Multiple-Time-Scale Dynamical Systems

1. Front Matter

   Pages i-xi

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2. Homoclinic Orbits to Invariant Tori in Hamiltonian Systems

   • Amadeu Delshams, Pere Gutiérrez

   Pages 1-27

3. Geometric Singular Perturbation Theory Beyond Normal Hyperbolicity

   • Freddy Dumortier, Robert Roussarie

   Pages 29-63

4. A Primer on the Exchange Lemma for Fast-Slow Systems

   • Tasso J. Kaper, Christopher K. R. T. Jones

   Pages 65-87

5. Geometric Analysis of the Singularly Perturbed Planar Fold

   • M. Krupa, P. Szmolyan

   Pages 89-116

6. Multiple Time Scales and Canards in a Chemical Oscillator

   • Alexandra Milik, Peter Szmolyan

   Pages 117-140

7. A Geometric Method for Periodic Orbits in Singularly-Perturbed Systems

   • Cristina Soto-Treviño

   Pages 141-202

8. The Phenomenon of Delayed Bifurcation and its Analyses

   • Jianzhong Su

   Pages 203-214

9. Synchrony in Networks of Neuronal Oscillators

   • D. Terman

   Pages 215-232

10. Metastable dynamics and exponential asymptotics in multi-dimensional domains

    • Michael J. Ward

    Pages 233-259

11. Back Matter

    Pages 261-273

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Systems with sub-processes evolving on many different time scales are ubiquitous in applications: chemical reactions, electro-optical and neuro-biological systems, to name just a few. This volume contains papers that expose the state of the art in mathematical techniques for analyzing such systems. Recently developed geometric ideas are highlighted in this work that includes a theory of relaxation-oscillation phenomena in higher dimensional phase spaces. Subtle exponentially small effects result from singular perturbations implicit in certain multiple time scale systems. Their role in the slow motion of fronts, bifurcations, and jumping between invariant tori are all explored here. Neurobiology has played a particularly stimulating role in the development of these techniques and one paper is directed specifically at applying geometric singular perturbation theory to reveal the synchrony in networks of neural oscillators.

• Volume
• bifurcation
• dynamical systems
• dynamics
• hamiltonian system

• Division of Applied Mathematics and Lefschetz Center for Dynamical Systems, Brown University, Providence, USA

  Christopher K. R. T. Jones

• United Technologies Research Center, East Hartford, USA

  Alexander I. Khibnik

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