Tricks of Jokers on One-Dimensional Maps

In many real systems there exist whimsical combinations of time intervals, regions in phase space or in parameter space, where deterministic, well forecasting processes alternating with probable and badly forecasting ones. These characters, associated with abrupt changes in forecast, are crucial for description of many systems. Moreover, in some cases these systems may undergo catastrophic jumps, moving the system from one point of phase space to another, which is not close to the first. To describe such systems, it is reasonable to introduce new class of mathematical models - dynamical systems with jokers, where joker – a region J in phase space, where system behavior stops to be deterministic. Joker region may correspond to higher dimensions during reconstruction of attractors, 'free will' or unpredictable actions of political leadership. The aim of this work is the analysis of qualitative and quantitative phenomenon, generating by the most simple jokers on one-dimensional maps. In detail two types of joke are examined.