of invariants in the literature on difference equations. Moreover, it turns out that invariants and Liapunov functions are strongly related concepts; in fact, invariants can be considered as special cases of Liapunov functions. For this reason, we shall extend the concept of Liapunov functions to nonautonomous discrete dynamical systems, and we shall supply a general definition for invariants that covers the nonautonomous case also.