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.
• Biological models using dynamical systems.
• Noninvertible maps

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.
   

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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.
   

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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.
   

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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 preprint. from Geometry Center. Look for GCG70.
   
   
   
   

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5. "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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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.
   

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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 compressed postscript text preprint (without figures). Download compressed postscript figures.






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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.
   



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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" with James Montaldi, International Journal of Bifurcation and Chaos, Vol. 10, No. 2, 391-414, 2000.
    See abstract. Download full text compressed postscript preprint. Uncompress with gunzip.
    

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

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12. "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.
    

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13. "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.
    

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14. • "A bifurcation analysis of a differential equations model for mutualism," with W. G. Graves and J. 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 JMB 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," with Frank Schilder. 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," 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.
    

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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), and the University of Bristol, Bristol England (Fall 2004).