Discrete Dynamical Systems From Real Valued Mutation

We introduce a family of discrete dynamical systems which includes, and generalizes, the mutation dynamics rank two cluster algebras. These exhibit behavior associated with integrability, namely preservation symplectic form, in tropical case, existence conserved quantity. show certain cases that orbits are unbounded. The related to matrix mutation, from theory able special cases, map is periodic. also explain how our imply asymptotic sign-coherence observed by Gekhtman Nakanishi $2$-dimensional situation.