Asymptotic behaviour of solutions of differential equations  Page description

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Details of project

 
Identifier
49516
Type K
Principal investigator Krisztin, Tibor
Title in Hungarian Differenciálegyenletek megoldásainak aszimptotikus viselkedése
Title in English Asymptotic behaviour of solutions of differential equations
Panel Mathematics and Computing Science
Department or equivalent Bolyai Institute (University of Szeged)
Participants Bartha, Mária
Hatvani, László
Karsai, János
Makay, Géza
Röst, Gergely
Terjéki, József
Starting date 2005-01-01
Closing date 2009-12-31
Funding (in million HUF) 13.553
FTE (full time equivalent) 9.77
state closed project





 

Final report

 
Results in Hungarian
Másodrendű közönséges differenciálegyenletek és funkcionál differenciálegyenletek megoldásainak aszimptotikus viselkedésére bizonyítottunk eredményeket. Többek között egy periodikusan perturbált fékezett inga mozgását leíró differenciálegyenlet megoldásainak a kaotikus viselkedését igazoltuk analitikus módszerek, topológiai eszközök és megbízható numerikus eljárások kombinálásával. Monoton késleltetett visszacsatoltást modellező funkcionál differenciálegyenletek globális attraktorai szerkezetének többé-kevésbé teljes leírását adtunk. Bizonyos nem monoton visszacsatolások esetére is kaptunk fontos eredményeket a globális attraktorra. Állapotfüggő késleltetéses funkcionál differenciálegyenletek geometriai elméletének az alapjait dolgoztuk ki lokális invariáns sokaságok létezésének az igazolásával.
Results in English
We studied the asymptotic behavior of solutions of second order ordinary differential equations and functional differential equations. Among others, combining analytical tools, topological methods and reliable numerical procedures, chaos was shown for the solutions of the equation modeling a periodically perturbed damped pendulum. For functional differential equations describing delayed monotone feedback we gave a more or less complete characterization of the global attractor. We obtained important results for the global attractor also in the non monotone feedback case. For functional differential equations with state-dependent delay we proved the existence of local invariant manifolds which has a fundamental role in the geometric theory for these type of problems.
Full text http://real.mtak.hu/1986/
Decision
Yes





 

List of publications

 
B. Bánhelyi, T. Csendes, L. Hatvani and B. Garay: A computer-assisted proof and location of chaos: the case of a forced damped pendulum equation, Proceedings of GO, 2005
J. Karsai and J.R. Graef: Behavior of solutions of second order differential equations with nonlinear damping, Nonlinear Oscillations 8, 2005
J. Karsai and J.R. Graef: Attractivity properties of oscillator equations with superlinear damping, Discrete and Continuous Dynamical Systems, 2005
G. Röst: Neimark-Sacker bifurcation for periodic delay differential equations, Nonlinear Analysis, 2005
G. Röst: Some applications of bifurcation formulae to the periodic map of delay differential equations, Dynamical Systems and Applications, Proc. Conf., Antalya, Turkey, 2005
B. Bánhelyi, T. Csendes, L. Hatvani and B. Garay: A computer-assisted proof and location of chaos: the case of a forced damped pendulum equation, Proceedings of GO, 2005
G Makay: Some notes on an almost periodic function, Acta Sci. Math. (Szeged), vol. 73, pp. 95-100, 2006
M. Bartha and J. Terjéki: On the convergence of solutions for an equation with state-dependent delay, Differential Equations and Dynamical Systems, vol. 14, pp. 195-206, 2006
B. Garay, L. Hatvani and J. Kolumbán: Lipot Fejér habilitated at Kolozsvár 100 years ago in stability theory, Alk. Mat. Lapok, vol. 23, pp. 163-189, 2006
L. Hatvani and L. Székely: On the existence of small solutions of systems of linear difference equations with varying coefficients, J. Difference Equ. Appl., vol. 12, pp. 837-845, 2006
T. Csendes, B. Bánhelyi and L. Hatvani: Towards a computer-assisted proof for chaos in a forced damped pendulum equation, J. Computational and Applied Mathematics, vol. 105, pp. 378-383, 2007
T. Krisztin: C^1-smoothness of center manifolds for differential equations with state-dependent delay, Fields Institute Communications, 48, pp.213-226, 2006
F. Hartung, T. Krisztin, H.-O. Walther and J. Wu: Functional differential equations with state-dependent delay: theory and applications, Handbook of Differential Equations: Ordinary Differential Equations, Vol. 3, Elsevier - North-Holland, pp. 435-545, 2006
G. Röst: Bifurcation of periodic delay differential equations at points of 1:4 resonance, Functional Differential Equations, vol. 13, pp. 585-602, 2006
L. Hatvani: Growth condition guaranteeing small solutions for linear oscillator with increasing elasticity coefficient, Georgian Mathematical Journal, 2007
Röst G.: On the global attractivity controversy for a delay model of hematopoiesis, Appl. Mat. Comput. vol. 190/1, pp. 846-850, 2007
Alexander M.E.; Bowman C.S.; Feng Zh.; Gardam M.; Moghadas S.M.; Röst G.; Wu J.; Yan P.: Emergence of drug-resistance: dynamical implications for pandemic influenza, Proc. R. Soc. London Ser. B-Biol. Sci. vol. 274, Nr. 1619, pp. 1675-1684, 2007
Röst G.; Wu J.: Domain-decomposition method for the global dynamics of dely diffreential equations with unimodal feedback, Proc. R. Soc. London Ser. A vol. 463, pp. 2655-2669, 2007
Móczár J.; Krisztin T.: A Harrod modell strukturális stabilitása, Szigma, 2006
B. Bánhelyi, T. Csendes, B.M. Garay and L. Hatvani: Computer-assisted proof of chaotic behaviour of the forced damped pendulum, Folia FSN Universitatis Masarykianae Bruensis, Mathematica 16 (2007), 9-20, 2007
B. Bánhelyi, T. Csendes, B.M. Garay and L. Hatvani: Computer-assisted proof for $\Sigma_3$ chaos in the forced damped pendulum equation, SIAM J. Appl. Dyn. Syst. 7(2008), 843-867, 2008
N. Guglielmi and L. Hatvani: On small oscillations of mechanical systems with time dependent kinetic and potential energy, Discrete Contin. Dyn. Syst. 20(2008), 911-926., 2008
L. Hatvani: Stability problems for the mathematical pendulum, Period. Math. Hungar. 56(2008), 71-82., 2008
M. Bartha and J. Terjéki: Uniqueness for retarded delay differential equations without Lipschitz condition, E.J. Qualitative Theory of Diff. Equations, Proc. 8th Coll. Qual. Theory of Diff. Eq., pp. 1-6, 2008
Alexander M.E.; Moghadas S.M.; Röst G. and Wu J.: A delay differential model for pandemic influenza with antiviral treatment, B. Mth. Biol. 70(2008), 382-397, 2008
Moghadas S.M.; Bowman C.S.; Röst G. and Wu J.: Population-wide emergence of antiviral resistance during pandemic influenza, PLOS ONE 3:(3) 1839-p. (2008), 2008
Röst G. and Wu J.: SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng. 5 (2008), 389-402, 2008
Rácz É. and Karsai J.: The effect of initial patterns on competitive exclusion, Community Ecology 7(2006), 23-33, 2006
Péics H. and Karsai J.: Existence of positive solutions of halflinear delay differential equation, J. Math. Anal. Appl. 323(2006), 1201-1212, 2006
T. Krisztin: Global dynamics of delay differential equations, Period. Math. Hung. 56(2008), 83-95, 2008
Karsai J.: Computer-aided study of mathematical models. CD-ROM, Univ. of Szeged, 2008
Karsai J.: Mathematical and visualization packages: Mathematica 6. CD-ROM, Univ. of Szeged, 2008
Karsai J.: Mathematica-aid to study impulsive system, Proceedings of the Int. Mathematica Symposium - Maastricht. http://www.ims08.org, 2008
Szimjanovszki I., Karsai J. and Rácz É.V.: Competition for territory: the Levins model for two species. Wolfram deomstration project, http://demonstrations.wolfram.com/CompetitionFor TerritoryTheLevinsModel For, 2008




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