Sungular hopf bifurcation to relaxation oscillations
SIAM Journal on Applied Mathematics
The slow passage through a Hopf bifurcation: delay, memory effects, and resonance
SIAM Journal on Applied Mathematics
Singular Hopf bifurcation to relaxation oscillations II
SIAM Journal on Applied Mathematics
A hierarchy of transitions to mixed mode oscillation in an electrochemical system
Proceedings of a NATO advanced research workshop held at the Centre for Nonlinear Phenomena and Complex Systems on Homoclinic Chaos
Mathematical physiology
Elements of applied bifurcation theory (2nd ed.)
Elements of applied bifurcation theory (2nd ed.)
MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs
ACM Transactions on Mathematical Software (TOMS)
Giant squid-hidden canard: the 3D geometry of the Hodgkin–Huxley model
Biological Cybernetics
NIST Handbook of Mathematical Functions
NIST Handbook of Mathematical Functions
Homoclinic Orbits of the FitzHugh-Nagumo Equation: Bifurcations in the Full System
SIAM Journal on Imaging Sciences
Low dimensional model of bursting neurons
Journal of Computational Neuroscience
Mixed-mode oscillations and chaotic solutions of jerk (Newtonian) equations
Journal of Computational and Applied Mathematics
Journal of Computational Neuroscience
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Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.