Dynamics of Piecewise Smooth Maps with Periodic and Stochastic Variation in the Functional Form

Mandal, Dhrubajyoti (2018) Dynamics of Piecewise Smooth Maps with Periodic and Stochastic Variation in the Functional Form. PhD thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

Piecewise smooth maps are frequently used to model the dynamics of many physical and engineering systems. Therefore it is very important to explore major dynamical properties of different types of piecewise smooth maps in order to understand the dynamics of those practical systems. There exists a lot of literature describing various kinds of dynamical features of different types of piecewise smooth maps. Most of these studies consider the functional form of the piecewise smooth map to be fixed, with variation allowed only in the values of the parameters. But there may be situations where we have to consider the variation in the functional form of a piecewise smooth map. In this thesis we mainly focus on finding 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. In chapter-2 and chapter-3, we have considered 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. Bifurcation from a stable fixed point attractor to a periodic attractor of period greater than one, due to the border collision bifurcation has been established in case of a one dimensional piecewise smooth maps having periodically varying functional form in any one compartment of the phase space. On the other hand it has been shown that if the time dependent variation is considered to be stochastic instead of periodic, then non-deterministic basins of attraction may exist. The dynamical behaviour of any two orbits may differ even if they start from the same initial point lying inside this non-deterministic basin of attraction. The next two chapters, i.e., chapter-4 and chapter-5, deal with a piecewise smooth map with stochastically varying border, retaining the deterministic dynamics in all the compartments of the phase space. This type of systems gives rise to piecewise smooth maps whose border varies stochastically inside a small region of the phase space. In case of such systems the existence of non-deterministic basin of attraction has been established. Similar to the case of stochastically varying functional form, the dynamical behaviour of any two orbits starting from the same initial point lying inside this non-deterministic basin of attraction, may be different. For example one of them may converge to a stable fixed point whereas another may diverge to infinity. Chapter-4 deals with this issue in one dimension while Chapter-5 considers 2D map. In chapter-6, we have considered a finite dimensional chaotic map and have shown that the chaotic nature of the map can be controlled by applying a time dependent feedback perturbation to the state of the original system. This control method is applicable not only for smooth but also nonsmooth maps, therefore in case of piecewise smooth maps the application of the proposed feedback perturbation makes the perturbed map to be a piecewise smooth map with periodically varying functional form. It has been shown that the dynamics of such perturbed map converges to some stable periodic orbit. This thesis has thus revealed that piecewise smooth maps with variable functional form display many novel dynamical features, such as bifurcation from a fixed point attractor to Milnor attractor, existence of non-deterministic basin of attraction etc., which cannot be found in other types of maps. Therefore the content of this thesis opens up new directions for further research in dynamics of piecewise smooth maps with periodic or stochastic perturbations in its functional form or in the position of the border.

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