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:

• Bifurcations that occur for two-parameter families of maps of the plane. Many such maps come from applications which can be grouped under the heading of PERIODICALLY FORCED OSCILLATORS. The picture below displays several "resonance surfaces," or sets of periodic points of a given period as they appear in a (projection to three dimensions of a) four-dimensional "phase cross parameter" space. Several of the publications below deal with the topic of "resonance surfaces."

  
  
  

• Numerical continuation of curves and surfaces related to bifurcation theory.
• Ecological models using dynamical systems.
• Noninvertible maps, especially two-dimensional

Links:

• Publications
• Software
• VCR Movies
• Acknowledgement of Support
• Go to the Math/Stats Dept. Home Page
• Go to Bruce Peckham's Home Page

Publications

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

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

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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. Download published version from SIAM JMA. Download reprint from Here.

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4. "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 reprint from Experimental Mathematics.
   
   
   
   

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5. "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. Download published version from SIAM JMA.
   

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6. "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.
   

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7. "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. Download published version from IJBC.






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8. "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. Download published version from IJBC.
   



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9. "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.
   



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

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11. "Lighting Arnold Flames: Resonance in Doubly Forced Periodic Oscillators," Bruce B. Peckham and I. G. Kevrekidis, Nonlinearity , Vol 15, No. 2, pp405--, 2002. See abstract. See full text pdf preprint. Download published version from IOP Nonlinearity.
    

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

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13. "A route to computational chaos revisited: noninvertibility and the breakup of an invariant circle," C. E. Frouzakis , I. G. Kevrekidis and Bruce B. Peckham, Physica D , Vol. 177, 101-121, 2003. See abstract. See full text pdf preprint or download from arXiv.org.
    

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14. • "A bifurcation analysis of a differential equations model for mutualism," W. G. Graves, Bruce B. Peckham and John Pastor, Bulletin of Mathematical Biology, Vol. 68, No. 8, November 2006, pp 1837-1850. See abstract.
    • "A 2D differential equations model for mutualism," with W. G. Graves and J. Pastor, University of Minnesota Department of Mathematics and Statistics Technical Report TR 2006-2. This paper is a longer version of the BMB paper. It includes more explanation and more bifurcation figures than the JMB paper. See full text pdf reprint.

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

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16. "Unfolding the cusp-cusp bifurcation of planar endomorphisms," Bernd Krauskopf, Hinke Osinga and Bruce Peckham, SIAM Journal on Dynamical Systems, Vol. 6 (2), 2007, pp403-440. See abstract. See full text preprint pdf preprint. Download published version from SIADS.
    

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17. "A Producer-Consumer Model with Stoichiometric Elimination Mechanisms," Laura Zimmermann, Bruce Peckham, John Pastor and Harlan Stech, UMD Technical Report, 2008. See abstract. See full text pdf preprint.
    

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18. • "A Stoichiometric Model of two Producers and one Consumer," with Laurence Lin, Harlan Stech and John Pastor, UMD Technical Report, 2010. (Expanded version of J Biological Dynamics paper.) See abstract. See full text pdf preprint.
      

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    • "Enrichment in a stoichiometric model of two producers and one consumer," Laurence Lin, Bruce Peckham, Harlan Stech and John Pastor (2012), Journal of Biological Dynamics. Vol. 6, No. 2, 97-116. (published online 27 Jan 24, 2011) See abstract. Free access to full text available at JBD link.
      

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19. ``Enrichment in a general class of stoichiometric producer-consumer population growth models,'' Harlan Stech, Bruce Peckham and John Pastor, Theoretical Population Biology, Volume 81, Issue 3, 2012, 210-222. See abstract. Download fulltext preprint fulltext PDF preprint. Get published version from TPB link.
    

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20. ``Enrichment Effects in a Simple Stoichiometric Producer-Consumer Population Growth Model,'' Harlan Stech, Bruce Peckham and John Pastor, Communications in Applied Analysis, Vol. 16, No. 4, 2012, 687-702. See abstract. Download fulltext preprint fulltext PDF preprint.
    

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21. ``Quasi-equilibrium Reduction in a General Class of Stoichiometric Producer-Consumer Models,'' Harlan Stech, Bruce Peckham and John Pastor, Journal of Biological Dynamics, Volume 6, No. 2, 2012, 992-1018. See abstract. Free access to full text available at JBD link.
    

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22. ``Dynamics of nonholomorphic singular continuations: a case with radial symmetry," Bozyk B D and Peckham B B, International Journal of Bifurcation and Chaos, Vol. 23, No. 11, 22 pages, 2013. See abstract. See full text pdf preprint. Download published version from IJBC.
    

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23. ``Classification of critical sets and their images for quadratic maps of the plane,'' C-H Nien, B B Peckham, and R P McGehee, Journal of Difference Equations and Applications, 2016. See full text preprint at arXiv.org. Download published version from JDEA.
    

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24. ``Affine map equivalence versus critical set equivalence for quadratic maps of the plane,'' C-H Nien, B B Peckham, and R P McGehee. See full text 2017 preprint at arXiv.org.
    

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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

• Much of work reported in this Web page was supported by the National Science Foundation, Grants DMS-9020220, DMS-9505051, DMS-9973926. Support during leaves was provided by The Geometry Center, University of Minnesota, Minneapolis (Fall 1992), Institute Nonlinear de Nice (Fall 1996), University of Colorado, Boulder (Spring 1997), the University of Bristol, Bristol England (Fall 2004), and the University Auckland, Auckland, New Zealand (Spring 2013).