Global dynamics of differential equations  Page description

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Details of project

 
Identifier
109782
Type K
Principal investigator Krisztin, Tibor
Title in Hungarian Differenciálegyenletek globális dinamikája
Title in English Global dynamics of differential equations
Keywords in Hungarian globális dinamika, késleltetett differenciálegyenlet, globális attraktor, visszacsatolás, állapotfüggő késleltetés, másodrendű egyenletek, nem-autonóm egyenletek
Keywords in English global dynamics, delay differential equations, global attractor, feedback, state-dependent delay, second order equations, non-autonomous equations
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 Balázs, István
Dénes, Attila
Dudás, János
Garab, Ábel
Hatvani, László
Karsai, János
Kiss, Gábor
Knipl, Diána
Röst, Gergely
Stachó, László
Székely, László
Van Leeuwen-Polner, Mónika
Vas, Gabriella Ágnes
Vízi, Zsolt
Starting date 2013-09-01
Closing date 2018-08-31
Funding (in million HUF) 21.312
FTE (full time equivalent) 27.70
state closed project
Summary in Hungarian
A kutatás összefoglalója, célkitűzései szakemberek számára
Itt írja le a kutatás fő célkitűzéseit a témában jártas szakember számára.

Különböző típusú differenciálegyenleteket vizsgálunk, elsősorban időkésleltetéseseket, de közönséges és parciális differenciálegyenleteket is. Az egyenletek megoldásai által general dinamikus rendszerek globális tulajdonságait tanulmányozzuk.

Mi a kutatás alapkérdése?
Ebben a részben írja le röviden, hogy mi a kutatás segítségével megválaszolni kívánt probléma, mi a kutatás kiinduló hipotézise, milyen kérdéseket válaszolnak meg a kísérletek.

A fő cél a globális attraktor szerkezetének leírása. Ez különböző egyenlettípusokra kölünböző technikák, módszerek kidolgozását követeli meg. A projekt célja ilyen eredmények bizonyítása.

Mi a kutatás jelentősége?
Röviden írja le, milyen új perspektívát nyitnak az alapkutatásban az elért eredmények, milyen társadalmi hasznosíthatóságnak teremtik meg a tudományos alapját. Mutassa be, hogy a megpályázott kutatási területen lévő hazai és a nemzetközi versenytársaihoz képest melyek az egyediségei és erősségei a pályázatának!

A vizsgált egyenlettípusok alkalmazások által motiváltak vagy az elmélet szempontjából érdekesek. Fontos alapkérdésekre keresünk választ, és új típusú egyenletekre dolgozunk ki módszereket.

A kutatás összefoglalója, célkitűzései laikusok számára
Ebben a fejezetben írja le a kutatás fő célkitűzéseit alapműveltséggel rendelkező laikusok számára. Ez az összefoglaló a döntéshozók, a média illetve az adófizetők tájékoztatása szempontjából különösen fontos az NKFI számára.

Időben változó fizikai, kémiai, biológiai, társadalmi jelenségek érthetők meg differenciálegyenletes modellek alkalmazásával. A modellegyenletek matematikai vizsgálatával az eredeti jelenség mélyebben megérhető, tervezhető, előre jelezhető. Ennek a projektnek a célja a modellegyenletek megértéséhez szükséges matematikai módszerek kidolgozása.
Summary
Summary of the research and its aims for experts
Describe the major aims of the research for experts.

We consider different types of differential equations, mainly equations with time delays, but also ordinary and partial differential equations. We study global properties of the dynamical systems generated by the solutions of the equations.

What is the major research question?
Describe here briefly the problem to be solved by the research, the starting hypothesis, and the questions addressed by the experiments.

The main goal is to describe the global attractor. This requires different mathematical tools, methods for different types of equations. The aim of the project is to prove these type of results.

What is the significance of the research?
Describe the new perspectives opened by the results achieved, including the scientific basics of potential societal applications. Please describe the unique strengths of your proposal in comparison to your domestic and international competitors in the given field.

The considered equations are motivated by applications or are interesting from a theoretical point of view. We study fundamental problems, and develop methods for new type of equations.

Summary and aims of the research for the public
Describe here the major aims of the research for an audience with average background information. This summary is especially important for NKFI in order to inform decision-makers, media, and the taxpayers.

It is possible to use differential equation to model physical , chemical, biological, social phenomena. The study of the model equations with mathematical tools may lead to deeper understanding of the problems, may help to plan or predict the phenomenon. The aim of this project to develop mathematical methods for the study of the model equations.





 

Final report

 
Results in Hungarian
Különböző típusú differenciálegyenletek megoldásainak aszimptotikus viselkedését, elsősorban globális kérdéseket vizsgáltunk. A vizsgált problémákat konkrét alkalmazások motiválták vagy elméleti szempontból voltak érdekesek. A kapott eredmények az adott problémák mélyebb megértését segítik, a vizsgált jelenségek előrejelzését, tervezését segíthetik. A kutatási időszak alatt 70 dolgozat született. Monoton visszacsatolási függvény esetén további nem-triviális attraktor-szerkezetek geometriai és dinamikai leírását adtuk. Wright egy 1955-ös sejtésének az igazolásához új technikai eszközöket dolgoztunk ki, ami végül fontos szerepet játszott a végső bizonyításban. Többek között állapotfüggő késleltetésű parabolikus parciális differenciálegyenletek egy osztályára kidolgoztuk az alapelméletet. Az ún. „hydra effektus” matematikai magyarázatát adtuk. Különböző járványterjedési modelleket vizsgáltunk, ezzel sikerült a jelenségeket mélyebben megérteni. Vizsgáltuk a Nicholson-egyenletek nagy rendszerére egyensúlyi helyzet létezését, globális stabilitását. Idegsejt hálózatok analitikus és numerikus vizsgálatának módszereit dolgoztuk ki.
Results in English
We studied the asymptotic behavior of solutions of different types of differential equations, mainly global problems. The considered equations were motivated by applications or were interesting from a theoretical point of view. The obtained results lead to a deeper understanding of the studied problems, may help to plan or predict the phenomenon. Within this project we published 70 research papers. For equations with monotone feedback we described additional nontrivial attractor structures. We developed new technical tools to study the Wright conjecture (from 1955). These tools played a vital role in the recent final proof of the conjecture. Among others, for parabolic partial differential equations with state-dependent delays we developed the fundamental theory. We explained mathematically the so called “hydra effect”. We studied different mathematical models for infectious diseases in order to understand the dynamics. Existence of steady states, global stability were shown for large systems of Nicholson equations modeling population dynamics. We obtained analytical and numerical tools to study neural network models.
Full text https://www.otka-palyazat.hu/download.php?type=zarobeszamolo&projektid=109782
Decision
Yes





 

List of publications

 
Dénes A; Röst G: Global dynamics for the spread of ectoparasite-borne diseases, Nonlinear Analysis: Real World Applications 18 (2014) 100-107., 2014
Hatvani L: An elementary method for the study of Meissner's equation and its application to proving the Oscillation Theorem, Acta Sci Math (Szeged) 79 (2013), 87--105., 2013
Csizmadia L; Hatvani L: An extension of the Levi-Weckesser method to the stabilization of the inverted pendulum under gravity, Meccanica, 49 (2014), 1091-1100., 2014
Bartha F; Garab A; Krisztin T: Local stability implies global stability for the 2-dimensional Ricker map, J. Difference Equ. Appl. 19 (2013), 2043-2078., 2013
Bánhelyi B;, Csendes T; Neumaier A; Krisztin T: Global attractivity of the zero solution for Wright’s equation, SIAM J. Appl. Dy. Syst. 13 (2014), 537-563., 2014
Krisztin T; Vas G: Erratum to: Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback, J. Dynam. Differential Equations 26 (2014), 401-403., 2014
Szimjanovszki I; Karsai J; Rácz ÉVP: On the asymptotic behavior of spatially implicit models of competition of two species with overcolonization, DYNAMIC SYSTEMS AND APPLICATIONS 23: pp. 677-690. (2014), 2014
Polner M; Van der Vegt JJW: A Hamiltonian vorticity-dilatation formulation of the compressible Euler equations, Nonlinear Analysis, Theory, Methods and Applications, 109 (2014), 113-135, 2014
Stachó LL: On strongly continuous one-parameter groups of automorphisme, ROMANIAN JOURNAL OF PURE AND APPLIED MATHEMATICS 4: pp. 1-11. (2014), 2014
Knipl DH; Röst G; Wu J: Epidemic Spread and Variation of Peak Times in Connected Regions Due to Travel-Related Infections -- Dynamics of an Antigravity-Type Delay Differential Model, SIAM J. Appl. Dyn. Syst., 12(4), 1722–1762. (2013), 2013
Knipl DH; Röst G: Backward bifurcation in SIVS model with immigration of non-infectives,, Biomath 2 (2013), 1312051, 2013
Pituk M; Röst G: Large Time Behavior of a Linear Delay Differential Equation with Asymptotically Small Coefficient, Boundary Value Problems, 2014:114, 2014
Gourley S; Röst G; Thieme HR: Uniform persistence in a model for bluetongue dynamics, SIAM J. Math. Anal., 46(2), 1160–1184, 2014
Röst G; Vizi Zs: Backward bifurcation for pulse vaccination, Nonlinear Analysis - Hybrid Systems, 14, pp 99-113, 2014, 2014
Faria T; Röst G: Persistence, permanence, and global stability for an n-dimensional Nicholson system, Journal of Dynamics and Differential Equations, 2014, 2014
Röst G: Baneling dynamics in the Legend of the Seeker, , Chapter 17 in: Mathematical Modelling of Zombies (ed. Robert Smith), pp 231-241, University of Ottawa Press, 2014 , ISBN: 9780776622101, 2014
F. A. Bartha; Á. Garab: Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model, Journal of Computational Dynamics 1(2):213-232, 2014., 2014
Krisztin T; Vas G: The Unstable Set of a Periodic Orbit for Delayed Positive Feedback, Journal of Dynamics and Differential Equations, DOI 10.1007/s10884-014-9375-0. pp 1-51, 2014
Nah K; Nakata Y; Röst G: Malaria dynamics with long incubation period in hosts, COMPUTERS AND MATHEMATICS WITH APPLICATIONS 68:(9) pp. 915-930. (2014), 2014
Knipl DH; Röst G: Large number of endemic equilibria for disease transmission models in patchy environment, MATHEMATICAL BIOSCIENCES 258: pp. 201-222. (2014), 2014
Dénes A, Röst G: Global dynamics of a compartmental system modeling ectoparasite-borne diseases, ACTA SCIENTIARUM MATHEMATICARUM - SZEGED 80:(3-4) pp. 553-572. (2014), 2014
Nakata Y; Röst G: Global dynamics of a delay differential system of a two-patch SIS-model with transport-related infections, MATHEMATICA BOHEMICA 140:(2) pp. 171-193. (2015), 2015
Nakata Y; Röst G: Global analysis for spread of infectious diseases via transportation networks, JOURNAL OF MATHEMATICAL BIOLOGY 70:(6) pp. 1411-1456. (2015), 2015
Liu M; Liz E; Röst G: Endemic bubbles generated by delayed behavioral response -- global stability and bifurcation switches in an SIS model, SIAM JOURNAL ON APPLIED MATHEMATICS 75:(1) pp. 75-91. (2015), 2015
Knipl DH; Pilarczyk P; Röst G: Rich bifurcation structure in a two-patch vaccination model, SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 14:(2) pp. 980-1017. (2015), 2015
Kiss IZ; Röst G; Vizi Zs: Generalization of pairwise models to non-Markovian epidemics on networks, PHYSICAL REVIEW LETTERS 115, 078701 (2015), 2015
Gourley S; Liu R; Röst G: Age-dependent intraspecific competition in pre-adult life stages and its effects on adult population dynamics, EUROPEAN JOURNAL OF APPLIED MATHEMATICS doi:10.1017/S0956792515000418 (2015), 2015
Dénes A; Röst G: Impact of excess mortality on the dynamics of diseases spread by ectoparasites, In: Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science, Springer (New York), 2015. pp. 177-182. (ISBN:978-3-319-12306-6), 2015
Barbarossa MV; Dénes A,; Kiss G,; Nakata Y,; Röst G; Vizi Zs: Transmission dynamics and final epidemic size of Ebola Virus Disease outbreaks with varying interventions, PLoS ONE 10(7): e0131398. doi:10.1371/journal.pone.0131398, 2015
Barbarossa MV; Röst G: Immuno-epidemiology of a population structured by immune status: a mathematical study of waning immunity and immune system boosting, JOURNAL OF MATHEMATICAL BIOLOGY doi:10.1007/s00285-015-0880-5 (2015), 2015
Nussbaum R; Vas G: Gevrey class regularity for analytic differential-delay equations, ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS 17: pp. 1-10. (2016), 2016
Tóbiás R; Stacho LL; Tasi G: First-order chemical reaction networks I: theoretical considerations, J Math Chem, DOI 10.1007/s10910-016-0655-2, 2016
Kiss IZ; Röst G; Vizi Zs: Impact of non-Markovian recovery on network epidemics, BIOMAT 2015: Proceedings of the International Symposium on Mathematical and Computational Biology, (2015) p.40-53, 2016
Garab Á; Kovács V; Krisztin T: Global stability of a price model with multiple delays, Discrete and Continuous Dynamical Systems 36(2016), 6855-6871, 2016
Dénes A; Röst G: Global stability for SIR and SIRS models with nonlinear incidence and removal terms via Dulac functions, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), 1101–1117, 2016
Dénes A; Hatvani L: On the asymptotic behaviour of solutions of an asymptotically Lotka–Volterra model, Elektron. J. Qual. Theory Differ. Equ. 2016, No. 67, 1–10., 2016
Hatvani L: Marachkov type stability conditions for non-autonomous functional differential equations with unbounded right-hand sides, Electron. J. Qual. Theory Differ. Equ. 2015, No. 64, 1-11., 2015
Csizmadia L; Hatvani L: On a linear model of swinging with a periodic step function coefficient, Acta Sci. Math. (Szeged), 81(2015), 483--502., 2015
El-Morshedy H; Röst G; Ruiz-Herrera A: Global dynamics of delay recruitment models with maximized lifespan, ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK (2016) 67: 56, 2016
Gourley S; Liu R; Röst G: Age-dependent intraspecific competition in pre-adult life stages and its effects on adult population dynamics, EUROPEAN JOURNAL OF APPLIED MATHEMATICS 27:(1) pp. 131-156. (2016), 2016
Knipl D; Röst G: Spatially heterogeneous populations with mixed negative and positive local density dependence, THEORETICAL POPULATION BIOLOGY 109: pp. 6-15. (2016), 2016
Röst G; Vizi Zs; Kiss I: Impact of Non-Markovian Recovery on Network Epidemics, BIOMAT 2015, 40-53 Singapore: World Scientific, 2016, 2016
Wu X; Röst G; Zou X: Impact of spring bird migration on the range expansion of Ixodes scapularis tick population, BULLETIN OF MATHEMATICAL BIOLOGY 78: pp. 138-168. (2016), 2016
Krisztin T; Vas G: The Unstable Set of a Periodic Orbit for Delayed Positive Feedback, Journal of Dynamics and Differential Equations 28 (2016), 805–855, 2016
Krisztin T; Rezounenko A: Parabolic partial differential equations with discrete state-dependent delay: classical solutions and solution manifold, J. Differential Equations 260 (2016), 4454–4472., 2016
Krisztin, T; Polner, M; Vas, G: Periodic solutions and hydra effect for delay differential equations with nonincreasing feedback, Qual. Theory Dyn. Syst. 16 (2017), 269–292., 2017
Hatvani L: On the global attractivity and asymptotic stability for autonomous systems of differential equations on the plane, Proc. Amer. Math. Soc. 145 (2017), no. 3, 1121–1129., 2017
Hatvani L: Asymptotic stability of non-autonomous functional differential equations with distributed delays, Electron. J. Differential Equations 2016, Paper No. 302,, 2016
Barbarossa, MV; Polner, M; Röst, G: Stability switches induced by immune system boosting in an SIRS model with discrete and distributed delays, SIAM J. Appl. Math. 77(2017), no. 3, 905–923., 2017
Dénes, A; Muroya, Y; Röst, G: Global stability of a multistrain SIS model with superinfection, Math. Biosci. Eng. 14 (2017), no. 2, 421–435., 2017
Nah, K; Röst, G: Stability threshold for scalar linear periodic delay differential equations, Canad. Math. Bull. 59 (2016), no. 4, 849-857., 2016
Vas, G: Configurations of periodic orbits for equations with delayed positive feedback, J. Differential Equations 262 (2017), no. 3, 1850–1896., 2017
Dénes A, Székely L: Global dynamics of a mathematical model for the possible re-emergence of polio, Math. Biosci. 293 (2017), 64–74., 2017
Dénes A, Székely L: Small solutions of the damped half-linear oscillator with step function coefficients, Electron. J. Qual. Theory Differ. Equ. 2018, No. 46, 1–13., 2018
Dénes A, Röst G: Dynamics of an infectious disease including ectoparasites, rodents and humans, Trends in Biomathematics: Modeling, Optimization and Computational Problems (ed. R. Mondaini), Springer, 2018, pp. 59–73., 2018
Hatvani L: On the damped harmonic oscillator with time dependent damping coefficient, J. Dynam. Differential Equations, 30(2018), 25—37, 2018
Hatvani L: Smith-type stability theorems for the damped linear oscillator, Dynam. Systems Appl., 27(2018), 299—318, 2018
Csizmadia L, Hatvani L: On the existence of periodic motions of the excited inverted pendulum by elementary methods, Acta Math. Hungar., 155(2018), 298-312, 2018
Polner M, van der Vegt J, van Gils S: A Space-Time Finite Element Method for Neural Field Equations with Transmission Delays, SIAM JOURNAL ON SCIENTIFIC COMPUTING 395) pp. B797-B818. (2017), 2017
Stachó L L: On C0-semigroups of holomorphic isometries with fixed point in JB*-triples, Roumain J. of Pure and Applied Math., 63 (2018), 211-235, 2018
Kiss G, Lessard J-P: Rapidly and slowly oscillating periodic solutions of a delayed Van der Pol oscillator, Journal of Dynamics and Differential Equations, 29(4):1233-1257, 2017, 2017
Kiss G, Röst G: Controlling Mackey-Glass chaos, Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(11):114321, 2017, 2017
Kiss G, Röst G: Controlling Mackey-Glass chaos, Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(11):114321, 2017, 2017
Bartha F A; Krisztin T: Global stability in a system using echo for position control, Electron. J. Qual. Theory Differ. Equ. 2018, Paper No. 40, 16 pp., 2018
Krisztin T; Walther H-O: Smoothness issues in differential equations with state-dependent delay, Rend. Istit. Mat. Univ. Trieste 49 (2017), 95–112., 2017
Győri I; Nakata Y; Röst G: Unbounded and blow-up solutions for a delay logistic equation with positive feedback, Commun. Pure Appl. Anal. 17 (2018), no. 6, 2845–2854., 2018
Röst G; Kuniya T; Moghadas S M; Wu J: Global dynamics of an epidemiological model with age-of-infection dependent treatment rate, Ric. Mat. 67 (2018), no. 1, 125–140., 2018
Győri I; Nakata Y; Röst G: Unbounded and blow-up solutions for a delay logistic equation with positive feedback, Commun. Pure Appl. Anal. 17 (2018), no. 6, 2845–2854., 2018
Nakata Y; Röst G: Global stability of an SIS epidemic model with a finite infectious period, Differential Integral Equations 31 (2018), no. 3-4, 161–172., 2018
Röst G, Kiss IZ, Vizi Z: Variance of Infectious Periods and Reproduction Numbers for Network Epidemics with Non-Markovian Recovery, Berlin: Springer-Verlag, 2017. (Mathematics in Industry), Progress in Industrial Mathematics at ECMI 2016, pp. 171-178, 2018





 

Events of the project

 
2018-11-29 16:09:05
Résztvevők változása
2017-02-28 20:55:36
Résztvevők változása
2014-08-26 11:23:27
Résztvevők változása




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