Date of Award
8-11-2015
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics and Statistics
First Advisor
Andrey Shilnikov
Second Advisor
Igor Belykh
Third Advisor
Vladimir Bondarenko
Fourth Advisor
Remus Osan
Abstract
A computational technique is introduced to reveal the complex intrinsic structure of homoclinic and heteroclinic bifurcations in a chaotic dynamical system. This technique is applied to several Lorenz-like systems with a saddle at the center, including the Lorenz system, the Shimizu-Morioka model, the homoclinic garden model, and the laser model. A multi-fractal, self-similar organization of heteroclinic and homoclinic bifurcations of saddle singularities is explored on a bi-parametric plane of those dynamical systems. Also a great detail is explored in the Shimizu-Morioka model as an example. The technique is also applied to a re exion symmetric dynamical system with a saddle-focus at the center (Chua's circuits). The layout of the homoclinic bifurcations near the primary one in such a system is studied theoretically, and a scalability ratio is proved. Another part of the dissertation explores the intrinsic mechanisms of escape in a reciprocally inhibitory FitzHugh-Nagumo type threecell network, using the phase-lag technique. The escape network can produce phase-locked states such as pace-makers, traveling-waves, and peristaltic patterns with recurrently phaselag varying.
DOI
https://doi.org/10.57709/7323199
Recommended Citation
Xing, Tingli, "Computational Study in Chaotic Dynamical Systems and Mechanisms for Pattern Generation in Three-Cell Networks." Dissertation, Georgia State University, 2015.
doi: https://doi.org/10.57709/7323199