Mathematics Department Research Areas

DSYREKI: Dynamical Systems and Reaction Kinetics Networks

In this project, in which Assoc. Prof. İlknur Kuşbeyzi Aybar from the Department of Mathematics serves as a researcher, the primary goal is to advance the theory of multi-parameter dynamical systems governed by differential equations and to leverage this understanding to examine mathematical models that reflect real-world phenomena and processes. Supported by the EU HORIZON-MSCA-2023-SE-01 (MSCA Staff Exchanges 2023) program from 2024 to 2028, the initiative is coordinated by Univerza v Mariboru (Slovenia) alongside a international consortium of partner institutions: Center za Uporabno Matematiko in Teoreticno Fiz (Slovenia), Budapesti Muszaki es Gazdasagtudomanyi Egyete (Hungary), Consiglio Nazionale delle Ricerche (Italy), Hochschule für Technik und Wirtschaft Berlin (Germany), Obudai Egyetem (Hungary), Sichuan University (China), Board of Trustees of Northern Illinois University (USA), Medizinische Universitaet Wien (Austria), Piri Reis Universitesi (Turkey), Yeditepe University (Turkey), Universitatea din Craiova (Romania), Universidade de Sao Paulo (Brazil), and Shanghai Jiao Tong University (China). The primary goal of the project is to explore both qualitative and quantitative attributes of these systems, including their general structure, solution properties, bifurcations, and stability. Employing a synergistic approach that melds symbolic and numerical computation, the project team aims to enhance the analysis of stability and bifurcations within intricate biochemical systems by developing bespoke symbolic-numeric tools designed to uncover instances of multi-stability and bifurcations. This inherently interdisciplinary endeavor fuses methodologies from mathematics, applied mathematics, reaction kinetics, and symbolic computation algorithms, while incorporating the principles of the Green Charter to emphasize sustainable practices, environmental impact mitigation, and substantial training opportunities for early-stage researchers.

Homotopy Theory for Frölicher Spaces and Higher Geometric Structures

This project aims to study homotopy theories and higher geometric structures on one of the models of generalized smooth spaces known as Frölicher spaces. The project is supported by the TÜBİTAK 3501 program, where from the Department of Mathematics Assist. Prof. Mehmet Akif Erdal is the project executive and Assist. Prof. Melike Işim Efe and Assist. Prof. Barış Efe are researchers. The primary goal is to find convenient homotopy theories (e.g. Quillen model structures) on the category of Frölicher spaces, which later will be used to contribute to their differential geometry.

Switching Dynamics in Neuronal Populations Coupled to Astrocytes (SDNPACTA)

This project, in which Assist. Prof. İlknur Kuşbeyzi Aybar serves as a researcher, spans the 2024–2026 period and is supported by the TÜBİTAK (Turkey) – NITRA (Serbia) 2559 Joint Research Program. The project is led by Assoc. Prof. Orhan Özgür Aybar from Piri Reis University, with Prof. Dr. Muhammet Uzuntarla from Gebze Technical University also serving as a researcher. The primary goal of the project is to investigate the switching dynamics and multistability that arise in neuronal populations due to feedback coupling with astrocytes, building on the fact that the functions of neuronal populations are strongly influenced by metabolic resources mediated via astrocytes.

Investigation of Some Geometric Properties of Orlicz Spaces According to s-norms

Res. Assist. Esra Başar is a researcher of the TUBITAK 1001 project entitled "Investigation of Some Geometric Properties of Orlicz Spaces According to s-norms", led by Serap Öztop Kaptanoğlu from Istanbul University. The project aims to examine the geometry of Orlicz spaces, whose elements are defined in a measurement space, through s-norms, which are newly defined and comprehensive in the literature, and to obtain the relationships between these geometric features. What is aimed to be achieved in the project will also form the basis for examining different geometric structures of Orlicz spaces according to s-norms, about which little is known.

Dynamical Transitions and Stability of Some Mathematical Biology Models (Completed)

Res. Assist. Esmanur Yıldız Akıl participated in the project as a researcher, supervised by Prof. M. Taylan Şengül from the Department of Mathematics at Marmara University, and supported by the Research Fund of Marmara University. The study investigates the first transition dynamics of one-dimensional reaction–diffusion equations in a spatial domain, focusing on cases where the nonlinear term involves higher-order nonlinearities. By applying center manifold theory, a reduced equation is derived, and a classification of dynamic transitions is established based on this reduced system. The results provide a theoretical framework for analyzing dynamic transitions in systems with higher-order nonlinearities.

2026

• F. Kızılaslan , G. Cüran. 2026. “Comment on: Estimation of stress–strength reliability for Marshall–Olkin distributions based on progressively Type-II censored samples”, Journal of Applied Statistics, 1–2. DOI:10.1080/02664763.2025.2567994.
• G. Cüran, F. Kızılaslan. 2026. “Statistical inference on stress-strength reliability for the Marshall-Olkin family of distributions”, Soft Computing, 30, 189-208. DOI:10.1007/s00500-025-10968-3, 2026.
• E. Başar, S. Öztop. 2026. "On local convexities of Orlicz spaces endowed with s-norms", Colloq. Math., 180(2), 145-163.
• B. N. Örnek, S. Aydemir, T. Düzenli, T. Akyel. 2026. "Activation function design in complex neural networks via boundary analysis of analytical functions", Neurocomputing, 689, 133821.
• S. Kaouache, R. A. Meskine, D. Coksak, I. K. Aybar, O. O. Aybar. 2026. "Fractional-Order Suppression of Chaos in a Memristor-Based Chua Circuit", International Journal of Modern Physics B.
• D. C. Er, G. Polat, I. K. Aybar, O. O. Aybar. 2026. "Reliability-constrained spatial b-value mapping via kernel-weighted kNN with the magnitude-difference estimator and effective sample size", The European Physical Journal Plus, 141(6), 658.
• D. Coksak Er, G. Polat, I. Kusbeyzi Aybar, O. O. Aybar. 2026. "A multi-lag transfer entropy framework for seismic network analysis: persistent marmara isolation following the 2023 Kahramanmaraş sequence", The European Physical Journal Plus, 141(5), 553.

2025

• O. Esen, A. Gezici, M. Grmela, H. Gümral vd. 2025. "Conformal and contact kinetic dynamics and their geometrization", Journal of Geometry and Physics, 209, 105369.
• E. Başar, S. Öztop, B. H. Uysal, Ş. Yaşar. 2025. "Strict convexity of Orlicz spaces under renorming", J. Math. Anal. Appl., 546(2), 129236.
• I. Kusbeyzi Aybar, F. Kocaman, M. C. Turkmen, Y. Yavuz, B. Yilmaz, A. A. Ergin. 2025. "Oscillatory dynamics in the adaptive exponential integrate-and-fire model", Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(12).
• O. O. Aybar, I. Kusbeyzi Aybar. 2025. "Managing Financial Instability: A Linear Control Approach for A Chaotic Model", International Journal of Applied and Computational Mathematics, 11(5), 196.

2024

• O. Esen, A. Gezici, H. Gümral. 2024. "Discrete Dynamics on Locally Conformal Framework", Proc. Inst. Math. Mech., 50(1), 133–151.
• O. Esen, A. Gezici, H. Gümral. 2024. "Variational aspect and kinetic theory of locally conformal dynamics", J. Phys. A: Math. Theor., 57(36), 365201.
• B. Çalışkan Desova, M. Polat. 2024. "Existence, uniqueness, and blow-up analysis of a quasi-linear bi-hyperbolic equation with dynamic boundary conditions", Electronic Research Archive, 32(5), 3363-3376.
• B. Çalışkan Desova, M. Polat. 2024. "On the local existence and blow-up solutions to a quasi-linear bi-hyperbolic equation with dynamic boundary conditions", Partial Differ. Equ. Appl., 12, 100925.
• İ. S. Yüce, A. N. Narman. 2024. "Isometries of Length 1 in Purely Loxodromic Free Kleinian Groups and Trace Inequalities", Turkish J. Math., 48(2), 186-209.
• M. A. Erdal. 2024. "Homotopy theory of monoid actions via group actions and an Elmendorf style theorem", Collect. Math., 75(1), 331-359.
• M. A. Erdal. 2024. "Fibration category structures induced by enrichments", Turk. J. Math., 48(6), 1138-1155.
• T. Şengül, B. Tiryakioğlu, E. Yıldız Akıl. 2024. "First transition dynamics of reaction–diffusion equations with higher order nonlinearity", Stud. Appl. Math., 153(3), 12735.

2023

• H. Gümral. 2023. "Symmetry and reduction for second order degenerate Lagrangians", J. of Phys.: Conf. Series, 2667, 012073.
• B. Ferčec, M. Mencinger, T. Petek, O. O. Aybar, İ. Kuşbeyzi Aybar. 2023. "Qualitative Analysis of the Minimal Higgins Model of Glycolysis", MATCH Commun. Math. Comput. Chem., 90, 563–580.
• D. Çoksak Er. 2023. "On the low Lagrangian formulation of Vlasov-Poisson equations", Hacettepe J. Math. Stat., 52(4), 983-994.
• E. Başar, S. Öztop, B. H. Uysal, Ş. Yaşar. 2023. "Extreme points in Orlicz spaces equipped with s-norms and closedness", Math. Nachr., 296(11), 5042-5062.
• E. Başar, B. H. Uysal, Ş. Yaşar. 2023. "Complex extreme points and complex rotundity in Orlicz spaces equipped with the s-norm", Istanbul Journal of Mathematics, 1(1), 40-47.
• S. Öztop, A. Osançlıol, R. Üster, B. Arıs, E. Başar, vd. 2023. Orlicz Uzaylarının Teorisi ve Uygulamaları(1). İstanbul: Istanbul University Press.

2022

• O. Esen, P. Guha, H. Gümral. 2022. "3D-flows Generated by the Curl of a Vector Potential and Maurer-Cartan Equations", Turkish J. Math., 46(8), 3234-3244.
• H. Gümral. 2022. "Dirac’s Analysis and Ostrogradskii’s theorem for a Class of Second Order Degenerate Lagrangians", Int. J. of Geom. Meth. in Mod. Phys., 19, 2250008.
• M. Polat. 2022. "On the blow-up of solutions to a fourth-order pseudoparabolic equation", Turkish J. Math., 46(3), 946–952.
• T. Akyel, M. Lanza de Cristoforis. 2022. "Microscopic behavior of the solutions of a transmission problem for the Helmholtz equation. A functional analytic approach", Stud. Univ. Babes-Bolyai Math., 67(2), 383–402.
• T. Akyel, M. Lanza de Cristoforis. 2022. "Asymptotic behavior of the solutions of a transmission problem for the Helmholtz equation: A functional analytic approach", Math. Meth. Appl. Sci., 45(9), 5360-5387.
• T. Akyel. 2022. "Estimates for λ-Spirallike Functions of Complex Order on the Boundary", Ukr. Math. J., 74, 1-14.

2021

• O. Esen, H. Gümral, S. Sütlü. 2021. "Tulczyjew's Triplets for Lie Groups III: higher order dynamics and reductions on iterated bundles", Theoretical and Applied Mechanics, 48(2), 201-236.
• İ. Kuşbeyzi Aybar. 2021. "Memristor-based oscillatory behavior in the FitzHugh-Nagumo and Hindmarsh-Rose models", Nonlinear Dynamics, 103(3), 2917–2929.
• B. Ferčec, M. Dukaric, O. O. Aybar, İ. Kuşbeyzi Aybar. 2021. "Supercritical Hopf Bifurcations in Two Biochemical Reaction Systems", MATCH Commun. Math. Comput. Chem., 85(3), 525-544.
• B. N. Örnek, T. Akyel. 2021. "On bounds for the derivative of analytic functions at the boundary", Korean Journal of Mathematics, 29(4), 785-800.
• T. Akyel. 2021. "Upper bound of Hankel determinant for a class of analytic functions", Filomat, 35(11), 3713-3720.
• T. Akyel, B. N. Örnek. 2021. "Sharpened forms for λ-spirallike function of complex order on the boundary", AIP Conference Proceedings.
• G. Cüran, F. Kızılaslan. 2021. "Statistical inference of the stress-strength reliability and mean remaining strength of series system with cold standby redundancy at system and component levels", Hacettepe J. Math. Stat., 50(6), 1793-1821.
• G. Cüran, F. Kızılaslan. 2021. "Estimation of stress-strength reliability of a parallel system with cold standby redundancy at component level", Journal of the Turkish Statistical Association, 13(2), 74–87.
• T. Şengül, E. Yıldız. 2021. "A dynamical systems approach to the interplay between tobacco smokers, electronic-cigarette smokers and smoking quitters", Chaos, Solitons & Fractals, 146, 110870.

2020

• F. Çağatay Uçgun, O. Esen, H. Gümral. 2020. "Reductions of Topologically Massive Gravity II: First Order Realizations of Second Order Lagrangians", J. Math. Phys., 61(7), 073504.

2019

• O. Esen, M. Grmela, H. Gümral, M. Pavelka. 2019. "Lifts of Symmetric Tensors: Fluids, Plasma and Grad Hierarchy", Entropy, 21(9), 907.
• M. Polat. 2019. "A blow-up result for nonlocal thin-film equation with positive initial energy", Turkish J. Math., 43(3), 1797–1807.
• İ. Kuşbeyzi Aybar. 2019. "A dynamical analysis of the virus replication epidemic model", TWMS J. App. and Eng. Math., 9(2), 206-219.
• İ. Kuşbeyzi Aybar. 2019. "Stability and Hopf Bifurcation in Three-Dimensional Predator-Prey Models with Allee Effect", Sakarya University Journal of Science, 23(5), 1005-1011.
• T. Akyel, B. N. Örnek. 2019. "Applications of the Jack's lemma for the meromorphic functions at the boundary", Boletim da Sociedade Paranaense de Matemática, 38(7), 219-226.
• B. N. Örnek, T. Akyel. 2019. "Some remarks for a certain class of holomorphic functions at the boundary of the unit disc", Sakarya University Journal of Science, 23(3), 446-452.
• İ. S. Yüce. 2019. "Jörgensen's Inequality and Purely Loxodromic 2-Generator Free Kleinian Groups", Turkish J. Math., 43(2), 833-861.
• M. A. Erdal, A. Güçlükan İlhan. 2019. "A model structure via orbit spaces for equivariant homotopy", J. Homotopy Relat. Struct., 14(4), 1131-1141.
• V. Kalantarov, A. Mostafazadeh, N. Oflaz. 2019. "Blow-up solutions of Helmholtz equation for a Kerr slab with a complex linear and nonlinear permittivity", J. Math. Phys., 60(4), 043508.

2018

• O. Esen, H. Gümral. 2018. "Tulczyjew's Triplet for Lie Groups. II: Dynamics", Journal of Lie Theory, 27(2), 329--356.
• F. Çağatay Uçgun, O. Esen, H. Gümral. 2018. "Reductions of topologically massive gravity I: Hamiltonian analysis of second order degenerate Lagrangians", J. Math. Phys., 59(1), 013510.
• İ. Kuşbeyzi Aybar, O. O. Aybar, M. Dukaric, B. Ferčec. 2018. "Dynamical analysis of a two prey-one predator system with quadratic self interaction", Applied Mathematics and Computation, 333, 118-132.
• İ. Kuşbeyzi Aybar. 2018. "A super-predator, predator, and prey model with logistic growth", Int. J. Appl. Math. Stat., 57(2), 26–32.
• T. Akyel, T. A. Azeroğlu. 2018. "Note on the Uniqueness Holomorphic Function on the Unit Disk", Filomat, 32(6), 2321–2325.
• İ. S. Yüce. 2018. "Symmetric decompositions of free Kleinian groups and hyperbolic displacements", Communications in Analysis and Geometry, 26(6), 1375–1447.
• M. A. Erdal. 2018. "On smooth manifolds with the homotopy type of a homology sphere", Topology and its Applications, 234, 82-91.
• M. A. Erdal, Ö. Ünlü. 2018. "Semigroup Actions on Sets and the Burnside Ring", Appl. Categor. Struct., 26, 7-28.
• N. Oflaz, A. Mostafazadeh, M. Ahmady. 2018. "Scattering due to geometry: Case of a spinless particle moving on an asymptoticallyflat embedded surface", Physical Review A, 98, 022126.

2017

• B. N. Örnek, T. Akyel. 2017. "The generalizations of the Carathèodory Inequality for the holomorphic functions", Sakarya University Journal of Science, 21(6), 1254-1257.
• B. N. Örnek, T. Akyel. 2017. "Uniqueness Part of the Schwarz Lemma at the Boundary", Filomat, 31(12), 3643–3650.
• B. N. Örnek, T. Akyel. 2017. "Some Remarks on Schwarz Lemma at the Boundary", Filomat, 31(13), 4139–4151.
• B. N. Örnek, T. Akyel. 2017. "Representation with majorant of the Schwarz Lemma at the boundary", Publications de l’Institut Mathématique, 101(115), 191–196.
• M. İşim Efe, E. Abadoğlu. 2017. "Global Existence of Bi-Hamiltonian Structures on Orientable Three-Dimensional Manifolds", Symmetry, Integrability and Geometry: Methods and Applications, 13, 055.
• A. Mostafazadeh, N. Oflaz. 2017. "Unidirectional reflection and invisibility in nonlinear media with an incoherent nonlinearity", Physics Letters A, 381, 3548-3552.

2016

• H. Gümral. 2016. "Lagrangian description, symplectization, and Eulerian dynamics of incompressible fluids", Turkish J. Math., 40(5), 925-940.
• O. Esen, A. G. Choudhury, P. Guha, H. Gümral. 2016. "Superintegrable Cases of Four-dimensional Dynamical Systems", Regular and Chaotic Dynamics, 21(2), 175-188.
• B. N. Örnek, T. Akyel. 2016. "An improved lower bound for Schwarz lemma", J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math., 23(1), 61–72.
• B. N. Örnek, T. Akyel. 2016. "Sharpened forms of the generalized Schwarz inequality on the boundary", Proc. Math. Sci., 126, 69–78.

2015

• İ. Kuşbeyzi Aybar, O. O. Aybar, B. Ferčec, V. G. Romanovski, S. S. Samal, A. Weber. 2015. "Investigation of invariants of a chemical reaction system with algorithms of computer algebra", MATCH Commun. Math. Comput. Chem., 74(3), 465-480.
• O. I. Shpotyuk, A. S. Hacinliyan, V. O. Balitska, A. Kozdras, Y. Skarlatos, İ. Kuşbeyzi Aybar, O. O. Aybar. 2015. "Low-dimentional deterministical chaos in thermally-induced physical aging of As10Se90 glass probed with nonlinear time-series method", 8th CHAOS Conference Proceedings, Henri Poincaré Institute, 55-607.
• B. N. Örnek, T. Akyel. 2015. "A sharp Schwarz Lemma at the boundary", J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math., 22(3), 263-273.
• Ö. Akarsu, T. Dereli, N. Oflaz. 2015. "Accelerated expansion of the Universe in a higher dimensional modified gravity with Euler-Poincaré terms", Class. Quantum Grav., 32, 215009.