We study the recently observed phenomena of torus canards. These are a higher-dimensional generalization of the classical canard orbits familiar from planar systems and arise in fast-slow systems of ordinary differential equations in which the fast subsystem contains a saddle-node bifurcation of limit cycles. Torus canards are trajectories that pass near the saddle-node and subsequently spend long times near a repelling branch of slowly varying limit cycles. In this article, we carry out a study of torus canards in an elementary third-order system that consists of a rotated planar system of van der Pol type in which the rotational symmetry is broken by including a phase-dependent term in the slow component of the vector field. In the regime of fast rotation, the torus canards behave much like their planar counterparts. In the regime of slow rotation, the phase dependence creates rich torus canard dynamics and dynamics of mixed mode type. The results of this elementary model provide insight into the torus canards observed in a higher-dimensional neuroscience model.
REFERENCES
1.
V. N.
Belykh
, I. V.
Belykh
, M.
Colding-Jørgensen
, and E.
Mosekilde
, “Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models
,” Eur. Phys. J. E
3
, 205
(2000
). 2.
E.
Benoît
, “Systemes lents-rapides dans R3 et leur canards
,” Asterisque
109–110
, 159
(1983
). 3.
E.
Benoît
, J. F.
Callot
, F.
Diener
, and M.
Diener
, “Chasse au canard
,” Collect. Math.
31–32
, 37
(1981
). 4.
K.
Bold
, C.
Edwards
, J.
Guckenheimer
, S.
Guharay
, K.
Hoffman
, J.
Hubbard
, R.
Oliva
, and W.
Weckesser
, “The forced van der Pol equation II: Canards in the reduced system
,” SIAM J. Appl. Dyn. Syst.
2
, 570
(2003
). 5.
B.
Braaksma
, “Critical phenomena in dynamical systems of van der Pol type
,” Ph.D. thesis (Utrecht University
, Netherlands, 1993
). 6.
M.
Brøns
and K.
Bar-Eli
, “Canard explosion and excitation in a model of the Belousov-Zhabotinskii reaction
,” J. Phys. Chem.
95
, 8706
(1991
). 7.
M.
Brøns
, T. J.
Kaper
, and H. G.
Rotstein
, “Introduction to focus issue: Mixed mode oscillations: Experiment, computation, and analysis
,” Chaos
18
, 015101
(2008
). 8.
M.
Brøns
, M.
Krupa
, and M.
Wechselberger
, “Mixed mode oscillations due to the generalized canard phenomenon
,” in Bifurcation Theory and Spatio-Temporal Pattern Formation, Fields Inst. Communications
(Amer. Math. Soc., Providence
, RI
), Vol. 49
, pp. 39
–63
. 9.
J.
Burke
, M.
Desroches
, and T.
Kaper
, “Folded singularities, mixed-mode oscillations, and torus canards
” (unpublished). 10.
P. F.
Byrd
and M. D.
Friedman
, “Handbook of elliptic integrals for scientists and engineers
,” in Grundlehren der Mathematische Wissenschaften, LXVII
(Springer-Verlag
, Berlin
, 1954
). 11.
G.
Cymbaluyk
and A.
Shilnikov
, “Co-existent tonic spiking modes in a leech neuron model
,” J. Comput. Neurosci.
18
, 269
(2005
). 12.
M.
Desroches
, B.
Krauskopf
, and H. M.
Osinga
, “The geometry of slow manifolds near a folded node
,” SIAM J. Appl. Dyn. Syst.
7
, 1131
(2008
). 13.
M.
Diener
, “The canard unchained or how fast/slow dynamical systems bifurcate
,” Math. Intell.
6
, 38
(1984
) 14.
M.
Diener
and F.
Diener
, “Chasse au canard: I–IV
,” Collect. Math.
31–32
(1–3
), 37
(1981
). 15.
E. J.
Doedel
, B. E.
Oldeman
, A. R.
Champneys
, F.
Dercole
, T.
Fairgrieve
, Y.
Kurnetsov
, R.
Paffenroth
, B.
Sandstede
, X.
Wang
, and C.
Zhang
, AUTO: Software for continuation and bifurcation problems in ordinary differential equations, available at http://indy.cs.concordia.ca/auto/. 16.
F.
Dumortier
, “Techniques in the theory of local bifurcations: Blow-up, normal forms, nilpotent bifurcations, singular perturbations
,” in Bifurcations and Periodic Orbits of Vector Fields
, edited by D.
Szlomiuk
(Kluwer
, Dordrecht
, 1993
). 17.
F.
Dumortier
and R.
Roussarie
, “Canard cycles and center manifolds
,” Mem. Am. Math. Soc.
121
, 577
(1996
). 18.
W.
Eckhaus
, “Relaxation oscillations including a standard chase on French ducks
,” in Asymptotic Analysis II, LNM 985
(Springer-Verlag
, New York
, 1983
), pp. 449
–494
. 19.
G. B.
Ermentrout
and M.
Wechselberger
, “Canards, clusters, and synchronization in a weakly coupled interneuron model
,” SIAM J. Appl. Dyn. Syst.
8
, 253
(2009
). 20.
N.
Fenichel
, “Persistence and smoothness of invariant manifolds for flows
,” Indiana Univ. Math. J.
21
, 193
(1971
). 21.
N.
Fenichel
, “Geometric singular perturbation theory for ordinary differential equations
,” J. Differ. Equations
31
, 53
(1979
). 22.
F. R.
Fernandez
, J. D. T.
Engbers
, and R. W.
Turner
, “Firing dynamics of cerebellar Purkinje cells
,” J. Neurophysiol.
98
, 278
(2007
). 23.
S.
Genet
and B.
Delord
, “A biophysical model of nonlinear dynamics underlying plateau potentials and calcium spikes in Purkinje cell dendrites
,” J. Neurophysiol.
88
, 2430
(2002
). 24.
S.
Genet
, L.
Sabarly
, E.
Guigon
, H.
Berry
, and B.
Delord
, “Dendritic signals command firing dynamics in a mathematical model of cerebellar Purkinje cells
,” Biophys J.
99
, 427
(2010
). 25.
J.
Grasman
and J. J.
Wentzel
, “Co-existence of a limit cycle and an equilibrium in Kaldor’s business cycle model and its consequences
,” J. Econ. Behav. Organ.
24
, 369
(1994
). 26.
J.
Guckenheimer
, “Singular Hopf bifurcation in systems with two slow variables
,” SIAM J. Appl. Dyn. Syst.
7
, 1355
(2008
). 27.
J.
Guckenheimer
, K.
Hoffman
, and W.
Weckesser
, “The forced van der Pol equation I: The slow flow and its bifurcations
,” SIAM J. Appl. Dyn. Syst.
2
, 1
(2003
). 28.
J.
Guckenheimer
and P.
Holmes
, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
, 2nd ed., Applied Mathematical Sciences Series 42 (Springer-Verlag
, New York
, 1990
). 29.
J.
Guckenheimer
and C.
Kuehn
, “Homoclinic orbits of the FitzHugh-Nagumo equation: Bifurcations in the full system
,” SIAM J. Appl. Dyn. Syst.
9
, 138
(2010
). 30.
E. M.
Izhikevich
, “Subcritical elliptic bursting of Bautin type
,” SIAM J. Appl. Math.
60
, 503
(2000
). 31.
E. M.
Izhikevich
, “Synchronization of elliptic bursters
,” SIAM Rev.
43
, 315
(2001
). 32.
33.
E. M.
Izhikevich
, N. S.
Desai
, E. C.
Walcott
, and F. C.
Hoppensteadt
, “Bursts as a unit of neural information: Selective communication via resonance
,” Trends Neurosci.
26
, 161
(2003
). 34.
C. K. R. T.
Jones
, “Geometric singular perturbation theory
,” in Dynamical Systems, Montecatini Terme, LNM 1609
(Springer-Verlag
, New York
, 1995
), pp. 44
–120
. 35.
C. K. T.
Jones
and S.-K.
Tin
, “Generalized exchange lemmas and orbits heteroclinic to invariant manifolds
,” Discrete Contin. Dyn. Syst., Ser. S
2
(4
), 967
(2009
). 36.
M. A.
Kramer
, R. D.
Traub
, and N. J.
Kopell
, “New dynamics in cerebellar Purkinje cells: Torus canards
,” Phys. Rev. Lett.
101
, 068103
(2008
). 37.
M.
Krupa
and P.
Szmolyan
, “Extending singular perturbation theory to non-hyperbolic points: Folds and canard points in two dimensions
,” SIAM J. Math. Anal.
33
, 286
(2001
). 38.
M.
Krupa
and P.
Szmolyan
, “Relaxation oscillation and canard exposition
,” J. Differ. Equations
174
, 312
(2001
). 39.
A. P.
Kuznetsov
, S. P.
Kuznetsov
, and N. V.
Stankevich
, “A simple autonomous quasiperiodic self-oscillator
,” Commun. Nonlinear Sci. Numer. Simul.
15
, 1676
(2010
). 40.
J.
Lisman
, “Bursts as a unit of neural information: Making unreliable synapses reliable
,” Trends Neurosci.
20
, 38
(1997
). 41.
A. I.
Neihstadt
, “The separation of motion of systems with rapidly rotating phase
,” Prikl. Mat. Mekh.
48
, 133
(1984
). 42.
A. I.
Neihstadt
, C.
Simo
, and D. V.
Treschev
, “On stability loss delay for a periodic trajectory
,” in Nonlinear Dynamical Systems and Chaos
, Prog. in Nonlin. Diff. Eq. and Their Applic. (Birkhauser
, Basel, Switzerland
, 1996
), Vol. 19
, pp. 253
–278
. 43.
H. G.
Rotstein
, N.
Kopell
, A. M.
Zhabotinsky
, and I. R.
Epstein
, “Canard phenomenon and localization of oscillations in the Belousov-Zhabotinsky reaction with global feedback
,” J. Chem. Phys.
119
, 8824
(2003
). 44.
H. G.
Rotstein
, M.
Wechselberger
, and N.
Kopell
, “Canard induced mixed-mode oscillations in a medial entorhinal cortex layer: II stellate cell model
,” SIAM J. Appl. Dyn. Syst.
7
, 1582
(2008
). 45.
J. A.
Sanders
and F.
Verhulst
, Averaging Methods in Nonlinear Dynamical Systems
, Applied Mathematical Sciences Vol. 59
(Springer-Verlag
, New York
, 1985
). 46.
A.
Shilnikov
and G.
Cymbaluyk
, “Transition between tonic-spiking and bursting in a neuron model via the blue-sky catastrophe
,” Phys. Rev. Lett.
94
, 048101
(2005
). 47.
A.
Shilnikov
and N.
Rulkov
, “Origin of chaos in a two-dimensional map modeling spiking-bursting neural activity
,” Int. J. Bifurication Chaos
13
, 3325
(2003
). 48.
A.
Shilnikov
and N.
Rulkov
, “Subthreshold oscillations in a map-based neuron model
,” Phys. Lett. A
328
, 177
(2004
). 49.
P.
Szmolyan
and M.
Wechselberger
, “Canards in R3
,” J. Differ Equations
177
, 419
(2001
). 50.
D.
Terman
, “The transition from bursting to continuous spiking in excitable membrane models
,” J. Nonlinear Sci.
2
, 135
(1992
). 51.
R. D.
Traub
, E. H.
Buhl
, T.
Gloveli
, and M. A.
Whittington
, “Fast rythmic bursting can be induced in layer 2/3 cortical neurons by enhancing persistent Na+ conductance or by blocking BK channels
,” J. Neurophysiol.
89
, 909
(2003
). 52.
X.
Wang
, “Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle
,” Physica D
62
, 263
(1993
). 53.
M.
Wechselberger
, “Existence and bifurcation of canards in R3 in the case of a folded node
,” SIAM J. Appl. Dyn. Syst.
4
, 101
(2005
). © 2011 American Institute of Physics.
2011
American Institute of Physics
You do not currently have access to this content.
