A survey of tools detecting the dynamical properties of one-dimensional families

Abstract

One-dimensional dynamical systems attract researches for more than half a century and the topic was inspired by many real problems. Mainly piecewise linear and polynomial maps were considered and researched under several motivations from different scientific fields. As a main aim of the paper, the Logistic (polynomials of the second order) and the Tent (piecewise linear maps with two pieces) families are considered and studied. The dynamical properties of both families are derived using bifurcation diagrams, Lyapunov exponents, and newly established techniques like the 0-1 test for chaos and recurrence matrices.