Qualitative theory of differential equations with applications  Page description

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Details of project

  
Identifier
75517
Type K
Principal investigator Krisztin, Tibor
Title in Hungarian Differenciálegyenletek kvalitatív elmélete alkalmazásokkal
Title in English Qualitative theory of differential equations with applications
Keywords in Hungarian Másodrendű differenciálegyenletek, funkcionál-differenciálegyenletek, állapotfüggő késleltetés, attraktorok, periodikus pályák, komplex viselkedés, alkalmazások
Keywords in English Second order differential equations, functional differential equations, state-dependent delay, attractors, periodic orbits, complex behaviour, applications
Discipline
Mathematics (Council of Physical Sciences)100 %
Ortelius classification: Differential equations
Panel Mathematics and Computing Science
Department or equivalent Bolyai Institute (University of Szeged)
Participants Bartha, Ferenc
Bartha, Mária
Dénes, Attila
Garab, Ábel
Hatvani, László
Karsai, János
Knipl, Diána
Röst, Gergely
Szimjanovszki, Irma
Terjéki, József
Van Leeuwen-Polner, Mónika
Vas, Gabriella Ágnes
Starting date 2009-01-01
Closing date 2012-12-31
Funding (in million HUF) 15.820
FTE (full time equivalent) 17.26
state closed project
Summary in Hungarian
A pályázat célja másodrendű közönséges differenciálegyenletek és funkcionál-differenciálegyenletek vizsgálata. Ezen egyenletek fontos fizikai, kémiai, biológiai , közgazdasági jelenségek modellegyenletei.
Mivel az egyenletek explicit megoldása többnyire lehetetlen, kvalitatív vizsgálatokkal tudunk a megoldások szerkezetéről információt adni. Klasszikus másodrendű egyenletek megoldásainak stabilitási tulajdonságait, kaotikus jelenségeit tanulmányozzuk. Visszacsatolási mechanizmusokat modellező funkcionál-differenciálegyenletek attraktorainak szerkezetét tervezzük leírni. Új típusú, ún. állapotfüggő késleltetésű egyenletek elméletének a kidolgozását folytatjuk. Az elméleti eredményekből várhatóan a modellezett jelenségre is fontos információ kapható.
A kutatás a senior kutatók mellett 2 PhD hallgató és 2 nemrég doktorált kutató munkáját segíti.
Summary
The aim of this project is the investigation of certain second order ordinary differential equations and functional differential equations. These type of equations model important phenomena of physics, chemistry, biology, economics. As the explicit solution of the equations is impossible in most of the cases, we use the qualitative theory to get information about the structure of solutions. We plan to study stability properties, chaotic behaviour for classical second order differential equations. We intend to describe the attractors of functional differential equations modelling feedback mechanisms. We continue the development of the fundamental theory of a new class of equations, the so called equations with state-dependent delays. It is expected that the theoretical results will have important impacts on the applications.
This project, in addition to the senior researchers, supports the research of 2 PhD students and 2 researchers
who obtained their PhD degrees within two years.




 

Final report

  
Results in Hungarian
Differenciálegyenletek kvalitatív elméletében végeztünk kutatásokat. Az elméleti eredményeket is fontos alkalmazások motiválták. Emellett alkalmazásokkal is foglalkoztunk. A főbb eredmények: másodrendű nem-autonóm differenciálegyenletek megoldásainak aszimptotikus vizsgálatára dolgoztunk ki új módszereket; bizonyos funkcionál-differenciálegyenletekre új típusú attraktorok szerkezetét írtuk le; járványterjedési jelenségek vizsgálatára differenciálegyenletes modelleket adtunk meg, és azok kvalitatív tulajdonságait leírva a járványok terjedéséről fontos információkat kaptunk.
Results in English
We studied the qualitative theory of differential equations. The theoretical results were motivated by important applications. In addition we considered applications, too. Some of the main results: we developed new methods to study the asymptotic behaviour of solutions of second order nonautonomous differential equations; we described the structure of new type of attractors for certain functional differenctial equations; different epidemic models were developed to describe the spread of infectious diseases, and we studied the qualitative properties of these models to get important information about the diseases.
Full text https://www.otka-palyazat.hu/download.php?type=zarobeszamolo&projektid=75517
Decision
Yes





 

List of publications

  
L. Hatvani: Stochastic parametric resonance in a linear oscillator at swuare-wave modulation, Problems of Analytical Mechanics and Stability Theory (Eds. F.L. Chernousko, A.V. Karapetyan, V.V. Kozlov, N.N. Krasovskii, V.N. Tkhai, S.N. Vasil'ev), Fzmatlit, Moscow, 2009
L. Hatvani, F. Toókos, G. Tusnády: A mutation-selection-recombination model in population genetics, Dynam. Systems Appl. 18(2009), 335-362, 2009
L. Hatvani: On the critical values of parametric resonance in Meissner's equation by the method of difference equations, Electronic J. Qualitative Theory of Diff. Equations, Spec. Ed. I, No. 1, 1-10, 2009
A. Dénes, L. Hatvani, L.L. Stachó: Eventual stability properties in a nonautonomous model of population dynamics, Nonlinear Anal. 73 (2010), 650-659., 2010
T. Krisztin: On the fundamental solution of a linear delay differential equation, Int. J. Qualitative Theory of Differential Equations and Appl. 3(2009), 1-7, 2009
G. Vas: Asymptotic constancy and periodicity for a single neuron model with delay, Nonlinear Analysis, vol. 71, 2268-2277, 2009
E. Liz, G. Röst: On the Global Attractor of Delay Differential Equations with Unimodal Feedback, Discrete and Continuous Dynamical Systems, 24:(4) 1215-1224, 2009
S.M. Moghadas, C.S. Bowman, G. Röst, D.N. Fisman, J. Wu: Post-exposure prophylaxis during pandemic, BMC Medicine 2009, 7:73 doi:10.1186/1741-7015-7-73, 2009
E. Liz, G. Röst: Dichotomy results for delay differential equations with negative, Nonlinear Anal. RWA, 2010 doi:10.1016/j.nonrwa.2009.02.030, 2010
S. Csörgő, L. Hatvani: Stability properties of solutions of linear second order differential equations with random coefficients, J. Differential Equations 248(2010), 21-49., 2010
Knipl D, Röst G.: Modelling the strategies for age specific vaccination scheduling during influenza pandemic outbreaks, Math. Biosci. Eng. 8(1), 123-139 (2011), 2011
Dénes, A. and Makay, G.: Attractors and basins of dynamical systems, E. J. Qualitative Theory of Diff. Equ., No. 20 (2011), pp. 1-11., 2011
G. Vas: Infinite number of stable periodic solutions for an equation with negative feedback,, E. J. Qualitative Theory of Diff. Equ., No. 18 (2011), pp. 1-20., 2011
T. Krisztin, E. Liz: Bubbles for a Class of Delay Differential Equations, Qualitative Theory of Dynamical Systems 10 (2011), 169-196, 2011
Á. Garab, T. Krisztin: The period function of a delay differential equation and an application,, Periodica Mathematica Hungarica 63 (2011), 173-190, 2011
T. Krisztin, G. Vas: On the fundamental solution of linear delay differential equations with multiple delays, E. J. Qualitative Theory of Diff. Equ., 36 (2011), 1-28, 2011
G. Vas: Infinite number of stable periodic solutions for an equation with negative feedback, E. J. Qualitative Theory of Diff. Equ. 18 (2011), 1-20., 2011
T. Krisztin, G. Vas: Large-amplitude periodic solutions for a differential equation with delayed positive feedback, Journal of Dynamics and Differential Equations 23 (2011), 727--790., 2011
J. Terjéki, M. Bartha: On the convergence of solutions of nonautonomous functional differential equations, E. J. Qualitative Theory of Diff. Equ., Proc. 9th Coll.QTDE, 2011, No. 1 1-10, 2011
D. Knipl, G. Röst: Influenza models with Wolfram Mathematica, Interesting Mathematical Problems in Sciences and Everyday Life, Chapter 4, pp 1-24 (Eds: J. Karsai, R. Vajda), Szeged - 2011 - Novi Sad, 2011
A. Dénes, G. Makay: Attractors and basins of dynamical systems, E. J. Qualitative Theory of Diff. Equ. 20 (2011), 1-11, 2011
J. Karsai: Investigations of Impulsive Systems with Mathematica, Interesting Mathematical Problems in Sciences and Everyday Life, 27 pages (Eds: J. Karsai, R. Vajda), Szeged - 2011 - Novi Sad, 2011, 2011
L. Székely, L. Hatvani: Asymptotic stability of two dimensional systems of linear difference equations and of second order half-linear differential equations with step function coefficients, E. J. Qualitative Theory of Diff. Equ., No. 38 (2011), pp. 1-17., 2011
Röst G; Huang ShY; Székely L: On a SEIR Epidemic Model with Delay, DYNAMIC SYSTEMS AND APPLICATIONS 21: pp. 33-48, 2012
Röst G: Global convergence and uniform bounds of fluctuating prices in a single commodity market model of Bélair and Mackey, ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS 2012:(26) pp. 1-9., 2012
Liu M; Röst G: Dynamics of an SIS Model on Homogeneous Networks with Delayed Reduction of Contact Numbers, BIOMATH 1:(2) pp. Article ID: 1210045-1-7., 2012
Knipl DH; Röst G: Multiregional SIR Model with Infection during Transportation, BIOMATH 1:(1) pp. Article ID: 1209255-1--7., 2012
Jones DA; Smith HL; Thieme HR; Röst G: Spread of Phage Infection of Bacteria in a Petri Dish, SIAM JOURNAL ON APPLIED MATHEMATICS 72:(2) pp. 670-688., 2012
Dénes A; Röst G: Structure of the global attractors in a model for ectoparasite borne diseases, BIOMATH 1:(1) pp. Article ID: 1209256-1--5., 2012
Röst G: SEI model with varying infectivity and mortality, MATHEMATICS IN SCIENCE AND TECHNOLOGY: Mathematical Methods, Models and Algorithms in Science and Technology, Proceedings of the Satellite Conference of ICM 2010 Konferen, 2011
Röst G: On an approximate method for the delay logistic equation COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 16:(9) pp. 3470-3474.(2011), COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 16:(9) pp. 3470-3474., 2011
Moghadas SM; Bowman CS,; Röst G; Fisman DN; Wu J: Post-exposure prophylaxis during pandemic outbreaks, BMC MEDICINE 7: 73doi:10.1186/1741-7015-7-73 p. 1.(2009), 2009
Garab Á: Unique periodic orbits of a delay differentail equation with piecewise linear feedback function, Discrete and Continuous Dynamical Systems, 33(6):2236-2387, 2013., 2013
Dénes A; Kevei P; Nishiura H; Röst G: Risk of infectious disease outbreaks by imported cases with application to the European Football Championship 2012, INTERNATIONAL JOURNAL OF STOCHASTIC ANALYSIS 2013: p. 1.(2013) online http://www.hindawi.com/journals/ijsa/aip/576381/, 2013
Liz E; Röst G: Global dynamics in a commodity market model, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 398:(2) pp. 707-714., 2013




 

Events of the project

  
2014-02-24 09:56:47
Kutatóhely váltás
A kutatás helye megváltozott. Korábbi kutatóhely: Alkalmazott és Numerikus Matematika Tanszék (Szegedi Tudományegyetem), Új kutatóhely: Bolyai Intézet (Szegedi Tudományegyetem).
2011-05-10 09:55:57
Résztvevők változása
2010-05-07 15:52:17
Résztvevők változása



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