One dimensional linear piecewise smooth discontinuous map Theory and Applications in Switching Circuits

Abstract

During the last three decades, development in the theory of piecewise-smooth discontinuous maps have played an important role in analyzing the atypical bifurcation phenomena occurring in the systems from various and#64257;elds e.g. impact oscillators, power electronic switching circuits etc. The bifurcations occurring in the piecewise-smooth maps are termed as border collision bifurcations. newlineIn this thesis, a 1-D linear piecewise-smooth discontinuous map with negative slopes newlinebetween 0 to -1 and one discontinuity either 1 or -1 is considered. The diand#64256;erent values of newlineslopes are considered to create regions. Within each region, there are six distinct scenarios considered for the analysis. The bifurcation parameter is varied to observe the behavior of the map. The existence of various types of stable, unstable orbits is analytically proved. Further, the eand#64256;ect of parameter variation on these orbits is also analyzed. Moreover, the coexistence of attractors is also shown. Additionally, the exact mathematical expression for the orbit to become periodic is derived. In chapter Analysis of Region P , it has been shown that stable periodic orbits of period-1 and period-2 orbits exist. Moreover, it is newlinealso shown that these are the only periodic orbit that exists. The coexistence of attractors has been observed. newlineIn chapter Analysis of Region O , a very peculiar type of period-2 orbits viz. both newlinethe points in a left plane or both the points in the right plane have been discovered. newlineAdditionally, it is shown that these peculiar period-2 orbits are eventually periodic in newlinenature. Further, the expression for an exact number of iterations is given after which a newlinepoint will jump to eventually periodic orbit. newlineIn chapter Analysis of Region PR , it has been proved that, there is an existence of newlineand#64257;xed point and inand#64257;nitely many period-2 orbits. It has been also proved that there is an existence of and#64257;xed point, period-2, and inand#64257;nitely many period-2 orbits, these things were not clear from the bifurcation diagram. The existence of the period-2 orbit, w