The DOI matches its Crossref record; 74 of the 74 references it cites were found in a registry or an index.
Mixed-Mode Oscillations with Multiple Time Scales
Abstract
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.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
Hodgkin, A. L., & Huxley, A. F. (1952) · The Journal of Physiology · doi:10.1113/jphysiol.1952.sp004764 · Cited by 23,191
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
Guckenheimer, J., & Holmes, P. (1983) · Applied Mathematical Sciences · doi:10.1007/978-1-4612-1140-2 · Cited by 16,144
Self-citationIntroduction to Functional Differential Equations
Hale, J. K., & Lunel, S. M. V. (1993) · Applied Mathematical Sciences · doi:10.1007/978-1-4612-4342-7 · Cited by 8,367
Elements of Applied Bifurcation Theory
Kuznetsov, Y. A. (2004) · Applied Mathematical Sciences · doi:10.1007/978-1-4757-3978-7 · Cited by 5,564
NIST Handbook of Mathematical Functions
Olver, F. W. J., Lozier, D. W., Boisvert, R. F., & Clark, C. W. (2010) · CERN Document Server (European Organization for Nuclear Research) · Cited by 5,206
An Active Pulse Transmission Line Simulating Nerve Axon
Nagumo, J., Arimoto, S., & Yoshizawa, S. (1962) · Proceedings of the IRE · doi:10.1109/jrproc.1962.288235 · Cited by 4,194
Geometric singular perturbation theory for ordinary differential equations
Fenichel, N. (1979) · Journal of Differential Equations · doi:10.1016/0022-0396(79)90152-9 · Cited by 2,384
Mathematical Physiology: I: Cellular Physiology
(2009) · Interdisciplinary Applied Mathematics · doi:10.1007/978-0-387-75847-3 · Cited by 2,175
NEURAL EXCITABILITY, SPIKING AND BURSTING
IZHIKEVICH, E. M. (2000) · International Journal of Bifurcation and Chaos · doi:10.1142/s0218127400000840 · Cited by 2,005
AUTO-07p: Continuation and bifurcation software for ordinary differential equations
Doedel, E. J., CHAMPNEYS, A. R., Dercole, F., Fairgrieve, T. F., Kuznetsov, Y. A., Oldeman, B. E., Paffenroth, R. C., Sandstede, B., Wang, X. J., & Zhang, C. H. (2007) · Cited by 1,631
Cited by
Demori, I., Siriya, S., & Burlando, B. (2025) · Applied Sciences · doi:10.3390/app151910483 · Cited by 2
Metadata
- Authors
- Desroches, M., Guckenheimer, J., Krauskopf, B., Kuehn, C., Osinga, H. M., & Wechselberger, M.
- Venue
- SIAM Review
- Published
- 2012-01
- Volume
- 54
- Issue
- 2
- Pages
- 211-288
- ISSN
- 0036-1445 (ISSN-L)
- 1095-7200
- OpenAlex
- W1967771793
- Cited by
- 554
- References
- 74
- FWCI
- 50.44
- Type
- journal article
- Language
- en
- Open access
- closed
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
How to cite
Cite CoChat for the checks on this page, and the paper for its own findings.
- This page
- CoChat. (2026). Citation check: Mixed-Mode Oscillations with Multiple Time Scales. https://app.cochat.ai/papers/10.1137/100791233
- The paper
- Desroches, M., Guckenheimer, J., Krauskopf, B., Kuehn, C., Osinga, H. M., & Wechselberger, M. (2012). Mixed-Mode Oscillations with Multiple Time Scales. SIAM Review. https://doi.org/10.1137/100791233
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
