Visualization of Central Manifolds of Slow Motions in Neuronal Models
Map Reductions in high-order Neuronal Models
Special issue of Bifurcations & Chaos
Dynamics of neurons, computational neuroscience.
Mathematical aspects of classical synchronization theory
Canards and Chaos  in a two time scale map
Bifurcation of Blue Sky Catastrophe in slow fast systems   [MMJ]
Noninvertible singularly perturbed maps

Books

Shilnikov, Leonid P., Shilnikov, Andrey L., Turaev, Dmitry, and Chua, Leon O., Methods of qualitative theory in nonlinear dynamics. Volume II. World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 5. World Scientific Publishing Co., Inc., River Edge, NJ, 2001. ISBN: 981-02-4072-4 Preface  Appendix

Shilnikov, Leonid P., Shilnikov, Andrey L., Turaev, Dmitry V., and Chua, Leon O. Methods of qualitative theory in nonlinear dynamics. Volume I. World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 4. World Scientific Publishing Co., Inc., River Edge, NJ, 1998. ISBN: 981-02-3382-5 Preface

Шильников Л.П., Шильников А.Л., Тураев Д.В., Чуа Л. Методы качественной теории в нелинейной динамике. Часть 1 5-93972-305-5 ИКИ 2004 г. Переплет. 416 стр. *** Russian Edition of Volume I.

Шильников Л.П., Шильников А.Л., Тураев Д.В., Чуа Л. Методы качественной теории в нелинейной динамике. Часть 2 *** Russian Edition of Volume II in preparation!

Book chapters

Shilnikov, A. L. Shilnikov, L.P. and Turaev, D.V., Mathematical theory of synchronization, in Nonlinear Oscillation and Waves. Eds Gaponov-Grekhov & Nekorkin, pp. 426-450, 2005.

Gavrilov, N. and Shilnikov, A.L., Example of a blue sky catastrophe, in Methods of qualitative theory of differential equations and related topics, Amer. Math. Soc. Transl. Ser. 2(200), Amer. Math. Soc., pp. 99-105, 2000. [pfd]  [ps]

Pisarevskii, A., Shilnikov, A.L., And Turaev, D.V., Asymptotic normal forms for equilibria with a triplet of zero characteristic exponents in systems with symmetry, in Iteration theory, Grazer Math. Ber., 339, Karl-Franzens-Univ. Graz, Graz, pp. 291--300, 1999.

Shilnikov, A.L. and Shilnikov, L.P., Bifurcation analysis of the asymmetric Lorenz model, in Nonlinear World, Naukova Dumka, Kiev, Ukraine, pp. 124-129, 1990.

Journal articles

Channell P., Cymbalyuk, G. and Shilnikov, A. L., Origin of bursting through homoclinic spike adding in a neuron model, Phys. Rev. Letters, PRL 98, 134101, 2007 [pdf] [gzip.ps]

Channell P., Cymbalyuk, G. and Shilnikov, A. L., Applications of the Poincare mapping technique to analysis of neuronal dynamics, Neurocomputing, 70 (10-12), 2007; doi:10.1016/j.neucom.2006.10.091 [pdf]

Shilnikov, A.L. and Turaev D., Blue Sky Catastrophe, Scholarpedia, 2006

Shilnikov, A. L., Bursting manifolds of neuronal models, submitted to J. Bifurcations and Chaos, 2006.

Channell P., Cymbalyuk, G. and Shilnikov, A. L., Applications of the Poincare mapping technique to neuronal dynamics, submitted to SIAM Applied Dynamical Systems, 2006.

Channell P., Cymbalyuk G. and Shilnikov A.L. Origin of bursting through homoclinic spike adding in a neuron model, submitted to Phys Review Letters, 2007, [pdf] in review

Shilnikov, A.L., Shilnikov, L.P. And Turaev, D.V. Blue sky catastrophe in singularly perturbed systems, AMS Moscow Mathematical Journal, vol. 5(1), 205-218, 2005 [pdf], [gzipped ps]

Shilnikov, A. L. and Cymbalyuk, G. Transition between tonic-spiking and bursting in a neuron model via the blue-sky catastrophe, Phys Review Letters, 94, 048101 (2005) [pdf] [gzipped ps], see also
Virtual Journal of Biological Physics Research, February issue, 2005.

Cymbalyuk, G. and Shilnikov, A. L. Co-existent tonic spiking modes in a leech neuron model, J. Computational Neuroscience18 (3), 255-263, 2005 [pdf] [gzipped ps]

Shilnikov, A. L., Calabrese R. and Cymbalyuk, G., How a neuron model can demonstrate coexistence of tonic spiking and bursting? Neurocomputing 65-66, 869-875, 2005 [pdf]

Mira, C. and Shilnikov, A.L., Slow and fast dynamics generated by non-invertible plane maps, Bifurcations and Chaos 15(11), 2005. [pdf]

Shilnikov, A. L., Calabrese R. and Cymbalyuk, G. Mechanism of bi-stability: tonic spiking and bursting in a neuron model, Phys Review E 71(5), 056214-046221, 2005, [pdf]

Shilnikov, A.L., And Turaev, D.V., Homoclinic SN-bifurcations of periodic orbits in slow-fast systems, in reparation, 2005.

Shilnikov, A. L. and Cymbalyuk, G. Homoclinic saddle-node orbit bifurcations en a route between tonic spiking and bursting in neuron models, Invited review. Regular & Chaotic Dynamics 9( 3), 281-297, 2004. [pdf]

Shilnikov, A. L. Shilnikov, L.P. and Turaev, D.V., Mathematical aspects of classical synchronization theory Tutorial.  Bifurcations and Chaos vol. 14, no. 7, 2004. [pdf]

Shilnikov, A.L. And Rulkov, N.F. Subthreshold oscillations in a map-based neuron model, Physics Letters , A 328 , 177-184, 2004. [pdf]

Shilnikov, A.L. And Rulkov, N.F. Origin of chaos in a two-dimensional map modelling spiking-bursting neural activity, Bifurcations and Chaos 13(11), 3325-3340, 2003. [pdf]

Pisarevskii, V., Shilnikov, A. and Turaev, D., Asymptotic normal forms for equilibria with a triplet of zero characteristic exponents in systems with symmetry. Regular & Chaotic Dynamics 3 (1998), 1, 19-27. [pdf]  [ps]

Shilnikov, A. L., Homoclinic phenomena in laser models. Computational tools of complex systems, I. Comput. Math. Appl. 34 (1997), 2-4, 245-251 [pdf]

Shilnikov, A., Nicolis, G. and Nicolis, C., Bifurcation and predictability analysis of a low-order atmospheric circulation model. Internat. Bifurcations & Chaos 5(6), 1701-1711, 1995. [pdf]

Shilnikov, A. L., Shilnikov, L. P. and Turaev, D. V., Normal forms and Lorenz attractors. Bifurcations & Chaos 3(5), 1123-1139, 1993. [pdf]

Shilnikov, A.L., On bifurcations of the Lorenz attractor in the Shimizu-Morioka model. Physica D 62 (1-4), 338-346, 1993. [pdf]

Bykov, V. V. and Shilnikov, A. L., On the boundaries of the domain of existence of the Lorenz attractor Selecta Math. Soviet. 11 (1992), no. 4, 375--382. [pdf]

Bykov, V. V.; Shilnikov, A. L. Boundaries of the domain of existence of a Lorenz attractor (Russian). in Methods in qualitative theory and bifurcation theory, 151--159, Gorkov. Gos. Univ., Gorki, 1989.

Shilnikov, A. L. Bifurcation and chaos in the Marioka-Shimizu system (Russian). in Methods of qualitative theory of differential equations, 180--193, 216, Gorkov. Gos. Univ., Gorki, 1986.

Papers in refereed proceedings

Shilnikov L.P. and Shilnikov A.L, Development of Synchronization Theory, 2nd IEEE International Conference on Circuits and Systems for Communications June 30 - July 2, 2004, Moscow, Russia [pdf]

Cymbalyuk G., Calabrese R. and Shilnikov A., How a neuron model can demonstrate co-existence of tonic spiking and bursting? Proc. of the Annual Computational Neuroscience Meeting (CNS-2003) , 2004 [pdf]

Rulkov N.F, Bazhenov M.V. and Shilnikov A.L.,  Modeling of spiking-bursting neural activity using map-based models, Proc. Topical Problems of Nonlinear Wave Physics (NWP-2003) [pdf]

      Preprints

Shilnikov, A.L., Shilnikov, L.P. And Turaev, D.V. , Blue sky catastrophe in singularly perturbed systems, Preprint no. 841, WIAS, Berlin, (2003) [pdf] [gzipped ps]
Shilnikov, A.L., Shilnikov, L.P. And Turaev, D.V., On some mathematical topics in classic synchronization. WIAS Preprint No. 892, (2003) [pdf] [gzipped ps]

Grants and Rewards

Xlinx FPGA hardware and software donation ($5280), 2006
Shilnikov A. ( Co-PI ), Belykh I., Functional significance of connectivity for information processing in dynamical networks, Brains & Behaviors program, 09/12/2006, $23,000.
Shilnikov A ( PI ), Belykh I. Advanced bifurcation theory for complex dynamics in neuronal models, NSF, $356,830, 11.15.2006 pending
Shilnikov A. ( Co-PI ), Belykh I. Functional significance of connectivity for information processing in dynamical networks, NSF, $ 323,196 pending
Shilnikov A. ( PI ), Turaev D. Complex nonlocal bifurcations in slow-fast systems, BSF, $ 158,800, 11/15/2006 , pending
Pallas S., Belykh I. and Shilnikov A. ( Co-PI ) Modeling circuits for stimulus velocity tuning in the superior colliculus, Brains & Behaviors program, 09/12/2006 , $30,000, declined
Shilnikov A. ( Co-PI ), Bourgeois A., Belykh I. Complex networks in bursting neuronal networks, James S. McDonnell Foundation, 11/15/2006, pending
Brains & Behaviors Ph.D. fellowship for Mokhov K ($88,000), with Cymbalyuk G., 2005-2009.
Bifurcation Analysis of Neuronal Rhythmogenesis, NSF ( $1.4M) pending, with Calabrese R. (Emory) Cymbalyuk G.
Bifurcation Analysis of Neuronal Rhythmogenesis, NIH ($1.25M) pending, with Calabrese R. (Emory) Cymbalyuk G.
Modeling circuits for stimulus velocity tuning in the superior colliculus, Brains & Behaviors program ($30,000), with Pallas S., Prasad S., 2005.
Applications of the Poincare mapping technique to analysis of neuronal dynamics, Brains & Behaviors program, ($23,000) with Cymbalyuk G., 2005.
Brains & Behaviors research supplement, $6,000, 2005.
P20 Implementation of neuronal models, dynamic clamp, and tools for the hybrid system analysis using FPGA technology," $51,000, with Drs. Bourgeuis (CompSci) & Cymbalyuk (Physics), 2005-2006.
United States-Israel Bi-National Science foundation "Advanced bifurcation theory for neuronal models", $174,800. Pending. with Drs. Cymbalyuk & Turaev, 2005.
NATO Reintegration grant, $35,000.
Brains & Behaviors seed grant "The Bifurcation Analysis of Neuronal Rhythmogenesis" $30,000. 2004-2005, with Cymbalyuk G.
Two Graduate Research fellowships, $22,000 anuualy, for 4 years Brains & Behaviors program ($88,000) with Cymbalyuk G.
Summer Brains & Behaviors assistantship. $2,000, 2004.
Mobile Classroom for Teaching Mathematics and Statistics. Tech Fee Grant Program at GSU, ($105,045.00) with Vidakovic, D. (PI), Arav, M., Miller, Patterson, N. V., and Smirnova, A., 2004.
GSU Research Team Grant “Dynamics of Bursting Behaviours in Neural Models, ( $15,000) with Cymbalyuk G., 2004.
INTAS Grant 99-000 “Numerical Methods for Local and Global Bifurcations in ODEs” ( $65,000), Belgium-Netherlands, 1999–2001.
World Scientific Publishing Grant, Singapore -New York - London, 1996, 1998
INTAS Grant 93-0570 “Dynamical Systems: Bifurcations and Chaotic Attractors”, Belgium, 1995–1998
RFFI (Russian Foundation for Fundamental Research) No. 960101135 “Homoclinic Bifurcations”, 96-99
International Science Foundation Project No. R98300, 1995-1997
International Science Foundation Project No. R9800, 1994-1995
EC-Russia Collaborative Project ESPRIT P9282-ACTS, Brussels, Belgium, 1993-1996
The Royal Society Postdoctoral Award, London-Cambridge, UK, 1993-95

 Meetings

SIAM Conference on Applications of Dynamical Systems, May 28-June 1, 2007. My mini-symposium Excotic bifurcations in beuronal models
Dynamics Days 2007
Neuroscience Meeting 2006, Atlanta, October 14-18, 2006 [presentation]
Fifteenth Annual Computational Neuroscience Meeting CNS*2006 July 16 - July 20, 2006, Edinburgh, UK
Nizhny Novgorod Mathematical Society, June 8, 2006
Nonlinear Dynamics, Chaos and Applications Workshop: Stochastic and Chaotic Dynamics in Action: from Laser
    to Brain, from Communication to Medicine, Crimea, Ukraine, May 15-26, 2006. [Photos]
Workshop Origin and Regulation of Bursting Activity in Neurons, Atlanta, April 7-8, 2006
Annual South East Nerve Net 2006, March 30-31
Annual AMS meeting, San Antonio TX, January 12-15, 2006
Neuroscience Meeting, Washington DC, November 12-16, 2005
2005 Fall Central Section Meeting: Special Session in Mathematical Ecology Lincoln, NE, October 21-23, 2005
Conference on control and synchronization of dynamical systems, Invited talk, Oct 4-7 2005 León Gto. México
GSU biotech symposium, Implementation of neuronal models, dynamic clamp, and tools for the hybrid system analysis using FPGA technology, with Bourgeois A., and Cymbalyuk G. Atlanta, June 5, 2005
SIAM Conference on Applications of Dynamical Systems, May 22-26, 2005. My mini-symposium Complex dynamics of systems with multiple time scales
Dynamical Systems in Neuroscience (AMS special session), Santa Barbara, April 16-17, 2005
Henri Poincare symposium, Brussels, October 8-9, 2004 [photos]
Coupled Map Lattices 2004, Paris, June 21 - July 2, 2004
Summer School and Conference on Dynamical Systems, ICTP, Trieste, July 18 - August 6, 2004
CNS *2004 (Computational Neuroscience) Baltimore July 18th - 20st, [pdf]
2004 SIAM Conference on the Life Sciences I co-organize a mini-symposium The Geometry of Spiking and Bursting
SECABC symposium. Neurobehavioral section, Atlanta, May 19-21.
20th Annual South East Nerve Net, March 26-27, 2005, Atlanta poster [pdf]
Symmetry and Bifurcation in Biology, Banff International Research Station, Canada, May 31- June 05, 2003 [Talk] Photos -Courtesy I Belykh
Workshop II: Chaotic Dynamics and Transport, UCLA, May 19 - 23, 2003
2003 SIAM Conference on Applications of Dynamical Systems, Bursting in Mappings, May 27-31, 2003 [Talk]
11th IEEE Workshop on Nonlinear Dynamics of Electronic Systems, Scuol, Switzerland, May 18-21, 2003 [Paper1] [Paper2] [photos][ph2]
NATO Advanced Study Institute: Synchronization: Theory and Application,  May 19 – June 1, 2002 [Photographs]
Weierstrass Institute  for Applied Analysis and Stochastics, Berlin, Germany, June 2002
Dynamical Systems Seminar at Boston University, Boston, November 26, 2001  [pfd]
School and Workshop on Dynamical Systems, ICTP, Trieste, Italy, 30 July - 17 August, 2001
International Conference Progress in Nonlinear Science, Nizhny Novgorod, Russia, July 2-6, 2001   [Photographs]
MRI Workshop Analysis and Continuation of Bifurcations, Utrecht University, the Netherlands, June 21-22, 2001
Blue Sky Catastrophe in Two Time-Scale Systems, SIAM meeting on Dynamical Systems, Snowbird, 2001  [pfd]
Weierstrass Institute  for Applied Analysis and Stochastics, Berlin, Germany, July 2000
Bifurcations: Numerical Methods, Software, Applications, University of Gent, Belgium, June 29-30, 2000
Dynamical Systems and Applications, Atlanta, May 26-29, 1999
Numerical Methods for Bifurcation Problems, IMA, Minnesota, September 15 - 19, 1997
Nonlinear Centre, University of Cambridge, UK, May 1997
Conference “Nonlinear Dynamics Electronic Systems”, Moscow, Russia, 08/1996
Conference “Contemporary. Problems in Dynamical Systems Theory”, N.Novgorod, Russia, 06/1996
Solvey Meeting, Brussels, Belgium, 03/1996
Conference “Iteration Theory”, Urbino, Italy, 09/1996
Conference “Contemporary Problems in Theory of Dynamical Systems”, Moscow, 06/ 1996
Noninvertible Dynamical Systems Workshop, Minneapolis, Minnesota, March, 1995
Conference “Nonlinear Dynamics Elec. Systems”, Dublin, Ireland, 08/1995
Workshop “Singular Vector Fields”, Triest, Italy, 09/1995
Workshop “Homoclinic Bifurcations”, Amsterdam, Netherlands, 05/1994
Workshop “Dynamics of Vector Fields”, Kyoto, Japan, 06/1994
Conference “Dynamical Systems and Chaos”, Tokyo, Japan, 05/1994
Conference “Differential Equations”, Katseveli, Ukraine, 05/1994
Solvey Meeting, Brussels, Belgium, 01/1994
Workshop “Volga-laser-tour”, Russia, Moscow-Nizhny Novgorod, 06/1993
NATO Workshop “Complexity in Optical Systems”, Edinburgh, Scotland, 08/1992
NATO Workshop “Homoclinic Chaos”, Brussels, Belgium, 05/1991
Conference “Qualitative Theory of Differential Equations”, Riga, Lithuania, 05/1986
Spring School “Nonlinear Oscillations and Waves”, Gorky, Russia, 03/1983, 1985, 1987
Workshop “Numerical Methods In Bifurcation Theory”, Puschino, Russia, 02/1984-1990

     Seminars
Ben-Gurion University, Israel, March 10, 2005, Weizmann Istitute, March 9, 2005
UC Berkeley, Institute Nonlinear Science UC San Diego, Cornell University, GaTech, Boston University
Salk Institute
United Technologies Research Center
University of Cambridge, Bath University, Bristol University, UK
Center for Informatics and Computers (CWI), Netherlands, University of Utrecht
Brussels Free University, Royal Meteorological Institute, Solvey Institute, Belgium
University of Urbino, University of Modena, Italy
University Paris-7, Poincare Institute, France
Steklov Math. Institute, Keldysh Applied Math. Institute, Space Research Institute,
Electronic Research Institute, Moscow State University, Leningrad State University,
Institute for Applied Mathematics, Russia
Technical University of Copenhagen, Denmark.

  A few illustrations from Appendix in Volume II
Horseshoes in Henon map
Unimodal map
Route to Chaos in the Lorenz equation
Homoclinic Bifurcation in a 3D normal form with a saddle
Homoclinic Bifurcations in a 3D normal form with a saddle-focus
Homoclinic period doubling
Scenarios of chaos in cubic  Chua's circuit.
A periodic orbit becomes a homoclinic one to a saddle point:  1  &   2
Gavrilov-Guckenheimer cod-2 bifurcation
Shilnikov chaos