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Thesis
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Studiul bifurcaţiilor cu metode de analiză complexă şi aplicaţii în sisteme dinamice
Bercia, Cristina Ruxandra (UPB)
2007-01-01
Abstract: The text begins in the first chapter with basic topics regarding the stability of continuous and discrete-time dynamical systems, the linearisation principle for the study of the asymptotic behavior of the solutions in the neighborhood of a hyperbolic equilibrium point, the Floquet theory of the complex characteristic exponents and complex dynamics. The second chapter deals with the types of bifurcations of equilibria in a dynamical system depending on parameters and the center manifold theorem that allows one to reduce the dimension of the state space near a local bifurcation. We apply the Poincare transformation to the study of the dynamics at infinity for a 2-dimensional differential system of Volterra type in order to obtain the global phase portrait. We use the reduction to the central manifold to study the system around the equilibrium points which are not hyperbolic and we deduce that these are of saddle-node type. We apply the bifurcation theory in the last chapter to the predator-prey dynamical systems, presenting models with two populations of Holling-Tanner and Michaelis-Menten type. Based on the ecological observations and analyzing the predictions of these models, we discuss the superiority of the Michaelis-Menten ratio-dependent model compared to the classical one. For the most realistic ones, we establish the conditions on the parameters when the species coexist or when only one species goes extinct, depending on initial data. It is shown that different kinds of bifurcations occur: such as the subcritical and supercritical Hopf bifurcation and if the ratio-dependent model is perturbed by a small constant term, more bifurcations appear, such as the saddle-node type and of codimension 2, Bogdanov-Takens bifurcation. By these local analysis, one can prove the existence of stable and unstable limit cycles and in the last case, the existence of a homoclinic orbit. We present bifurcation diagrams and phase portraits for different types of dynamics. We also obtained original results, studying two dynamical systems in three variables with nine parameters, which model the acute inflammation and septic shock of the human body due to an infection or a trauma. We established the domains in the parameters’ space where the equilibrium points exist and the set of conditions for these to be asymptotic stable. We also performed bifurcation analysis, we proved analytical results and if it wasn’t possible, we appealed to numerical simulations. For both systems we proved static bifurcation of equilibrium points, namely transcritical bifurcation and of saddle-node type. We detected a Hopf bifurcation point which was subcritical, since we found an unstable limit cycle emerging from this point. Bifurcation diagrams are presented, varying one of the parameters, which allowed us to obtain phase portraits topologically unequivalent. These were obtained by numerical integration in Matlab. In consequence we established all the types of dynamics of the systems which capture clinically known scenarios.
Keyword(s): Ecuaţii diferenţiale -- Sisteme dinamice -- Teză de doctorat
Note: Octavian Stanasila
OPAC: See record in BC-UPB Web OPAC
Full Text: see files
Record created 2010-12-09, last modified 2011-11-11
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