Ebook: Multiple-Time-Scale Dynamical Systems
Author: Amadeu Delshams Pere Gutiérrez (auth.) Christopher K. R. T. Jones Alexander I. Khibnik (eds.)
- Tags: Analysis, Topology, Geometry
- Series: The IMA Volumes in Mathematics and its Applications 122
- Year: 2001
- Publisher: Springer-Verlag New York
- Edition: 1
- Language: English
- pdf
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.
Content:
Front Matter....Pages i-xi
Homoclinic Orbits to Invariant Tori in Hamiltonian Systems....Pages 1-27
Geometric Singular Perturbation Theory Beyond Normal Hyperbolicity....Pages 29-63
A Primer on the Exchange Lemma for Fast-Slow Systems....Pages 65-87
Geometric Analysis of the Singularly Perturbed Planar Fold....Pages 89-116
Multiple Time Scales and Canards in a Chemical Oscillator....Pages 117-140
A Geometric Method for Periodic Orbits in Singularly-Perturbed Systems....Pages 141-202
The Phenomenon of Delayed Bifurcation and its Analyses....Pages 203-214
Synchrony in Networks of Neuronal Oscillators....Pages 215-232
Metastable dynamics and exponential asymptotics in multi-dimensional domains....Pages 233-259
Back Matter....Pages 261-273
Content:
Front Matter....Pages i-xi
Homoclinic Orbits to Invariant Tori in Hamiltonian Systems....Pages 1-27
Geometric Singular Perturbation Theory Beyond Normal Hyperbolicity....Pages 29-63
A Primer on the Exchange Lemma for Fast-Slow Systems....Pages 65-87
Geometric Analysis of the Singularly Perturbed Planar Fold....Pages 89-116
Multiple Time Scales and Canards in a Chemical Oscillator....Pages 117-140
A Geometric Method for Periodic Orbits in Singularly-Perturbed Systems....Pages 141-202
The Phenomenon of Delayed Bifurcation and its Analyses....Pages 203-214
Synchrony in Networks of Neuronal Oscillators....Pages 215-232
Metastable dynamics and exponential asymptotics in multi-dimensional domains....Pages 233-259
Back Matter....Pages 261-273
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