Research Interests and Activities

Bruce B. Peckham


Research Overview

My research is mostly in the general area of DYNAMICAL SYSTEMS. More specifically, I study the iteration of maps and ask how the dynamical properties of the maps change as parameters in the defining equations change. This is called BIFURCATION THEORY.

Primary topics of interest:

Links:



Publications


  1. "The Closing of Resonance Horns for Periodically Forced Oscillators," Thesis, University of Minnesota, 1988.
    See abstract.

  2. "The Necessity of the Hopf Bifurcation in Families of Periodically Forced Oscillators," Nonlinearity (3) , pp. 261-280, 1990.
    See abstract.

  3. "Period Doubling with Higher Order Degeneracies ," with I. G. Kevrekidis, SIAM Journal of Mathematical Analysis, Vol. 22, No. 6, pp. 1552-1574, 1991.
    See abstract.

  4. Picture Resonance Surfaces

  5. "Resonance Surfaces for Forced Oscillators," with R. P. McGehee, Geometry Center Research Report GCG70, Feb. 1995, and Experimental Mathematics, Vol.3, No. 3, pp. 221-244, 1994.
    See abstract.
    Download preprint. from Geometry Center. Look for GCG70.




  6. "Bananas and banana splits: A parametric degeneracy in the Hopf bifurcation for maps," with C. E. Frouzakis and I. G. Kevrekidis, SIAM J Math Analysis, Vol. 26, No. 1, pp. 190-217, 1995.
    See abstract. Download full text pdf preprint.

  7. Picture Resonance Surfaces
  8. "Determining the global topology of resonance surfaces for periodically forced oscillator families" with R. P. McGehee 1992 Normal Forms and Homoclinic Chaos Conference Proceedings, Fields Institute Communications, Vol. 4, Americal Mathematical Society, pp. 233-251, 1995.
    See abstract. See full text pdf preprint.

  9. "Arnold Flames and Resonance Surface Folds" with R. P. McGehee, Geometry Center Research Report GCG84, July 1995, and International Journal of Bifurcation and Chaos, Vol. 6, No. 2 (1996) 315-336.
    See abstract.
    Download pdf text preprint (without figures). Download pdf of figures.





  10. "Real Perturbation of Complex Analytic Maps: Points to Regions" International Journal of Bifurcation and Chaos, Vol. 8, No. 1, (1998) 73-93.
    See abstract. See full text pdf preprint.


  11. "Global parametrization and computation of resonance surfaces for periodically forced oscillators" Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, E. Doedel and L. Tuckerman, Editors, IMA Volumes in Mathematics and its Applications, Vol. 119, Springer-Verlag, 385-406, 2000.
    See abstract. See full text pdf preprint.


  12. "Real Continuation from the Complex Quadratic Family: Fixed Point Bifurcation Sets" with James Montaldi, International Journal of Bifurcation and Chaos, Vol. 10, No. 2, 391-414, 2000.
    See abstract. View full text pdf preprint.

  13. "Lighting Arnold Flames: Resonance in Doubly Forced Periodic Oscillators," with I. G. Kevrekidis, Nonlinearity , Vol 15, 2002. See abstract. See full text pdf preprint.

  14. "Plant community dynamics, nutrient cycling and alternative stable equilibria in peatlands," with John Pastor, Scott Bridgham, Jake Weltzin, and Jiquan Chen, The American Naturalist, Vol. 160, No. 5, 553-568, 2002. See abstract.

  15. "A route to computational chaos revisited: noninvertibility and the breakup of an invariant circle," with C. E. Frouzakis and I. G. Kevrekidis, Physica D , Vol. 177, 101-121, 2003. See abstract. See full text pdf preprint.


  16. "Computing Arnold Tongue Scenarios," with Frank Schilder. Journal of Computational Physics 220 (2007), 932-951. Electronic version posted 2006. See abstract. See full text pdf preprint.

  17. "Unfolding the cusp-cusp bifurcation of planar endomorphisms," with Bernd Krauskopf and Hinke Osinga, SIAM Journal on Dynamical Systems, Vol. 6 (2), 2007, pp403-440. See abstract. See full text preprint pdf preprint.

  18. "A Producer-Consumer Model with Stoichiometric Elimination Mechanisms," with Laura Zimmermann, John Pastor and Harlan Stech, submitted to Bulletin of Mathematical Biology, July 2008. See abstract. See full text pdf preprint.

  19. "A Stoichiometric Model of two Producers and one Consumer," with Laurence Lin, Harlan Stech and John Pastor, to be submitted to Journal of Biological Dynamics. See abstract. See full text pdf preprint.


Software


  1. "To Be Continued ... " A discrete dynamical systems software package. TBC home page..

VCR Movies


  1. "Resonance City," a 15 minute video of various resonance surfaces for a periodically forced oscillator model, made with R. P. McGehee at the Geometry Center, Minneapolis, MN. For related discussion see "Resonance Surfaces for Forced Oscillators" abstract.

Acknowledgement of Support



This page (http://www.d.umn.edu/~bpeckham/www) is maintained by Bruce Peckham (bpeckham@d.umn.edu) and was last modified on Thursday, 23-Oct-2008 12:33:30 CDT.