The phenomenon of slow passage through a Hopf bifurcation is ubiquitous in multiple-timescale dynamical systems, where a slowly varying quantity replacing a static parameter induces the solutions of the resulting slow–fast system to feel the effect of the Hopf bifurcation with a delay. This phenomenon is well understood in the context of smooth slow–fast dynamical systems; in the present work, we study it for the first time in piecewise linear (PWL) slow–fast systems. This special class of systems is indeed known to reproduce all features of their smooth counterpart while being more amenable to quantitative analysis and offering some level of simplification, in particular, through the existence of canonical (linear) slow manifolds. We provide conditions for a PWL slow–fast system to exhibit a slow passage through a Hopf-like bifurcation, in link with possible connections between canonical attracting and repelling slow manifolds. In doing so, we fully describe the so-called way-in/way-out function. Finally, we investigate this slow passage effect in the Doi–Kumagai model, a neuronal PWL model exhibiting elliptic bursting oscillations.

1.
V.
Carmona
,
S.
Fernández-García
, and
A. E.
Teruel
, “
Saddle-node of limit cycles in planar piecewise linear systems and applications
,”
Discrete Contin. Dyn. Syst.
39
(
9
),
5275
5299
(
2019
).
2.
V.
Carmona
,
S.
Fernández-García
,
F.
Fernández-Sánchez
,
E.
Garcia-Medina
, and
A. E.
Teruel
, “
Reversible periodic orbits in a class of 3d continuous piecewise linear systems of differential equations
,”
Nonlinear Anal.: Theory Methods Appl.
75
(
15
),
5866
5883
(
2012
).
3.
J.
Llibre
,
E.
Ponce
, and
A. E.
Teruel
, “
Horseshoes near homoclinic orbits for piecewise linear differential systems in R3
,”
Int. J. Bifurcation Chaos
17
,
1171
1184
(
2007
).
4.
V.
Carmona
,
S.
Fernández-García
, and
A. E.
Teruel
, “Saddle-node canard cycles in planar piecewise linear differential systems,” arXiv:2003.14112v2.
5.
M.
Desroches
,
E.
Freire
,
S. J.
Hogan
,
E.
Ponce
, and
P.
Thota
, “
Canards in piecewise-linear systems: Explosions and super-explosions
,”
Proc. R. Soc. A
469
(
2154
),
20120603
(
2013
).
6.
M.
Desroches
,
A.
Guillamon
,
E.
Ponce
,
R.
Prohens
,
S.
Rodrigues
, and
A. E.
Teruel
, “
Canards, folded nodes and mixed-mode oscillations in piecewise-linear systems
,”
SIAM Rev.
58
(
4
),
653
691
(
2016
).
7.
S.
Fernández-García
,
M.
Desroches
,
M.
Krupa
, and
A. E.
Teruel
, “
Canard solutions in planar piecewise linear systems with three zones
,”
Dyn. Syst.
31
(
2
),
173
197
(
2016
).
8.
M.
Desroches
,
M.
Krupa
, and
S.
Rodrigues
, “
Inflection, canards and excitability threshold in neuronal models
,”
J. Math. Biol.
67
(
4
),
989
1017
(
2013
).
9.
J.
Mitry
,
M.
McCarthy
,
N.
Kopell
, and
M.
Wechselberger
, “
Excitable neurons, firing threshold manifolds and canards
,”
J. Math. Neurosci.
3
(
1
),
1
32
(
2013
).
10.
S.
Doi
and
S.
Kumagai
, “Complicated slow oscillations with simple switching dynamics in piecewise linear neuronal model,” in The 2004 47th Midwest Symposium on Circuits and Systems, 2004. MWSCAS’04 (IEEE, 2004), Vol. 2, p. II.
11.
S.
Doi
and
S.
Kumagai
, “
Generation of very slow neuronal rhythms and chaos near the Hopf bifurcation in single neuron models
,”
J. Comput. Neurosci.
19
(
3
),
325
356
(
2005
).
12.
J.
Honerkamp
,
G.
Mutschler
, and
R.
Seitz
, “
Coupling of a slow and a fast oscillator can generate bursting
,”
Bull. Math. Biol.
47
(
1
),
1
21
(
1985
).
13.
E. M.
Izhikevich
, “
Neural excitability, spiking and bursting
,”
Int. J. Bifurcation Chaos
10
(
06
),
1171
1266
(
2000
).
14.
J.
Rinzel
, “A formal classification of bursting mechanisms in excitable systems,” in Mathematical Topics in Population Biology, Morphogenesis and Neurosciences, Lecture Notes in Biomathematics Vol. 71, edited by E. Teramoto and M. Yumaguti (Springer, 1987), pp. 267–281.
15.
C. K. R. T.
Jones
, “Geometric singular perturbation theory,” in Dynamical Systems (Montecatini Terme, 1994), Lecture Notes in Mathematics Vol. 1609, edited by R. Johnson (Springer, Berlin, 1995), pp. 44–118.
16.
J. E.
Marsden
and
M.
McCracken
,
The Hopf Bifurcation and Its Applications
(
Springer
,
1976
).
17.
A.
Neishtadt
, “
On stability loss delay for dynamical bifurcations
,”
Discrete Contin. Dyn. Syst. Ser. S
2
(
4
),
897
(
2009
).
18.
C.
Lobry
, “Dynamic bifurcations,” in Dynamic Bifurcations, Lecture Notes in Mathematics Vol. 1493, edited by E. Benoît (Springer, Berlin, 1991), pp. 1–13.
19.
A. I.
Neishtadt
, “
Persistence of stability loss for dynamical bifurcations I
,”
Differ. Equ.
23
,
1385
1391
(
1987
).
20.
A. I.
Neishtadt
, “
Persistence of stability loss for dynamical bifurcations II
,”
Differ. Equ.
24
,
171
176
(
1988
).
21.
M. A.
Shishkova
, “
Examination of a system of differential equations with a small parameter in the highest derivatives
,”
Dokl. Akad. Nauk
209
(
3
),
576
579
(
1973
).
22.
S.
Baer
,
T.
Erneux
, and
J.
Rinzel
, “
The slow passage through a Hopf bifurcation: Delay, memory effects, and resonance
,”
SIAM J. Appl. Math.
49
(
1
),
55
71
(
1989
).
23.
D.
Premraj
,
K.
Suresh
,
T.
Banerjee
, and
K.
Thamilmaran
, “
An experimental study of slow passage through Hopf and pitchfork bifurcations in a parametrically driven nonlinear oscillator
,”
Commun. Nonlinear Sci. Numer. Simul.
37
,
212
221
(
2016
).
24.
Yu. A.
Kuznetsov
and
S.
Rinaldi
, “
Remarks on food chain dynamics
,”
Math. Biosci.
134
(
1
),
1
33
(
1996
).
25.
L.
Holden
and
T.
Erneux
, “
Slow passage through a Hopf bifurcation: From oscillatory to steady state solutions
,”
SIAM J. Appl. Math.
53
(
4
),
1045
1058
(
1993
).
26.
S.
Coombes
and
P.
Bressloff
,
Bursting: The Genesis of Rhythm in the Nervous System
(
World Scientific
,
2005
).
27.
L.
Holden
and
T.
Erneux
, “
Understanding bursting oscillations as periodic slow passages through bifurcation and limit points
,”
J. Math. Biol.
31
(
4
),
351
365
(
1993
).
28.
J.
Rinzel
, “A formal classification of bursting mechanisms in excitable systems,” in International Congress of Mathematicians, Berkeley, CA, USA, August 3–11, 1986 (American Mathematical Society, 1987), Vol. II, pp. 1578–1593.
29.
F.
Diener
and
M.
Diener
, “Maximal delay,” in Dynamic Bifurcations, Lecture Notes in Mathematics Vol. 1493, edited by E. Benoît (Springer, Berlin, 1991), pp. 71–86.
30.
X.
Han
and
Q.
Bi
, “
Slow passage through canard explosion and mixed-mode oscillations in the forced van der Pol’s equation
,”
Nonlinear Dyn.
68
(
1
),
275
283
(
2012
).
31.
W.
Huagan
,
Y.
Ye
,
M.
Chen
,
X.
Quan
, and
B.
Bao
, “
Extremely slow passages in low-pass filter-based memristive oscillator
,”
Nonlinear Dyn.
97
,
2339
2353
(
2019
).
32.
E.
Benoît
,
J.-L.
Callot
,
F.
Diener
, and
M.
Diener
, “
Chasse au canard
,”
Collect. Math.
32
(
1-2
),
37
119
(
1981
).
33.
M.
Desroches
,
J.
Guckenheimer
,
B.
Krauskopf
,
C.
Kuehn
,
H. M.
Osinga
, and
M.
Wechselberger
, “
Mixed-mode oscillations with multiple time scales
,”
SIAM Rev.
54
(
2
),
211
288
(
2012
).
34.
M.
Desroches
,
S.
Fernández-García
,
M.
Krupa
,
R.
Prohens
, and
A. E.
Teruel
, “Piecewise-linear (pwl) canard dynamics: Simplifying singular perturbation theory in the canard regime using piecewise-linear systems,” in Nonlinear Systems, Mathematical Theory and Computational Methods Vol. 1 (Springer, 2018).
35.
R.
Prohens
and
A. E.
Teruel
, “
Canard trajectories in 3D piecewise linear systems
,”
Discrete Contin. Dyn. Syst. - Ser. A
33
(
10
),
4595
4611
(
2013
).
36.
R.
Prohens
,
A. E.
Teruel
, and
C.
Vich
, “
Slow–fast n-dimensional piecewise linear differential systems
,”
J. Differ. Equ.
260
(
2
),
1865
1892
(
2016
).
37.
H. G.
Rotstein
,
S.
Coombes
, and
A. M.
Gheorghe
, “
Canard-like explosion of limit cycles in two-dimensional piece-wise linear models of Fitzhugh-Nagumo type
,”
SIAM J. Appl. Dyn. Syst.
11
,
135
180
(
2012
).
38.
E.
Freire
,
E.
Ponce
, and
F.
Torres
, “
Hopf-like bifurcations in planar piecewise linear systems
,”
Publ. Mat.
41
(
1
),
135
148
(
1997
).
39.
D. J. W.
Simpson
, “
A compendium of Hopf-like bifurcations in piecewise-smooth dynamical systems
,”
Phys. Lett. A
382
(
35
),
2439
2444
(
2018
).
40.
E. M.
Izhikevich
, “
Subcritical elliptic bursting of Bautin type
,”
SIAM J. Appl. Math.
60
(
2
),
503
535
(
2000
).
41.
J.
Su
,
J.
Rubin
, and
D.
Terman
, “
Effects of noise on elliptic bursters
,”
Nonlinearity
17
(
1
),
133
157
(
2004
).
42.
D. J. W.
Simpson
,
Bifurcations in Piecewise-Smooth Continuous Systems
(
World Scientific
,
2010
), Vol. 70.
43.
M.
Krupa
and
M.
Wechselberger
, “
Local analysis near a folded saddle-node singularity
,”
J. Differ. Equ.
248
(
12
),
2841
2888
(
2010
).
44.
S. M.
Baer
and
E. M.
Gaekel
, “
Slow acceleration and deacceleration through a Hopf bifurcation: Power ramps, target nucleation, and elliptic bursting
,”
Phys. Rev. E
78
(
3
),
036205
(
2008
).
45.
S.
Venugopal
,
S.
Seki
,
D. H.
Terman
,
A.
Pantazis
,
R.
Olcese
,
M.
Wiedau-Pazos
, and
S. H.
Chandler
, “
Resurgent Na+ current offers noise modulation in bursting neurons
,”
PLoS Comput. Biol.
15
(
6
),
e1007154
(
2019
).
46.
J.
Wojcik
and
A.
Shilnikov
, “
Voltage interval mappings for activity transitions in neuron models for elliptic bursters
,”
Phys. D
240
(
14–15
),
1164
1180
(
2011
).
47.
B.
Deng
, “
Conceptual circuit models of neurons
,”
J. Integr. Neurosci.
8
(
03
),
255
297
(
2009
).
48.
M.
Desroches
,
S.
Fernández-García
, and
M.
Krupa
, “
Canards in a minimal piecewise-linear square-wave burster
,”
Chaos
26
(
7
),
073111
(
2016
).
49.
S.
Fernández-García
,
M.
Desroches
,
M.
Krupa
, and
F.
Clément
, “
A multiple time scale coupling of piecewise linear oscillators. Application to a neuroendocrine system
,”
SIAM J. Appl. Dyn. Syst.
14
(
2
),
643
673
(
2015
).
50.
E.
Baspinar
,
D.
Avitabile
, and
M.
Desroches
, “
Canonical models for torus canards in elliptic bursters
,”
Chaos
31
(
6
),
063129
(
2021
).
51.
G. N.
Benes
,
A. M.
Barry
,
T. J.
Kaper
,
M. A.
Kramer
, and
J.
Burke
, “
An elementary model of torus canards
,”
Chaos
21
(
2
),
023131
(
2011
).
52.
H.
Ju
,
A. B.
Neiman
, and
A.
Shilnikov
, “
Bottom-up approach to torus bifurcation in neuron models
,”
Chaos
28
(
10
),
106317
(
2018
).
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