Nonautonomous transcritical and pitchfork bifurcations in impulsive/hybrid systems

Download

index.pdf

Date

2016

Author

Kashkynbayev, Ardak

Metadata

Show full item record

Item Usage Stats

195
views

66
downloads

The main purpose of this thesis is to study nonautonomous transcritical and pitchfork bifurcations in continuous and discontinuous dynamical systems. Two classes of discontinuity, impulsive differential equations and differential equations with an alternating piecewise constant argument of generalized type, are addressed. Moreover, the Bernoulli equation in impulsive as well as hybrid systems is introduced. For the former one, the corresponding jump equation is chosen so that after Bernoulli transformation the original system is reduced to a linear non-homogeneous system. For the latter, this is achieved by constructing a special type of transformation. Sufficient conditions are obtained for the existence of bounded solutions of the Bernoulli equations. Next, it is shown that different types of convergence analysis, such as pullback and forward remain as a fruitful idea in impulsive and hybrid systems. Furthermore, bifurcation scenarios are obtained depending on the sign of Lyapunov exponent by using these convergence analysis. Attraction and transition approaches are used to study bifurcation patterns in impulsive systems which cannot be solved explicitly. In other words, qualitative change in the attractor/reppeller pair is observed as a parameter goes though bifurcation value. Besides, finite-time analogues of nonautonomous transcritical and pitchfork bifurcations are investigated in impulsive systems. Illustrative examples with numerical simulations are provided to demonstrate the theoretical results.

Subject Keywords

Bifurcation theory., Differential equations, Partial., Impulsive differential equations., Differentiable dynamical systems.

URI

http://etd.lib.metu.edu.tr/upload/12619954/index.pdf
https://hdl.handle.net/11511/25623

Collections

Graduate School of Natural and Applied Sciences, Thesis

Suggestions

OpenMETU
Core

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....
Lyapunov type inequalities and their applications for linear and nonlinear systems under impulse effect Kayar, Zeynep; Ağacık, Zafer; Department of Mathematics (2014) In this thesis, Lyapunov type inequalities and their applications for impulsive systems of linear/nonlinear differential equations are studied. Since systems under impulse effect are one of the fundamental problems in most branches of applied mathematics, science and technology, investigation of their theory has developed rapidly in the last three decades. In addition, Lyapunov type inequalities have become a popular research area in recent years due to the fact that they provide not only better understandi...
Periodic solutions and stability of linear impulsive delay differential equations ALZabut, Jehad; Ağacık, Zafer; Department of Mathematics (2004) In this thesis, we investigate impulsive differential systems with delays of the form And more generally of the form The dissertation consists of five chapters. The first chapter serves as introduction, contains preliminary considerations and assertions that will be encountered in the sequel. In chapter 2, we construct the adjoint systems and obtain the variation of parameters formulas of the solutions in terms of fundamental matrices. The asymptotic behavior of solutions of systems satisfying the Perron co...
Oscillation of second order matrix equations on time scales Selçuk, Aysun; Ağacık, Zafer; Department of Mathematics (2004) The theory of time scales is introduced by Stefan Hilger in his PhD thesis in 1998 in order to unify continuous and discrete analysis. In our thesis, by making use of the time scale calculus we study the oscillation of nonlinear matrix differential equations of second order. the first chapter is introductory in nature and contains some basic definitions and tools of the time scales calculus, while certain well-known results have been presented with regard to oscillation of the solutions of second order matr...
Finite-time nonautonomous bifurcation in impulsive systems Akhmet, Marat (UNIV SZEGED, BOLYAI INSTITUTE, ARADI VERTANUK TERE 1, 6720 SZEGED, HUNGARY, 2016-01-01) The purpose of this article is to investigate nonautonomous bifurcation in impulsive differential equations. The impulsive finite-time analogues of transcritical and pitchfork bifurcation are provided. An illustrative example is given with numerical simulations which support theoretical results.

Citation Formats

A. Kashkynbayev, “Nonautonomous transcritical and pitchfork bifurcations in impulsive/hybrid systems,” Ph.D. - Doctoral Program, Middle East Technical University, 2016.