Analytical and Computational Tools for the Study of Grazing Bifurcations of Periodic Orbits and Invariant Tori
The objective of this dissertation is to develop theoretical and computational tools for the study of qualitative changes in the dynamics of systems with discontinuities, also known as nonsmooth or hybrid dynamical systems, under parameter variations. Accordingly, this dissertation is divided into t...
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Virginia Tech
2014
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| Online Access: | http://hdl.handle.net/10919/26196 http://scholar.lib.vt.edu/theses/available/etd-02142007-140350/ |
| Summary: | The objective of this dissertation is to develop theoretical and
computational tools for the study of qualitative changes in the dynamics of
systems with discontinuities, also known as nonsmooth or hybrid dynamical
systems, under parameter variations. Accordingly, this dissertation is
divided into two parts.
The analytical section of this dissertation discusses mathematical tools for
the analysis of hybrid dynamical systems and their application to a series
of model examples. Specifically, qualitative changes in the system dynamics
from a nonimpacting to an impacting motion, referred to as grazing
bifurcations, are studied in oscillators where the discontinuities are
caused by impacts. Here, the study emphasizes the formulation of conditions
for the persistence of a steady state motion in the immediate vicinity of
periodic and quasiperiodic grazing trajectories in an impacting mechanical
system. A local analysis based on the discontinuity-mapping approach is
employed to derive a normal-form description of the dynamics near a grazing
trajectory. Also, the results obtained using the discontinuity-mapping
approach and direct numerical integration are found to be in good agreement.
It is found that the instabilities caused by the presence of the square-root
singularity in the normal-form description affect the grazing bifurcation
scenario differently depending on the relative dimensionality of the state
space and the steady state motion at the grazing contact.
The computational section presents the structure and applications of a
software program, TC-HAT, developed to study the bifurcation analysis of
hybrid dynamical systems. Here, we present a general boundary value problem
(BVP) approach to locate periodic trajectories corresponding to a hybrid
dynamical system under parameter variations. A methodology to compute the
eigenvalues of periodic trajectories when using the BVP formulation is
illustrated using a model example. Finally, bifurcation analysis of four
model hybrid dynamical systems is performed using TC-HAT. === Ph. D. |
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