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JOURNALS // Moscow Mathematical Journal

Mosc. Math. J., 2005, Volume 5, Number 1, Pages 91–103 (Mi mmj185)

Canards at folded nodes
J. Guckenheimer, R. Haiduc

This publication is cited in the following articles:
  1. Yaru Liu, Yinyan Zhang, Shenquan Liu, Manchun Tan, “Asymptotic return maps and geometric dynamics in multi-timescale neuronal systems: Decoding mixed-mode oscillations”, Eur. Phys. J. Spec. Top., 2026  crossref
  2. Bruno F.F. Gonçalves, Isabel S. Labouriau, Alexandre A.P. Rodrigues, “Synchrony and canards in two coupled FitzHugh–Nagumo equations”, Physica D: Nonlinear Phenomena, 2025, 134853  crossref
  3. Bingrong Song, Yaru Liu, “Review on mixed-mode oscillations dynamics based on neuron models and neural networks”, Eur. Phys. J. Spec. Top., 2025  crossref
  4. Feibiao Zhan, Yingteng Zhang, Jian Song, Shenquan Liu, “Canard Mechanism and Rhythm Dynamics of Neuron Models”, Mathematics, 11:13 (2023), 2874  crossref
  5. M. Paul Asir, D. Premraj, K. Sathiyadevi, “Complex mixed-mode oscillations in oscillators sharing nonlinearity”, Eur. Phys. J. Plus, 137:2 (2022)  crossref
  6. Benjamin Letson, Jonathan E. Rubin, “On the Nonexistence of Terrestrial Canards: Linking Canards and Rivers”, SIAM J. Appl. Dyn. Syst., 21:4 (2022), 2432  crossref
  7. Irina Bashkirtseva, Lev Ryashko, “Canard oscillations in the randomly forced suspension flows”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 31:3 (2021)  crossref
  8. Hildeberto Jardón-Kojakhmetov, Christian Kuehn, Contemporary Mathematics, 775, Mexican Mathematicians in the World, 2021, 115  crossref
  9. I. Bashkirtseva, A. Pankratov, E. Slepukhina, I. Tsvetkov, “Constructive role of noise and diffusion in an excitable slow–fast population system”, Phil. Trans. R. Soc. A., 378:2171 (2020), 20190253  crossref
  10. Li Ya., “a Data-Driven Method For the Steady State of Randomly Perturbed Dynamics”, Commun. Math. Sci., 17:4 (2019), 1045–1059  mathscinet  zmath  isi
  11. Ginoux J.-M., Llibre J., Tchizawa K., “Canards Existence in the Hindmarsh-Rose Model”, Math. Model. Nat. Phenom., 14:4 (2019), UNSP 409  crossref  mathscinet  isi  scopus
  12. Hasan C.R., Krauskopf B., Osinga H.M., “Saddle Slow Manifolds and Canard Orbits in R-4 and Application to the Full Hodgkin-Huxley Model”, J. Math. Neurosci., 8 (2018), 5  crossref  mathscinet  zmath  isi
  13. Mujica J., Krauskopf B., Osinga H.M., “Tangencies Between Global Invariant Manifolds and Slow Manifolds Near a Singular Hopf Bifurcation”, SIAM J. Appl. Dyn. Syst., 17:2 (2018), 1395–1431  crossref  mathscinet  zmath  isi  scopus
  14. Hasan C.R., Krauskopf B., Osinga H.M., “Mixed-Mode Oscillations and Twin Canard Orbits in An Autocatalytic Chemical Reaction”, SIAM J. Appl. Dyn. Syst., 16:4 (2017), 2165–2195  crossref  zmath  isi  scopus
  15. Ginoux J.-M., Llibre J., “Canards Existence in Memristor'S Circuits”, Qual. Theor. Dyn. Syst., 15:2 (2016), 383–431  crossref  mathscinet  zmath  isi  scopus
  16. Desroches M., Guillamon A., Ponce E., Prohens R., Rodrigues S., Teruel A.E., “Canards, Folded Nodes, and Mixed-Mode Oscillations in Piecewise-Linear Slow-Fast Systems”, SIAM Rev., 58:4 (2016), 653–691  crossref  mathscinet  zmath  isi  scopus
  17. Gutschke M., Vais A., Wolter F.-E., “Differential Geometric Methods For Examining the Dynamics of Slow-Fast Vector Fields”, Visual Comput., 31:2 (2015), 169–186  crossref  isi  scopus
  18. Ginoux J.-M., Llibre J., “Canards Existence in Fitzhugh-Nagumo and Hodgkin-Huxley Neuronal Models”, Math. Probl. Eng., 2015, 342010  crossref  mathscinet  isi  elib  scopus
  19. Kristiansen K.U., “Computation of Saddle-Type Slow Manifolds Using Iterative Methods”, SIAM J. Appl. Dyn. Syst., 14:2 (2015), 1189–1227  crossref  mathscinet  zmath  isi  scopus
  20. Richard Bertram, Joël Tabak, Wondimu Teka, Theodore Vo, Martin Wechselberger, Frontiers in Applied Dynamical Systems: Reviews and Tutorials, 1, Mathematical Analysis of Complex Cellular Activity, 2015, 1  crossref
  21. Christian Kuehn, Applied Mathematical Sciences, 191, Multiple Time Scale Dynamics, 2015, 197  crossref
  22. Cunningham S.W., Kwakkel J.H., “Tipping Points in Science: a Catastrophe Model of Scientific Change”, J. Eng. Technol. Manage., 32:SI (2014), 185–205  crossref  isi  scopus
  23. De Saedeleer B., Crucifix M., Wieczorek S., “Is the Astronomical Forcing a Reliable and Unique Pacemaker for Climate? a Conceptual Model Study”, Clim. Dyn., 40:1-2 (2013), 273–294  crossref  isi  scopus
  24. Berglund N., Gentz B., Kuehn Ch., “Hunting French ducks in a noisy environment”, J Differential Equations, 252:9 (2012), 4786–4841  crossref  mathscinet  zmath  isi  scopus
  25. Wechselberger M., “About Canards”, Trans. Am. Math. Soc., 364:6 (2012), 3289–3309  crossref  mathscinet  zmath  isi  scopus
  26. Guckenheimer J., Meerkamp Ph., “Unfoldings of Singular Hopf Bifurcation”, SIAM J. Appl. Dyn. Syst., 11:4 (2012), 1325–1359  crossref  mathscinet  zmath  isi
  27. Krupa M., Vidal A., Desroches M., Clement F., “Mixed-Mode Oscillations in a Multiple Time Scale Phantom Bursting System”, SIAM J. Appl. Dyn. Syst., 11:4 (2012), 1458–1498  crossref  mathscinet  zmath  isi
  28. Desroches M., Guckenheimer J., Krauskopf B., Kuehn Ch., Osinga H.M., Wechselberger M., “Mixed-Mode Oscillations with Multiple Time Scales”, SIAM Rev., 54:2 (2012), 211–288  crossref  mathscinet  zmath  isi  elib  scopus
  29. Kwakkel J.H., Cunningham S.W., “Tipping Points in Science: a Catastrophe Model of Scientific Change”, Picmet `12: Proceedings - Technology Management for Emerging Technologies, eds. Kocaoglu D., Anderson T., Daim T., IEEE, 2012, 173–184  isi
  30. Guckenheimer J., Scheper Ch., “A geometric model for mixed-mode oscillations in a chemical system”, SIAM J. Appl. Dyn. Syst., 10:1 (2011), 92–128  crossref  mathscinet  zmath  isi  scopus
  31. Kuehn Ch., “On decomposing mixed-mode oscillations and their return maps”, Chaos, 21:3 (2011), 033107  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
  32. Wondimu Teka, Joël Tabak, Theodore Vo, Martin Wechselberger, Richard Bertram, “The dynamics underlying pseudo-plateau bursting in a pituitary cell model”, J. Math. Neurosc., 1:1 (2011)  crossref
  33. Desroches M., Krauskopf B., Osinga H.M., “Numerical continuation of canard orbits in slow-fast dynamical systems”, Nonlinearity, 23:3 (2010), 739–765  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
  34. Krupa M., Wechselberger M., “Local analysis near a folded saddle-node singularity”, J. Differential Equations, 248:12 (2010), 2841–2888  crossref  mathscinet  zmath  isi  scopus
  35. Thiessen T., Gutschke M., Blanke Ph., Mathis W., Wolter F.-E., “Numerical Analysis of Relaxation Oscillators Based on a Differential Geometric Approach”, International Conference on Signals and Electronic Systems (ICSES '10): Conference Proceedings, 2010, 209–212  isi
  36. Wechselberger M., Weckesser W., “Bifurcations of mixed-mode oscillations in a stellate cell model”, Phys. D, 238:16 (2009), 1598–1614  crossref  zmath  adsnasa  isi  scopus
  37. Ermentrout B., Wechselberger M., “Canards, clusters, and synchronization in a weakly coupled interneuron model”, SIAM J. Appl. Dyn. Syst., 8:1 (2009), 253–278  crossref  mathscinet  zmath  adsnasa  isi  scopus
  38. Desroches M., Krauskopf B., Osinga H.M., “The geometry of slow manifolds near a folded node”, SIAM J. Appl. Dyn. Syst., 7:4 (2008), 1131–1162  crossref  mathscinet  zmath  adsnasa  isi  scopus
  39. Guckenheimer J., “Singular Hopf bifurcation in systems with two slow variables”, SIAM J. Appl. Dyn. Syst., 7:4 (2008), 1355–1377  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
  40. Desroches M., Krauskopf B., Osinga H.M., “Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh-Nagumo system”, Chaos, 18:1 (2008), 015107, 8 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
  41. Guckenheimer J., “Return maps of folded nodes and folded saddle-nodes”, Chaos, 18:1 (2008), 015108, 9 pp.  crossref  mathscinet  adsnasa  isi  elib  scopus
  42. Popovic N., “Mixed-mode dynamics and the canard phenomenon: towards a classification”, International Workshop on Multi-Rate Processes and Hysteresis, Journal of Physics Conference Series, 138, 2008  crossref  isi  scopus
  43. Rubin J., Wechselberger M., “Giant squid-hidden canard: the 3D geometry of the Hodgkin-Huxley model”, Biol. Cybernet., 97:1 (2007), 5–32  crossref  mathscinet  zmath  isi  scopus
  44. Guckenheimer J., Hoffman K., Weckesser W., “Bifurcations of relaxation oscillations near folded saddles”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15:11 (2005), 3411–3421  crossref  mathscinet  zmath  isi  scopus
  45. Martin Wechselberger, “Existence and Bifurcation of Canards in $\mathbbR^3$ in the Case of a Folded Node”, SIAM J. Appl. Dyn. Syst., 4:1 (2005), 101  crossref


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