Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities

This book is devoted to bifurcation theory for autonomous and nonautonomous differential equations with discontinuities of different types. That is, those with jumps present either in the right-hand-side or in trajectories or in the arguments of solutions of equations. The results obtained in this book can be applied to various fields such as neural networks, brain dynamics, mechanical systems, weather phenomena, population dynamics, etc. Without any doubt, bifurcation theory should be further developed to different types of differential equations. In this sense, the present book will be a leading one in this field. The reader will benefit from the recent results of the theory and will learn in the very concrete way how to apply this theory to differential equations with various types of discontinuity. Moreover, the reader will learn new ways to analyze nonautonomous bifurcation scenarios in these equations. The book will be of a big interest both for beginners and experts in the field. For the former group of specialists, that is, undergraduate and graduate students, the book will be useful since it provides a strong impression that bifurcation theory can be developed not only for discrete and continuous systems, but those which combine these systems in very different ways. The latter group of specialists will find in this book several powerful instruments developed for the theory of discontinuous dynamical systems with variable moments of impacts, differential equations with piecewise constant arguments of generalized type and Filippov systems. A significant benefit of the present book is expected to be for those who consider bifurcations in systems with impulses since they are presumably nonautonomous systems


Finite Volume Method For Hyperbolic Conservation Laws On Manifolds
Okutmuştur, Baver (LAP LAMBERT Academic Publishing, 2017-03-01)
The purpose of this book is to lay out a mathematical framework for the convergence and error analysis of the finite volume method for the discretization of hyperbolic conservation laws on manifolds. Finite Volume Method (FVM) is a discretization approach for the numerical simulation of a wide variety physical processes described by conservation law systems. It is extensively employed in fluid mechanics, meteorology, heat and mass transfer, electromagnetic, models of biological processes and m...
Prolongation structures, backlund transformations and painleve analysis of nonlinear evolution equations
Yurduşen, İsmet; Karasu, Emine Ayşe; Department of Physics (2004)
The Wahlquist-Estabrook prolongation technique and the Painleve analysis, used for testing the integrability of nonlinear evolution equations, are considered and applied both to the Drinfel'd-Sokolov system of equations, indeed known to be one of the coupled Korteweg-de Vries (KdV) systems, and Kersten-Krasil'shchik coupled KdV-mKdV equations. Some new Backlund transformations for the Drinfel'd-Sokolov system of equations are also found.
Asymptotic integration of impulsive differential equations
Doğru Akgöl, Sibel; Ağacık, Zafer; Özbekler, Abdullah; Department of Mathematics (2017)
The main objective of this thesis is to investigate asymptotic properties of the solutions of differential equations under impulse effect, and in this way to fulfill the gap in the literature about asymptotic integration of impulsive differential equations. In this process our main instruments are fixed point theorems; lemmas on compactness; principal and nonprincipal solutions of impulsive differential equations and Cauchy function for impulsive differential equations. The thesis consists of five chapters....
Oscillation of even order nonlinear delay dynamic equations on time scales
Erbe, Lynn; Mert, Raziye; Peterson, Allan; Zafer, Ağacık (Institute of Mathematics, Czech Academy of Sciences, 2013-03-01)
One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor's Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality is...
Multi-symplectic integration of coupled non-linear Schrodinger system with soliton solutions
AYDIN, AYHAN; Karasözen, Bülent (2009-01-01)
Systems of coupled non-linear Schrodinger equations with soliton solutions are integrated using the six-point scheme which is equivalent to the multi-symplectic Preissman scheme. The numerical dispersion relations are studied for the linearized equation. Numerical results for elastic and inelastic soliton collisions are presented. Numerical experiments confirm the excellent conservation of energy, momentum and norm in long-term computations and their relations to the qualitative behaviour of the soliton sol...
Citation Formats
M. Akhmet, Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities. 2017.