An approximation formula and parameter-dependence of statistical quantities in low-dimensional chaotic systems

We derive an approximation formula for obtaining parameter dependence of statistical quantities in chaotic systems on the basis of a Frobenius -Perron equation. The formula is characterized by three integers; One is the number of application of the Frobenius - Perron operator, the second is the number of the delta-peaks of the initial density, and the third is the number of the integration domains. We find numerically that this formula is applicable to a wide varaiety of states of a system ranging from stable cycles to band-chaos and totaly spread chaotic regime by merely changing the system parameter., except for the very narrow windows representing stable cycles with large: periodicity, where the concrete: examples are a logistic map, a Henon map and so on. We finally consider how we extend our theory to ODE's.