Bifurcation of Bounded Solutions of Impulsive Differential Equations

Electronic version of an article published as International Journal of Bifurcation and Chaos, Volume 26, No. 14, 2016, 1-20 doi:10.1142/S0218127416502424 © copyright World Scientific Publishing Company, http://dx.doi.org/10.1142/S0218127416502424In this article, we examine nonautonomous bifurcation patterns in nonlinear systems of impulsive differential equations. The approach is based on Lyapunov–Schmidt reduction applied to the linearization of a particular nonlinear integral operator whose zeroes coincide with bounded solutions of the impulsive differential equation in question. This leads to sufficient conditions for the presence of fold, transcritical and pitchfork bifurcations. Additionally, we provide a computable necessary condition for bifurcation in nonlinear scalar impulsive differential equations. Several examples are provided illustrating the results

Numerical methods for a PDE system modeling tumor angiogenesis

This thesis is centered on the foundation needed for performing numerical simulations of a partial differential system (PDE) of reaction-diffusion equations modeling tumor growth in two dimensions. It includes the derivation of a particular PDE system modeling chemotaxis with a generalized logistic growth of the cell population density, and original results on the one-dimensional case. In particular, we show the effect of the generalized logistic growth on the population density of the organism. The second part of this thesis lays down the foundation needed to extend the numerical simulations to a two-dimensional Euclidean framework, confirming the accuracy of results through various numerical and analytical methods. The results of this work may pave the way to a deeper understanding of tumor angiogenesis over curved organ surfaces.</p