John Guckenheimer

Professor of Mathematics and Theoretical and Applied Mechanics

565 Malott Hall
(607) 255-8290
gucken@cam.cornell.edu

 
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    • Publications

  • J. Guckenheimer, Endomorphisms of the Riemann sphere, Proc. Am. Math. Soc. Symposia in Pure Math. XIV, 95-124, 1970. 
  • J. Guckenheimer, Holomorphic vector fields on CP(2), An. Acad. Brasil. Cienc. 42, 415-420, 1970. 
  • J. Guckenheimer, Axiom A + No Cycles implies zeta(f) rational. Bulletin Am. Math. Soc. 76, 592-594, 1970. 
  • J. Guckenheimer, Hartman's theorem for complex flows in the Poincare domain. Compositio, 24, 75-82, 1972. 
  • J. Guckenheimer, Absolutely omega-stable diffeomorphisms, Topology 11, 195-197, 1972. 
  • J. Guckenheimer, Bifurcation and catastrophe, Dynamical Systems, ed. M.M. Peixoto, Academic Press, 99-110, 1973. 
  • J. Guckenheimer, One parameter families of vector fields: another non-density theorem, Dynamical Systems, ed. M.M. Peixoto, Academic Press, 111-128, 1973. 
  • J. Guckenheimer, Catastrophes and partial differential equations. Annales de l'Institut Fourier 23, 31-59, 1073. 
  • J. Guckenheimer, Review of Stabilite Structurelle et Morphogenese by Rene Thom, Bulletin Am. Math. Soc. 79, 878-890, 1973. 
  • J. Guckenheimer, Caustics, Global Analysis and Its Applications II, International Atomic Energy Agency, 281-289, 1974. 
  • J. Guckenheimer, Caustics and non-degenerate hamiltonians, Topology 13, 127-133, 1974. 
  • J. Guckenheimer, Solving a single conservation law, Lecture Notes in Mathematics, 468, 108-134, 1974. 
  • J. Guckenheimer, Isochrons and phaseless sets, J. Math. Biology 1, 259-273, 1975. 
  • J. Guckenheimer, Shocks and rarefactions in two space dimensions, Arch. Rat. Mech. Anal. 59, 281-291, 1975. 
  • G. Oster and J. Guckenheimer, Bifurcation phenomena in population models, in The Hopf Bifurcation Theorem and its Applications, ed. J. Marsden and M. McCracken, Springer-Verlag, 327-353, 1976. 
  • J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation Theorem and its Applications, ed. J. Marsden and M. McCracken, Springer-Verlag, 368-381, 1976. 
  • J. Guckenheimer, G. Oster and A. Ipaktchi, Dynamics of deterministic population models, J. Math. Biology 4, 101-147, 1977. 
  • J. Guckenheimer, Constant velocity waves in oscillating chemical reactions, Lecture Notes in Mathematics, 525, 99-103, 1976. 
  • J. Guckenheimer, The cusps of Zeeman's catastrophe machine, Topology 16, 177-180, 1977. 
  • D. Auslander, J. Guckenheimer and G. Oster, Random evolutionarily stable strategies, J. Theoretical Population Biology 13, 276-293, 1978. 
  • J. Guckenheimer, On the bifurcation of maps of the interval, Inventiones Math. 39, 165-178, 1977. 
  • J. Guckenheimer, Comments on catastrophe and chaos, Some Mathematical Questions in Biology IX, Am. Math. Soc. 1-48, 1978. 
  • J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Publ. I. H. E. S. 50, 60-72, 1979. 
  • J. Guckenheimer, The bifurcation of quadratic functions, Annals. N.Y. Acad. Sci. 316, 78-85, 1979. 
  • J. Guckenheimer, The catastrophe controversy, Mathematical Intelligencer 1, 15-20, 1978. 
  • J. Guckenheimer, J. Moser and S. Newhouse, Dynamical Systems, CIME Lectures, Birkhauser, 289pp., 1980. 
  • J. Guckenheimer, A brief introduction to dynamical systems, Lectures in Appl. Math. 17, 187-253, 1979. 
  • J. Guckenheimer, Instabilities and chaos in nonhydrodynamic systems, in Hydrodynamic Instabilities and the Transition to Turbulence, ed. H. L. Swinney, Springer-Verlag, 271-287, 1980. 
  • J. Guckenheimer, Sensitive dependence to initial conditions for one dimensional maps, Comm. Math. Phys. 70, 133-160, 1979. 
  • J. Guckenheimer, One dimensional dynamics, Annals N. Y. Acad. Sci. 316, 76-85, 1979. 
  • J. Guckenheimer, Symbolic dynamics and relaxation oscillations, Physica D1, 227-235, 1980. 
  • J. Guckenheimer, Growth of topological entropy for one dimensional maps, Lecture Notes in Mathematics 819, 216-223, 1980. 
  • L. Block, J. Guckenheimer, M. Misiurewicz and Lai Sang Young, Periodic points of one dimensional maps, Lecture Notes in Mathematics 819, 18-34, 1980. 
  • J. Guckenheimer, Patterns of bifurcation, New Approaches to Nonlinear Problems in Dynamics, ed. P. Holmes, SIAM, 71-104, 1980. 
  • J. Guckenheimer, Dynamics of the van der Pol equation, IEEE Trans. Circuits and Systems 27, 983-989, 1980. 
  • J. Guckenheimer, On a codimension two bifurcation, Lecture Notes in Mathematics 898, 99-142, 1981. 
  • J. Guckenheimer, Noise in chaotic systems, Nature 298, 358-361, 1982. 
  • J. Guckenheimer and E. Knobloch, Nonlinear convection in a rotating layer: amplitude expansions and normal forms, Geophys. Astrophys. Fluid Dynamics 23, 247-272, 1983. 
  • E. Knobloch and J. Guckenheimer, Convective transitions induced by varying aspect ratio, Phys. Rev. A27, 408-417, 1983. 
  • J. Guckenheimer, An introduction to chaotic motion and strange attractors, Lectures in Appl. Math. 20, 220-227, 1983. 
  • J. Guckenheimer, Persistent properties of bifurcations, Physica D7, 105-110, 1983. 
  • J. Guckenheimer, Strange attractors in fluid dynamics, Lecture Notes in Physics 179, 149-157, 1983. 
  • J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, 453pp., 1983. 
  • J. Guckenheimer, Toolkit for nonlinear dynamics, IEEE Trans. Circuits and Systems 30, 586-590, 1983. 
  • J. Guckenheimer and G. Buzyna, Dimension measurements for geostrophic turbulence, Phys. Rev. Lett. 51, 1438-1441, 1983. 
  • J. Guckenheimer, Multiple bifurcations of codimension two, SIAM J. Math. Anal. 15, 1-49, 1984. 
  • J. Guckenheimer, Dimension estimates for attractors, Contemporary Mathematics 28, 357-367, 1984. 
  • J. Guckenheimer and R. McGehee, A proof of the Mandelbrot N-squared conjecture, Mittag-Leffler Institute Report 15, 12pp., 1984. 
  • R. Tycko, A. Pines and J. Guckenheimer, Fixed point theory of iterative excitation schemes in NMR, J. Chem. Phys. 83, 2775-2802, 1985. 
  • J. Guckenheimer, Multiple bifurcation problems for chemical reactors, Physica 20D, 1-20, 1986. 
  • J. Guckenheimer, Global Bifurcations in simple models of a chemical reactor, Lectures in Applied Mathematics, 24, 163-174, 1986. 
  • J. Guckenheimer, A codimension two bifurcation with circular symmetry, Contemporary Mathematics, 56, 175-184, 1986. 
  • J. Guckenheimer, M. Golubitsky and J. Guckenheimer, eds., Multiparameter Bifurcation Theory, Contemporary Mathematics 56, American Mathematical Society, xvii + 387 pp., 1986. 
  • J. Guckenheimer, H. Cho, R. Tycko, A. Pines, and J. Guckenheimer, Iterative maps for bistable excitation of two-level systems, Phys. Rev. Lett. 56, 1905-1908, 1986. 
  • J. Guckenheimer, Strange attractors in fluids: another view, Ann. Rev. Fluid Mech., 18, 15-31, 1986. 
  • J. Guckenheimer, G. Dangelmayr and J. Guckenheimer, On a four parameter family of planar vector fields, Arch. Rat. Mech. Anal., 97, 321-352, 1987. 
  • J. Guckenheimer, Renormalization of one dimensional mappings and strange attractors, Contemporary Mathematics 58 part III, 143--160, 1987. 
  • J. Guckenheimer, Limit Sets of S-unimodal maps with zero entropy, Comm. Math. Phys. 110, 655--659, 1987. 
  • J. Guckenheimer, Comments on multiple bifurcations, Proceedings of 1986 Army Research Conference. 
  • J. Guckenheimer and P. Holmes, Structurally stable heteroclinic cycles, Proc. Camb. Phil. Soc. 103, 189--192, 1988. 
  • Dieter Armbruster, J. Guckenheimer and P. Holmes, Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry, Physica D29, 257--282, 1988. 
  • J. Guckenheimer, R. Rand and D. Schlomiuk, Degenerate homoclinic cycles in perturbations of quadratic hamiltonian systems, Nonlinearity, 2, 405--418, 1989. 
  • Dieter Armbruster, J. Guckenheimer and P. Holmes, Kuramoto-Sivashinsky Dynamics on the center-unstable manifold, SIAM J. Appl. Math, 49, 676--691, 1989. 
  • J. Guckenheimer, Obstacles to modelling large dynamical systems, Mathematical Approaches to Problems in Resource Management and Ecology, C. Chavez et al., ed. Lecture Notes in Mathematical Biology 81, 319--327, 1989. 
  • S. Kim, R. MacKay and John Guckenheimer, Resonance regions for families of torus maps, Nonlinearity, 2, 391--404, 1989. 
  • D. Armbruster, J. Guckenheimer and S. Kim, Chaotic Dynamics in Systems with Square Symmetry, Physics Letters A, 140, 416--420, 1989. 
  • S. Kim, J. Guckenheimer, FunTorus, interactive explorations of the two dimensional torus, MSI Technical Report, 1989. 
  • J. Guckenheimer and S. Johnson, Distortion of S-unimodal maps, Annals of Mathematics, 132, 71--130, 1990. 
  • J. Guckenheimer, R. Rand and D. Schlomiuk, Integrability of plane quadratic vector fields, Expositiones Mathematicae, 8, 3--25, 1990. 
  • S. Kim and J. Guckenheimer, The two dimensional torus: Numerical explorations and mathematical conjectures, Chaos/XAOC Soviet-American Perspectives on Nonlinear Science, AIP, 145--152, 1990. 
  • G. Berkooz, J. Guckenheimer, P. Holmes, J. Lumley, J. Marsden, N. Aubry and E. Stone, Dynamical Systems Theory Approach to the Wall Region, AIAA Fluid Dynamics, Plasma Dynamics and Lasers Conference, 1990. 
  • J. Guckenheimer and S. Kim, kaos, Dynamical System Toolkit with Interactive Graphic Interface, MSI Technical report, 1990. 
  • C. Baesens, J. Guckenheimer, S. Kim and R. MacKay, Simple Resonance Regions of Torus Diffeomorphisms, Patterns and Dynamics of Reactive Media (Aris, Aronson & Swinney eds.), IMA Volumes in Mathematics and its Applications, 37, 1--11, 1991. 
  • C. Baesens, J. Guckenheimer, S. Kim and R. MacKay, Three Coupled Oscillators: Mode-locking, Global Bifurcations and Toroidal Chaos, Physica D, 49, 387--485, 1991. 
  • D. Armbruster, J. Guckenheimer, and S. Kim, Resonant Surface Waves in a Square Container, Differential Equations and Computer Algebra, M. Singer, ed., Academic Press, 61--76, 1991. 
  • J. Guckenheimer, Computational Environments for Exploring Dynamical Systems, Int., Journal of Bifurcation and Chaos, 1, 269--276, 1991. 
  • J. Guckenheimer, Symmetry Properties of Confined Convective States, Z. angew Math Phys, 43, 422--429, 1992. 
  • A. Back, J. Guckenheimer, and M. Myers, A Dynamical Simulation Facility for Hybrid Systems, MSI Technical Report 92-6, 1992. 
  • J. Guckenheimer, Chaos: Science or Non-science, Nonlinear Science Today, 1(2), 6-8, 1992. 
  • J. Guckenheimer and A. Mahalov, Resonant Triad Interactions in Systems with O(2) Symmetry, Physica D, 54, 267--310, 1992. 
  • J. Guckenheimer, A. Back, J. Guckenheimer, M. Myers, F. Wicklin and P. Worfolk, dstool: Computer Assisted Exploration of Dynamical Systems, Notices of the American Mathemtical Society, 39, 303--309, 1992. 
  • J. Guckenheimer and A. Mahalov, Instability induced by symmetry reduction, Phys. Rev. Lett., 68, 2257--2260, 1992. 
  • J. Guckenheimer and P. Worfolk, Instant Chaos, Nonlinearity 5, 1211--1222, 1992. 
  • J. Guckenheimer and I. Labouriau, Bifurcation of the Hodgkin Huxley Equations: A New Twist, Bulletin of Math. Biol., 55, 937--952, 1993. 
  • D. Golomb, J. Guckenheimer, and S. Gueron, Reduction of a Channel Based Model for a Stomatogastric Ganglion LP Neuron, Biol. Cybernetics, 69, 129--137, 1993. 
  • J. Guckenheimer and P. Worfolk, Dynamical Systems: Some Computational Problems, in Bifurcations and Periodic Orbits of Vector Fields, D. Schlomiuk, ed. pp241-278, Kluwer 1993. 
  • J. Guckenheimer, S. Gueron and R. Harris-Warrick, The Dynamics of a Conditionally Bursting Neuron, Philosophical Transactions of Royal Society, 341, 345-359, 1993. 
  • J. Guckenheimer and S. Johnson, Beyond Hyperbolicity: Expansion Properties of One Dimensional Mappings,in From Topology to Computation, Hirsch, Marsden and Shub, editors, 227-236, Springer-Verlag, 1993. 
  • J. Guckenheimer, M. Myers and B. Sturmfels, Computing Hopf Bifurcations I, SIAM J. Num. Anal., in press. 
  • J. Guckenheimer, and M. Myers, Computing Hopf Bifurcations II, SIAM J. Sci. Comp, in press. 
  • J. Guckenheimer, A Robust Hybrid Stabilization Strategy for Equilibria, IEEE Trans. Automatic Control, 40, 321-326, 1995. 
  • J. Guckenheimer and S. Malo, Computer-generated Proofs of Phase Portraits for Planar Systems, , Int. J. Bif. and Chaos, 6, 889-892, 1996. 
  • E. J. Doedel, M. J. Friedman and J. Guckenheimer, On computing connecting orbits in the Sine-Gordon and Hodgkin-Huxley equations, IEICE Trans. fundamentals, E77-A, 1-4, 1994. 
  • J. Guckenheimer, R. Harris-Warrick, L. Coniglio, N. Barazangi, J. Guckenheimer and S. Gueron, Dopamine modulation of transient potassium current evokes phase shifts in a central pattern generator network, J. Neuroscience, 15, 342-358, 1995. 
  • J. Guckenheimer, The role of geometry in computational dynamics, Dynamics, Bifurcation and Symmetry, P. Chossat, ed. 159-166, 1994. 
  • J. Guckenheimer, A general review of dynamical systems, in Ecological Time Series, T. Powell and J. Steele, eds., Chapman-Hall, 54-69, 1994. 
  • J. Guckenheimer, Phase portraits of planar vector fields: computer proofs, J. Experimental Mathematics, 4, 153-164, 1995. 
  • J. Guckenheimer and S. Johnson, Planar hybrid systems, Hybrid Systems II, Lecture Notes in Computer Science, 999, 202-225, 1995. 
  • W. Govaerts, J. Guckenheimer and A. Khibnik, Defining functions for multiple Hopf bifurcations, SIAM J. Num. Anal., in press. 
  • J. Guckenheimer, R. Harris-Warrick, L. Coniglio, R. Levini, S. Gueron and J. Guckenheimer, Dopamine modulation of two subthreshold currents produces phase shifts in activity of an identified neuron, J. Neurophysiology, 74, 1404-1420, 1995. 
  • J. Guckenheimer, Numerical analysis of dynamical systems, in Chaotic, Fractal and Nonlinear Signal Processing, R. Katz ed., AIP Press, 3-13, 1996. 
  • J. Guckenheimer, Phase portraits of planar vector fields: computer proofs, J. Experimental Math., 4, 153-165, 1995. 
  • J. Guckenheimer, Towards a global theory of singularly perturbed systems, Progress in Nonlinear Differential Equations and Their Applications, 19, 214-225, 1996. 
  • J. Guckenheimer and P. Rowat, Dynamical systems analyses of real neuronal networks, in Neurons, Networks nad Motor Behavior, P. Stein, S. grillner and A. Seleverston, eds., MIT Press, in press. 
  • J. Guckenheimer, R. Harris-Warrick, J Peck and A. Willms, Bifurcation, bursting and spike frequency adaptation, to appear. 
  • J. Guckenheimer, Computer proofs for bifurcations of planar dynamical systems, in Computational Differentiation, Techniques, Applications and Tools, eds. M. Berz, C. Bischof, G. Corliss and A. Griewank, SIAM, 229-238, 1996.
  • J. Guckenheimer, Complexity of hybrid switching models, in Control using Logic-based Switching, A. Morse., ed. Springer-Verlag, Lecture Notes in Control and Information Sciences, 13-16, 1997. 
  • J. Guckenheimer and W. G. Choe, Using dynamical systems tools in Matlab, Proceedings of IMA Workshops, in press.
  • A. R. Willms, D. J. Baro, R. M. Harris-Warwick and J. Guckenheimer, An improved parameter estimation method for Hodgkin-Huxley models, J. Comp. Neuroscience, in press.
  • J. Guckenheimer, Book Review: Dynamical Systems and Numerical Analysis by Stuart and Humphries, Ergodic Theory and Dynamical Systems, in press.
  • J. Guckenheimer, Computer simulation and beyond - for the 21st Century, Notices of the Am. Math Soc. 45, 1120-1123, 1998.
  • J. Guckenheimer and A. Willms, Analysis of a subcritical Hopf-homoclinic bifurcation, submitted.
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