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A Geometric Method for Periodic Orbits in Singularly-Perturbed Systems

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Multiple-Time-Scale Dynamical Systems

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 122))

Abstract

In this work, we establish a modular geometric method to demonstrate the existence of periodic orbits in singularly perturbed systems of differential equations. These orbits have alternating fast and slow segments, reflecting the two time scales in the problems. The method involves converting the periodic orbit problem into a boundary value problem in an appropriately augmented system, and it employs several versions of the exchange lemmas due to Jones, Kopell, Kaper and Tin. It is applicable to models that arise in a wide variety of scientific disciplines, and applications are given to the FitzHugh-Nagumo, Hodgkin-Huxley, and Gray-Scott systems.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Boston University, Boston, MA, 02215, USA

    Cristina Soto-Treviño

  2. Volen Center, Brandeis University, Waltham, MA, 02454, USA

    Cristina Soto-Treviño

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  1. Cristina Soto-Treviño
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Editors and Affiliations

  1. Division of Applied Mathematics and Lefschetz Center for Dynamical Systems, Brown University, Providence, RI, 02912, USA

    Christopher K. R. T. Jones

  2. United Technologies Research Center, East Hartford, CT, 06108, USA

    Alexander I. Khibnik

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Soto-Treviño, C. (2001). A Geometric Method for Periodic Orbits in Singularly-Perturbed Systems. In: Jones, C.K.R.T., Khibnik, A.I. (eds) Multiple-Time-Scale Dynamical Systems. The IMA Volumes in Mathematics and its Applications, vol 122. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0117-2_6

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  • DOI: https://doi.org/10.1007/978-1-4613-0117-2_6

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6529-0

  • Online ISBN: 978-1-4613-0117-2

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