A Discontinuous Irreversible Image Kinetics

In recent years, the dynamic of the discontinuous map has received much attention. Because there are some practical systems show a discontinuous change behavior, such as the DC-DC converter, relaxation oscillator, impact oscillator mechanical systems, biological systems, Power converter and many others. The discontinuous map as a simplified model for studying the discontinuous dynamics systems has been of great importance in both basic theory and application for the studying of the dynamics.In this paper, we study the dynamics of a discontinuous and inconvertible map under the influence of the forbidden area.First, we observe the period-adding phenomenon in the specific control parameters space of the forbidden area, which is due to the border collision bifurcation at two boundaries of the forbidden area with critical parameter, meanwhile the orbits can iterate to another region and still retain the original region. In order to quantitatively describe the period-adding phenomenon,we define the rotation number of the periodic orbits and then study the relationship between the period-adding sequences and the phase-locked devil's staircase determined by the fary rule.The analytical expressions of the both end points of each phase-locking plateau in the staircase are given in last.Second, a new type of the crisis is reported in the different control parameters space of the forbidden area. Careful studies show that:due to the influence of the forbidden area, the map forms the special structure, which decides the border of the chaotic attractor. The crisis emerges when the border of the chaotic attractor collides with the unstable point. The iteration orbits escape from the original chaotic attractor by the unstable point which lead to the sudden widening of the chaotic attractor. The critical condition and the characteristic time of the crisis and are analyzed numerically.