`What I saw, I wrote down; what I didn't see, I left out.'

       Mikhail Bulgakov, A Theatrical Novel

* H. Chiba, G.S. Medvedev, M.S. Mizuhara, Bifurcations and patterns in the Kuramoto model with inertia, submitted, ( arxiv ).

* G.S. Medvedev and G. Simpson, A Numerical Method for a Nonlocal Diffusion Equation with Additive Noise, submitted, ( arxiv ).

* P. Dupuis and G.S. Medvedev, The large deviation principle for interacting dynamical systems on random graphs, Comm. in Math. Phys., 390, (2022) ( arxiv).

* H. Chiba and G.S. Medvedev, Stability and bifurcation of mixing in the Kuramoto model with inertia, SIAM J. Math. Anal., 54(2), 2022 ( arxiv ).

* G.S. Medvedev and M. Mizuhara, Chimeras unfolded, J. Stat. Phys., 86, 2022.

* G.S. Medvedev and M. Mizuhara, Stability of Clusters in the Second-Order Kuramoto Model on Random Graphs , J. Stat. Physics, 2021, ( arxiv).

* H. Chiba, G.S. Medvedev, M.S. Mizuhara, Instability of mixing in the Kuramoto model: From bifurcations to patterns, Pure and Applied Functional Analysis, accepted, ( arxiv).

* D. Kaliuzhnyi-Verbovetskyi and G.S. Medvedev, Sparse Monte Carlo method for nonlocal diffusion equations, submitted arxiv

* G.S. Medvedev, The continuum limit of the Kuramoto model on sparse random graphs, Communications in Mathematical Sciences, vol. 17 (2019), no. 4,pp. 883- 898.

* H. Chiba and G.S. Medvedev, The mean field analysis for the Kuramoto model on graphs I. The mean field equation and transition point formulas, Discrete and Continuous Dynamical Systems - A, 39(1), 2019.   ( abstract )

* H. Chiba and G.S. Medvedev, The mean field analysis for the Kuramoto model on graphs II. Asymptotic stability of the incoherent state, center manifold reduction, and bifurcations, Discrete and Continuous Dynamical Systems - A, 39(7), 2019.   ( abstract )   arxiv

* H. Chiba, G.S. Medvedev, and M. Mizuhara, Bifurcations in the Kuramoto model on graphs, Chaos 28, 073109 (2018),   abstract,   arxiv

* D. Kaliuzhnyi-Verbovetskyi and G.S. Medvedev, The mean field equation for the Kuramoto model on graph sequences with non-Lipschitz limit, SIAM J. Math. Anal., 50 (2018), no. 3, 2441-2465, abstract   pdf

* G.S. Medvedev and X. Tang, The Kuramoto model on power law graphs: Synchronization and Contrast States, Journal of Nonlinear Science, 2018,   abstract   arxiv

* D. Kaliuzhnyi-Verbovetskyi and G.S. Medvedev, The semilinear heat equation on sparse random graphs, SIAM J. Math. Anal., 49(2), 1333-1355, 2017.   abstract   pdf

* G.S. Medvedev and J.D. Wright, Stability of twisted states in the continuum Kuramto model, SIAM J. Appl. Dyn. Syst., 16(1), 188-203, 2017.   abstract   pdf

* P. Hitczenko and G.S. Medvedev, Stability of equilibria of randomly perturbed maps, Discrete and Continuous Dynamical Systems - B, 22(2), 2017.   abstract   pdf

* D. Kaliuzhnyi-Verbovetskyi and G.S. Medvedev, The mean field equation for the Kuramoto model on graph sequences with non-Lipschitz limit, SIAM J. Math. Anal., 50 (2018), no. 3, 2441-2465, abstract   pdf

* G.S. Medvedev and X. Tang, The Kuramoto model on power law graphs: Synchronization and Contrast States, Journal of Nonlinear Science, 2018,   abstract   arxiv

* G.S. Medvedev and J.D. Wright, Stability of twisted states in the continuum Kuramto model, SIAM J. Appl. Dyn. Syst., 16(1), 188-203, 2017.   abstract   pdf

* P. Hitczenko and G.S. Medvedev, Stability of equilibria of randomly perturbed maps, Discrete and Continuous Dynamical Systems - B, 22(2), 2017.   abstract   pdf

G.S. Medvedev and X. Tang, Synchronization of coupled chaotic maps, Physica D, 2015.   abstract   arxiv

G.S. Medvedev and X. Tang, Stability of twisted states in the Kuramoto model on Cayley and random graphs, Journal of Nonlinear Science, 2015.   abstract   arxiv

G.S. Medvedev, The nonlinear heat equation on dense graphs and graph limits, SIAM J. Math. Analysis, 46(4), 2743-2766, 2014. abstract   pdf

G.S. Medvedev, The nonlinear heat equation on W-random graphs, Archive for Rational Mechanics and Analysis June 2014, Volume 212, Issue 3, pp 781-803, 2014.   abstract   pdf

G.S. Medvedev, Small-world networks of Kuramoto oscillators, Physica D 266 (2014), 13-22. abstract   arXiv preprint

P. Hitczenko and G.S. Medvedev, The Poincare map of randomly perturbed periodic motion, J. Nonlin. Sci., J. Nonlin. Sci., Vol. 23(5), pp. 835-861, 2013. (abstract) (arXiv preprint)

G.S. Medvedev, Stochastic stability of continuous time consensus protocols, SIAM J. Control Optim., Vol. 50, No. 4, pp. 1859-1885, 2012. PDF

G.S. Medvedev and S. Zhuravytska, Shaping bursting by electrical coupling and noise, Biol. Cybernatics, 2012 arXiv:1111.0642.

G.S. Medvedev and S. Zhuravytska, The geometry of spontaneous spiking in neuronal networks, J. Nonl. Sci., 2012. ( arXiv:1105.2801).

G.S. Medvedev, Synchronization of coupled limit cycles, Journal of Nonlinear Science, 2011. PDF

G.S. Medvedev, Synchronization and spontaneous dynamics in the Locus Coeruleus network. BMC Neuroscience 2011 12 (Suppl 1):P219.

G.S. Medvedev, Synchronization of coupled stochastic limit cycle oscillators, Physics Letters A,Volume 374, Issues 15-16, 1712-1720, 2010. PDF.

E. Manica, G.S. Medvedev, and J.E. Rubin, First return maps for the dynamics of synaptically coupled conditional bursters, Biological Cybernetics, 103:87-104, 2010. ( PDF )

G.S. Medvedev, Electrical coupling promotes fidelity of responses in the networks of model neurons, Neural Computation, 21(11) 3057-3078, 2009. ( PDF )

P. Hitczenko, G.S. Medvedev, Bursting oscillations induced by small noise, SIAM J. Appl. Math., Volume 69, Issue 5, pp. 1359-1392, 2009. ( PDF )

G. Medvedev, Noise-induced bursting in stochastic models of single cells and electrically coupled ensembles, BMC Neuroscience 2008, 9(Suppl 1):O5.

G.S. Medvedev and Yun Yoo, Chaos at the border of criticality, CHAOS 18, 033105 (2008) ( PDF )

G.S. Medvedev and Yun Yoo, Multimodal oscillations in systems with strong contraction, Physica D, 228(2), 87-106, 2007. ( PDF )

G.S. Medvedev, Transition to bursting via deterministic chaos, Phys. Rev. Lett. 97, 048102 (2006) ( PDF )

G.S. Medvedev, Reduction of a model of an excitable cell to a one-dimensional map, Physica D, 202(1-2), 37-59, 2005. ( PDF )

G.S. Medvedev, J.E. Cisternas, Multimodal regimes in a compartmental model of the dopamine neuron, Physica D, 194(3-4), 333-356, 2004. ( PDF )

G.S. Medvedev, C.J. Wilson, J.C. Callaway, and N. Kopell, Dendritic synchrony and transient dynamics in a coupled oscillator model of the dopaminergic neuron, Journal of Computational Neuroscience, 15(1), pp. 53-69, 2003. ( PDF )

G.S. Medvedev, K. Ono, and P. Holmes, Traveling wave solutions of the degenerate Kolmogorov-Petrovski-Piskunov equation, European Journal of Applied Mathematics, 14(3), 343-367, 2003. ( PDF )

G.S. Medvedev and N. Kopell, Synchronization and transient dynamics in the chains of electrically coupled FitzHugh-Nagumo oscillators, SIAM J. Appl. Math., vol. 61, No. 5, pp. 1762-1801. ( PDF )

G.S. Medvedev, T.J. Kaper, and N. Kopell, A reaction-diffusion system with periodic front dynamics, SIAM J. Appl. Math., Vol. 60, No. 5, pp. 1601-1638. ( PDF )

G.S. Medvedev and V.G.Prikazchikov, Two-sided eigenvalue estimates for some spectral problems, in J. DeSanto, editor, Mathematical and numerical aspects of wave propagation, SIAM, Philadelphia, 1998.

V.G. Prikazchikov and G.M., On the asymptotic error estimate for a discrete problem of fourth-order accuracy. J. Math. Sci. (New York) 84 (1997), no. 4, 1304--1309. 65N25 (65N15)

V.G. Prikazchikov and G.S. Medvedev, An approximate method for solving a spectral problem in a nonconvex domain. (Russian) Differentsial\cprime nye Uravneniya 30 (1994), no. 12,2153--2161, 2207 (1995); translation in Differential Equations 30, no. 12, 1979--1986 (1995) 65N25