Let a nonlinear dynamical system be described by the difference equations $$x(n+1)=f(n,x(n)),\ n\ge 0$$ with the function $f$ being nonlinear and non-lipschitz and bounded over a domain $D$. My question is
Is there any general set of methods which can be used for stability analysis of this system?
I know that if $f$ is Lipschitz then there are many such methods, but I could not find any for the non-lipschitz case. I am just a novice in the field of nonlinear dynamical systems and I must admit I do not have the mathematical machinery required to start to think solving this kind of system from the scratch. This kind of thing is required for my research and after spending a lot of time I could not find examples of such systems analyzed in literature. So it would be very kind if anyone can give me some reference regarding this problem. Thanks in advance.
Edit: I think I should properly mention the type of analysis I want to find. Specifically, it would be quite helpful to me if I know some techniques regarding analysis of the behavior of the sequence $$\{\|x(n+1)-x(n)\|\}_{n\in \mathbb{N}}$$