Differenciál- és differenciaegyenletek kvalitatív és kvantitatív elmélete alkalmazásokkal = Qualitative and quantitative theory of differential and difference equations with applications

Kutatásaink a következő témakörökhöz kapcsolódtak: Megoldások aszimptotikus jellemzése és stabilitása; integrálegyenletek és egyenlőtlenségek mértékterekben; differenciálegyenletek megoldásainak paraméterektől való differenciálható függése, paraméterek becslése; állapotfüggő késleltetésű differenciálegyenletek stabilitása. A 2004-2007 kutatási időszakban 31 publikációnk jelent meg. Dolgoztainkra az elmúlt négy évben 575, ezen belül a kutatási periódusban megjelent 31 publikációnkra pedig 57 hivatkozást regisztráltunk. Eredményeinkről 9 plenáris, 33 meghívott szekció és 16 szekció előadásban számoltunk be nemzetközi konferenciákon. Ezeken kívül 34 meghívott előadást tartottunk különböző hazai és külföldi egyetemek szakmai szemináriumain. | Our research is related to the following topics: Asymptotic characterization and stability of solutions; integral equations and inequalities in measure spaces; differentiability of the solutions with respect to the parameters, and parameter estimation methods; stability of differential equations with state-dependent delays. 31 of our pubications have appeared in the research period 2004-2007. We have counted 575 citations of our papers in the last four years, including 57 citations of our 31 papers published in this period. We gave 9 plenary, 33 invited, and 16 contributed talks at international conferences, and 34 invited talks at research seminars of national and foreign universities

Minimal sets in monotone and sublinear skew-product semiflows I: The general case

AbstractThe dynamics of a general monotone and sublinear skew-product semiflow is analyzed, paying special attention to the long-term behavior of the strongly positive semiorbits and to the minimal sets. Four possibilities arise depending on the existence or absence of strongly positive minimal sets and bounded semiorbits, as well as on the coexistence or not of bounded and unbounded strongly positive semiorbits. Previous results are unified and extended, and scenarios which are impossible in the autonomous or periodic cases are described

Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems

Producción CientíficaWe determine sufficient conditions for uniform and strict persistence in the case of skew-product semiflows generated by solutions of non-autonomous families of cooperative systems of ODEs or delay FDEs in terms of the principal spectrums of some associated linear skew-product semiflows which admit a continuous separation. Our conditions are also necessary in the linear case. We apply our results to a noncooperative almost periodic Nicholson system with a patch structure, whose persistence turns out to be equivalent to the persistence of the linearized system along the null solution.MINECO/FEDER MTM2015-6633

Breathers in inhomogeneous nonlinear lattices: an analysis via centre manifold reduction

We consider an infinite chain of particles linearly coupled to their nearest neighbours and subject to an anharmonic local potential. The chain is assumed weakly inhomogeneous. We look for small amplitude discrete breathers. The problem is reformulated as a nonautonomous recurrence in a space of time-periodic functions, where the dynamics is considered along the discrete spatial coordinate. We show that small amplitude oscillations are determined by finite-dimensional nonautonomous mappings, whose dimension depends on the solutions frequency. We consider the case of two-dimensional reduced mappings, which occurs for frequencies close to the edges of the phonon band. For an homogeneous chain, the reduced map is autonomous and reversible, and bifurcations of reversible homoclinics or heteroclinic solutions are found for appropriate parameter values. These orbits correspond respectively to discrete breathers, or dark breathers superposed on a spatially extended standing wave. Breather existence is shown in some cases for any value of the coupling constant, which generalizes an existence result obtained by MacKay and Aubry at small coupling. For an inhomogeneous chain the study of the nonautonomous reduced map is in general far more involved. For the principal part of the reduced recurrence, using the assumption of weak inhomogeneity, we show that homoclinics to 0 exist when the image of the unstable manifold under a linear transformation intersects the stable manifold. This provides a geometrical understanding of tangent bifurcations of discrete breathers. The case of a mass impurity is studied in detail, and our geometrical analysis is successfully compared with direct numerical simulations