Show/Hide Menu
Hide/Show Apps
Logout
Türkçe
Türkçe
Search
Search
Login
Login
OpenMETU
OpenMETU
About
About
Open Science Policy
Open Science Policy
Communities & Collections
Communities & Collections
Help
Help
Frequently Asked Questions
Frequently Asked Questions
Guides
Guides
Thesis submission
Thesis submission
MS without thesis term project submission
MS without thesis term project submission
Publication submission with DOI
Publication submission with DOI
Publication submission
Publication submission
Supporting Information
Supporting Information
General Information
General Information
Copyright, Embargo and License
Copyright, Embargo and License
Contact us
Contact us
Differential equations with discontinuities and population dynamics
Download
index.pdf
Date
2009
Author
Aruğaslan Çinçin, Duygu
Metadata
Show full item record
Item Usage Stats
120
views
40
downloads
Cite This
In this thesis, both theoretical and application oriented results are obtained for differential equations with discontinuities of different types: impulsive differential equations, differential equations with piecewise constant argument of generalized type and differential equations with discontinuous right-hand sides. Several qualitative problems such as stability, Hopf bifurcation, center manifold reduction, permanence and persistence are addressed for these equations and also for Lotka-Volterra predator-prey models with variable time of impulses, ratio-dependent predator-prey systems and logistic equation with piecewise constant argument of generalized type. For the first time, by means of Lyapunov functions coupled with the Razumikhin method, sufficient conditions are established for stability of the trivial solution of differential equations with piecewise constant argument of generalized type. Appropriate examples are worked out to illustrate the applicability of the method. Moreover, stability analysis is performed for the logistic equation, which is one of the most widely used population dynamics models. The behaviour of solutions for a 2-dimensional system of differential equations with discontinuous right-hand side, also called a Filippov system, is studied. Discontinuity sets intersect at a vertex, and are of the quasilinear nature. Through the B−equivalence of that system to an impulsive differential equation, Hopf bifurcation is investigated. Finally, the obtained results are extended to a 3-dimensional discontinuous system of Filippov type. After the existence of a center manifold is proved for the 3-dimensional system, a theorem on the bifurcation of periodic solutions is provided in the critical case. Illustrative examples and numerical simulations are presented to verify the theoretical results.
Subject Keywords
Mathematics.
,
Differential equations.
URI
http://etd.lib.metu.edu.tr/upload/3/12610574/index.pdf
https://hdl.handle.net/11511/18790
Collections
Graduate School of Natural and Applied Sciences, Thesis
Suggestions
OpenMETU
Core
Periodic solutions and stability of differential equations with piecewise constant argument of generalized type
Büyükadalı, Cemil; Akhmet, Marat; Department of Mathematics (2009)
In this thesis, we study periodic solutions and stability of differential equations with piecewise constant argument of generalized type. These equations can be divided into three main classes: differential equations with retarded, alternately advanced-retarded, and state-dependent piecewise constant argument of generalized type. First, using the method of small parameter due to Poincaré, the existence and stability of periodic solutions of quasilinear differential equations with retarded piecewise constant...
Inverse problems for a semilinear heat equation with memory
Kaya, Müjdat; Çelebi, Okay; Department of Mathematics (2005)
In this thesis, we study the existence and uniqueness of the solutions of the inverse problems to identify the memory kernel k and the source term h, derived from First, we obtain the structural stability for k, when p=1 and the coefficient p, when g( )= . To identify the memory kernel, we find an operator equation after employing the half Fourier transformation. For the source term identification, we make use of the direct application of the final overdetermination conditions.
Studies on the perturbation problems in quantum mechanics
Koca, Burcu; Taşeli, Hasan; Department of Mathematics (2004)
In this thesis, the main perturbation problems encountered in quantum mechanics have been studied.Since the special functions and orthogonal polynomials appear very extensively in such problems, we emphasize on those topics as well. In this context, the classical quantum mechanical anharmonic oscillators described mathematically by the one-dimensional Schrodinger equation have been treated perturbatively in both finite and infinite intervals, corresponding to confined and non-confined systems, respectively.
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....
Differential equations on variable time scales
Akhmet, Marat (2009-02-01)
We introduce a class of differential equations on variable time scales with a transition condition between two consecutive parts of the scale. Conditions for existence and uniqueness of solutions are obtained. Periodicity, boundedness and stability of solutions are considered. The method of investigation is by means of two successive reductions: B-equivalence of the system [E. Akalfn, M.U. Akhmet, The principles of B-smooth discontinuous flows, Computers and Mathematics with Applications 49 (2005) 981-995; ...
Citation Formats
IEEE
ACM
APA
CHICAGO
MLA
BibTeX
D. Aruğaslan Çinçin, “Differential equations with discontinuities and population dynamics,” Ph.D. - Doctoral Program, Middle East Technical University, 2009.