Reinjection probability density in type-III intermittency

► We model the reinjection probability density (RPD) in type-III intermittency (Int). ► We present a new RPD with and without the lower bound of the reinjection (LBR). ► We present a new characteristic relations that depends on the RPD and LBR. ► The main classical results can be extracted from our RPD as particular cases. ► Our theory can be applied to a many non-linear maps with type-III and type-II Int.