The evolution of crisis: Bifurcation and demise of a snapback repeller

Abstract

This paper studies the dynamics of members of the two-parameter family of maps x → μx(1 − x v ), emphasizing the evolution from snapback repeller to crisis bifurcations. The example of the square root map v = 1 2 is taken to represent the subfamily where v is fixed and taken from the range 1 2 ≤ v ≤ 1 . A map from such a subfamily is shown to be conjugate with a map with negative Schwarzian derivative. This allows a characterization of crisis as the demise of a snapback repeller on a proper subinterval.