Dynamics in chaotic zones of area preserving maps: Close to separatrix and global instability zones

► We study chaotic regions of area-preserving maps using suitable return models. ► Chaotic regions around separatrices are widely investigated. ► The separatrix map is used to quantitatively describe the dynamics in these regions. ► We introduce the biseparatrix map to study the dynamics in larger chaotic zones. ► Several examples illustrate how to use these return models for general systems.