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Piecewise smooth maps are frequently used to model switching systems
encountered in physics and engineering. The dynamics and bifurcations
in such maps have been studied in reasonable detail. However, most of
these studies do not consider practical effects like noise, parameter
fluctuations, and variation of the functional form. In this thesis we
focus on the novel dynamical features of piecewise smooth maps under
periodic or stochastic variation of its functional form. These
variations may arise due to coexistence of state- and time-dependent
switching or due to some special kinds of noise which affect the
system dynamics significantly. We first consider a piecewise smooth
map which contains state dependent as well as time dependent
switching. Due to the presence of both kinds of switching in a system,
the functional form of the piecewise smooth map varies either
periodically or stochastically. The existence of a new type of border
collision bifurcation, where an invariant attractor bifurcates into a
non-invariant attractor has been established in case of a piecewise
smooth map having periodically varying functional form. On the other
hand it has been shown that if the time dependent variation is
considered to be stochastic instead of periodic, then a
non-deterministic basin of attraction may exist. Next we take up the
piecewise smooth map with stochastically varying border, retaining the
deterministic dynamics in all the compartments of the phase space. We
prove the existence of non-deterministic basins of attraction in such
systems. Finally, we consider a finite dimensional chaotic map and
have shown that the chaotic nature of such a map can be controlled by
the introduction of a time dependent feedback perturbation.
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