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Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation
Saddle-node of limit cycles in planar piecewise linear systems and applications
1. | Dpto. Matemática Aplicada Ⅱ & IMUS, University of Seville, Escuela Superior de Ingenieros, Avenida de los Descubrimientos s/n, 41092 Sevilla, Spain |
2. | Departamento EDAN & IMUS, University of Seville, Facultad de Matemáticas, C/ Tarfia, s/n., 41012 Sevilla, Spain |
3. | Departament de Matemàtiques i Informàtica & IAC3, Universitat de les Illes Balears, Carretera de Valldemossa km 7.5, 07122 Palma de Mallorca, Spain |
In this article, we prove the existence of a saddle-node bifurcation of limit cycles in continuous piecewise linear systems with three zones. The bifurcation arises from the perturbation of a non-generic situation, where there exists a linear center in the middle zone. We obtain an approximation of the relation between the parameters of the system, such that the saddle-node bifurcation takes place, as well as of the period and amplitude of the non-hyperbolic limit cycle that bifurcates. We consider two applications, first a piecewise linear version of the FitzHugh-Nagumo neuron model of spike generation and second an electronic circuit, the memristor oscillator.
References:
[1] |
I. Alcalá, F. Gordillo and J. Aracil, Saddle-node bifurcation of limit cycles in a feedback system with rate limiter, in 2001 European Control Conference (ECC), (2001), 354–359. |
[2] |
A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillators, Pergamon Press, Oxford, 1966.
![]() |
[3] |
C. A. Buzzi and J. Torregrosa,
Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst., 33 (2013), 3915-3936.
doi: 10.3934/dcds.2013.33.3915. |
[4] |
V. Carmona, E. Freire, E. Ponce and F. Torres,
On simplifying and classifying piecewise-linear systems, IEEE Transactions on Circuits and Systems Ⅰ: Fundamental Theory and Applications, 49 (2002), 609-620.
doi: 10.1109/TCSI.2002.1001950. |
[5] |
V. Carmona, E. Freire, E. Ponce, F. Torres and F. Ros,
Limit cycle bifurcation in 3D continuous piecewise linear systems with two zones. Application to {Chua's} circuit, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3153-3164.
doi: 10.1142/S0218127405014027. |
[6] |
V. Carmona, S. Fernández-García, F. Fernández-Sánchez, E. García-Medina and A. E. Teruel,
Noose bifurcation and crossing tangency in reversible piecewise linear systems, Nonlinearity, 27 (2014), 585-606.
doi: 10.1088/0951-7715/27/3/585. |
[7] |
V. Carmona, S. Fernández-García and E. Freire,
Saddle–node bifurcation of invariant cones in 3D piecewise linear systems, Phys. D: Nonlinear Phenomena, 241 (2012), 623-635.
doi: 10.1016/j.physd.2011.11.020. |
[8] |
V. Carmona, S. Fernández-García, E. Freire and F. Torres,
Melnikov theory for a class of planar hybrid systems, Phys. D: Nonlinear Phenomena, 248 (2013), 44-54.
doi: 10.1016/j.physd.2013.01.002. |
[9] |
V. Carmona, F. Fernández-Sánchez, E. García-Medina and A. E. Teruel, Existence of Homoclinic Connections in Continuous Piecewise Linear Systems, Chaos (Woodbury, N.Y.), 20 (2010), 013124, 8 pp.
doi: 10.1063/1.3339819. |
[10] |
V. Carmona, F. Fernández-Sánchez, E. García-Medina and A. E. Teruel,
Noose structure and bifurcations of periodic orbits in reversible three-dimensional piecewise linear differential systems, Journal of Nonlinear Science, 25 (2015), 1209-1224.
doi: 10.1007/s00332-015-9251-z. |
[11] |
C. Chicone,
Bifurcations of nonlinear oscillations and frequency entrainment near resonance, SIAM J. Math. Anal., 23 (1992), 1577-1608.
doi: 10.1137/0523087. |
[12] |
S. N. Chow, B. Deng and B. Fiedler,
Homoclinic bifurcation at resonant eigenvalues, J. Dyn. Diff. Equat., 2 (1990), 177-244.
doi: 10.1007/BF01057418. |
[13] |
L. O. Chua,
Memristor: the missing circuit element, IEEE Trans. Circuit Theory, 18 (1971), 507-519.
doi: 10.1109/TCT.1971.1083337. |
[14] |
F. Corinto, A. Ascoli and M. Gilli,
Nonlinear dynamics of memristor oscillators, IEEE Trans. Ciruits Syst. Ⅰ: Regul. Pap., 58 (2011), 1323-1336.
doi: 10.1109/TCSI.2010.2097731. |
[15] |
C. A. Del Negro, C. F. Hsiao, S. H. Chandler and A. Garfinkel,
Evidence for a novel bursting mechanism in rodent trigeminal neurons, Biophysical Journal, 75 (1998), 174-182.
|
[16] |
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 (2013), 20120603, 18pp.
doi: 10.1098/rspa.2012.0603. |
[17] |
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications, Springer-Verlag London, London, 2008. |
[18] |
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 (2015), 643-673.
doi: 10.1137/140984464. |
[19] |
S. Fernández-García, M. Desroches, M. Krupa and A. E. Teruel,
Canard solutions in planar piecewise linear systems with three zones, Dynam. Syst., 31 (2016), 173-197.
doi: 10.1080/14689367.2015.1079304. |
[20] |
R. FitzHugh,
Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.
doi: 10.1016/S0006-3495(61)86902-6. |
[21] |
E. Freire, E. Ponce, F. Rodrigo and F. Torres,
Bifurcation sets of continuous piecewise linear systems with two zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 2073-2097.
doi: 10.1142/S0218127498001728. |
[22] |
E. Freire, E. Ponce and J. Ros,
Limit cycle bifurcation from center in symmetric piecewise- linear systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 895-907.
doi: 10.1142/S0218127499000638. |
[23] |
E. Freire, E. Ponce and J. Ros,
A biparametric bifurcation in 3D continuous piecewise linear systems with two zones. Application to Chua's circuit, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 445-457.
doi: 10.1142/S0218127407017367. |
[24] |
E. Freire, E. Ponce and J. Ros,
Following a saddle-node of periodic orbits bifurcation curve in Chua's circuit, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 487-495.
doi: 10.1142/S0218127409023147. |
[25] |
W. Govaerts, Y. Kuznetsov and A. Dhooge,
Numerical continuation of bifurcations of limit cycles in Matlab, SIAM J. Sci. Comput., 27 (2005), 231-252.
doi: 10.1137/030600746. |
[26] |
M. Guevara, Bifurcations Involving Fixed Points and Limit Cycles in Biological Systems, in Nonlinear Dynamics in Physiology and Medicine (eds. A. Beuter, L. Glass, MC. Mackey, MS. Titcombe) Springer-Verlag, New York, 25 (2003), 41–85.
doi: 10.1007/978-0-387-21640-9_3. |
[27] |
M. Guevara and H. Jongsma, Three ways of abolishing automaticity in sinoatrial node: ionic modeling and nonlinear dynamics, Am. J. Physiol., 262 (1992), H1268–H1286. |
[28] |
A. Guillamon, R. Prohens, A. E. Teruel and C. Vich,
Estimation of synaptic conductance in the spiking regime for the mckean neuron model, SIAM J. Appl. Dyn. Syst, 16 (2017), 1397-1424.
doi: 10.1137/16M1088326. |
[29] |
E. M. Izhikevich,
Neural Excitability, Spiking and Bursting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1171-1266.
doi: 10.1142/S0218127400000840. |
[30] |
E. M. Izhikevich,
Synchronization of elliptic bursters, SIAM Rev., 43 (2001), 315-344.
doi: 10.1137/S0036144500382064. |
[31] |
M. Krupa and P. Szmolyan,
Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.
doi: 10.1006/jdeq.2000.3929. |
[32] |
M. Kunze, Non-smooth dynamical systems, Lecture Notes in Mathematics, 1744, Springer, 2000.
doi: 10.1007/BFb0103843. |
[33] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd edition, Springer, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[34] |
J. Llibre, E. Ponce and C. Valls,
Uniqueness and non-uniqueness of limit cycles for piecewise linear differential systems with three zones and no symmetry, J. Nonlinear Sci., 25 (2015), 861-887.
doi: 10.1007/s00332-015-9244-y. |
[35] |
J. Llibre and A. E. Teruel, Introduction to the Qualitative Theory of Differential Systems: Planar, Symmetric and Continuous Piecewise Linear Systems, Springer Basel, Birkhäuser Advanced Texts, Basel, 2014.
doi: 10.1007/978-3-0348-0657-2. |
[36] |
O. Makarenkov,
Bifurcation of Limit Cycles from a Fold-Fold Singularity in Planar Switched Systems, SIAM J. Appl. Dyn. Syst, 16 (2017), 1340-1371.
doi: 10.1137/16M1070943. |
[37] |
H. P. McKean,
Nagumo's equation, Adv. Math, 4 (1970), 209-223.
doi: 10.1016/0001-8708(70)90023-X. |
[38] |
J. Nagumo, S. Arimoto and S. Yoshizawa,
An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070.
doi: 10.1109/JRPROC.1962.288235. |
[39] |
L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, 2001. |
[40] |
E. Ponce, J. Ros and E. Vela, Limit cycle and boundary equilibrium bifurcations in continuous planar piecewise linear systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1530008, 18pp.
doi: 10.1142/S0218127415300086. |
[41] |
M. Ringkvist and Y. Zhou,
On Existence and Nonexistence of Limit Cycles for FitzHugh-Nagumo Class Models, LNCIS, 321 (2005), 337-351.
doi: 10.1007/10984413_21. |
[42] |
C. Rocșoreanu, A. Georgescu and N. Giurgiteanu, The FitzHugh-Nagumo Model. Bifurcation and Dynamics, , Springer Science+Business Media Dordrecht, 2000. |
[43] |
J. Stensby,
Saddle node bifurcation at a nonhyperbolic limit cycle in a phase locked loop, J. of Franklin Inst., 330 (1993), 775-786.
doi: 10.1016/0016-0032(93)90076-7. |
[44] |
A. Tonnelier,
The McKean caricature of the FitzHugh-Nagumo model Ⅰ. The space-clamped system, SIAM J. Appl. Math., 63 (2002), 459-484.
doi: 10.1137/S0036139901393500. |
[45] |
A. T. Winfree, The Geometry of Biological Time, Interdisciplinary Applied Mathematics, 12, Springer-Verlag, New York, 2001. |
show all references
References:
[1] |
I. Alcalá, F. Gordillo and J. Aracil, Saddle-node bifurcation of limit cycles in a feedback system with rate limiter, in 2001 European Control Conference (ECC), (2001), 354–359. |
[2] |
A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillators, Pergamon Press, Oxford, 1966.
![]() |
[3] |
C. A. Buzzi and J. Torregrosa,
Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst., 33 (2013), 3915-3936.
doi: 10.3934/dcds.2013.33.3915. |
[4] |
V. Carmona, E. Freire, E. Ponce and F. Torres,
On simplifying and classifying piecewise-linear systems, IEEE Transactions on Circuits and Systems Ⅰ: Fundamental Theory and Applications, 49 (2002), 609-620.
doi: 10.1109/TCSI.2002.1001950. |
[5] |
V. Carmona, E. Freire, E. Ponce, F. Torres and F. Ros,
Limit cycle bifurcation in 3D continuous piecewise linear systems with two zones. Application to {Chua's} circuit, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3153-3164.
doi: 10.1142/S0218127405014027. |
[6] |
V. Carmona, S. Fernández-García, F. Fernández-Sánchez, E. García-Medina and A. E. Teruel,
Noose bifurcation and crossing tangency in reversible piecewise linear systems, Nonlinearity, 27 (2014), 585-606.
doi: 10.1088/0951-7715/27/3/585. |
[7] |
V. Carmona, S. Fernández-García and E. Freire,
Saddle–node bifurcation of invariant cones in 3D piecewise linear systems, Phys. D: Nonlinear Phenomena, 241 (2012), 623-635.
doi: 10.1016/j.physd.2011.11.020. |
[8] |
V. Carmona, S. Fernández-García, E. Freire and F. Torres,
Melnikov theory for a class of planar hybrid systems, Phys. D: Nonlinear Phenomena, 248 (2013), 44-54.
doi: 10.1016/j.physd.2013.01.002. |
[9] |
V. Carmona, F. Fernández-Sánchez, E. García-Medina and A. E. Teruel, Existence of Homoclinic Connections in Continuous Piecewise Linear Systems, Chaos (Woodbury, N.Y.), 20 (2010), 013124, 8 pp.
doi: 10.1063/1.3339819. |
[10] |
V. Carmona, F. Fernández-Sánchez, E. García-Medina and A. E. Teruel,
Noose structure and bifurcations of periodic orbits in reversible three-dimensional piecewise linear differential systems, Journal of Nonlinear Science, 25 (2015), 1209-1224.
doi: 10.1007/s00332-015-9251-z. |
[11] |
C. Chicone,
Bifurcations of nonlinear oscillations and frequency entrainment near resonance, SIAM J. Math. Anal., 23 (1992), 1577-1608.
doi: 10.1137/0523087. |
[12] |
S. N. Chow, B. Deng and B. Fiedler,
Homoclinic bifurcation at resonant eigenvalues, J. Dyn. Diff. Equat., 2 (1990), 177-244.
doi: 10.1007/BF01057418. |
[13] |
L. O. Chua,
Memristor: the missing circuit element, IEEE Trans. Circuit Theory, 18 (1971), 507-519.
doi: 10.1109/TCT.1971.1083337. |
[14] |
F. Corinto, A. Ascoli and M. Gilli,
Nonlinear dynamics of memristor oscillators, IEEE Trans. Ciruits Syst. Ⅰ: Regul. Pap., 58 (2011), 1323-1336.
doi: 10.1109/TCSI.2010.2097731. |
[15] |
C. A. Del Negro, C. F. Hsiao, S. H. Chandler and A. Garfinkel,
Evidence for a novel bursting mechanism in rodent trigeminal neurons, Biophysical Journal, 75 (1998), 174-182.
|
[16] |
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 (2013), 20120603, 18pp.
doi: 10.1098/rspa.2012.0603. |
[17] |
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications, Springer-Verlag London, London, 2008. |
[18] |
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 (2015), 643-673.
doi: 10.1137/140984464. |
[19] |
S. Fernández-García, M. Desroches, M. Krupa and A. E. Teruel,
Canard solutions in planar piecewise linear systems with three zones, Dynam. Syst., 31 (2016), 173-197.
doi: 10.1080/14689367.2015.1079304. |
[20] |
R. FitzHugh,
Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.
doi: 10.1016/S0006-3495(61)86902-6. |
[21] |
E. Freire, E. Ponce, F. Rodrigo and F. Torres,
Bifurcation sets of continuous piecewise linear systems with two zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 2073-2097.
doi: 10.1142/S0218127498001728. |
[22] |
E. Freire, E. Ponce and J. Ros,
Limit cycle bifurcation from center in symmetric piecewise- linear systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 895-907.
doi: 10.1142/S0218127499000638. |
[23] |
E. Freire, E. Ponce and J. Ros,
A biparametric bifurcation in 3D continuous piecewise linear systems with two zones. Application to Chua's circuit, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 445-457.
doi: 10.1142/S0218127407017367. |
[24] |
E. Freire, E. Ponce and J. Ros,
Following a saddle-node of periodic orbits bifurcation curve in Chua's circuit, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 487-495.
doi: 10.1142/S0218127409023147. |
[25] |
W. Govaerts, Y. Kuznetsov and A. Dhooge,
Numerical continuation of bifurcations of limit cycles in Matlab, SIAM J. Sci. Comput., 27 (2005), 231-252.
doi: 10.1137/030600746. |
[26] |
M. Guevara, Bifurcations Involving Fixed Points and Limit Cycles in Biological Systems, in Nonlinear Dynamics in Physiology and Medicine (eds. A. Beuter, L. Glass, MC. Mackey, MS. Titcombe) Springer-Verlag, New York, 25 (2003), 41–85.
doi: 10.1007/978-0-387-21640-9_3. |
[27] |
M. Guevara and H. Jongsma, Three ways of abolishing automaticity in sinoatrial node: ionic modeling and nonlinear dynamics, Am. J. Physiol., 262 (1992), H1268–H1286. |
[28] |
A. Guillamon, R. Prohens, A. E. Teruel and C. Vich,
Estimation of synaptic conductance in the spiking regime for the mckean neuron model, SIAM J. Appl. Dyn. Syst, 16 (2017), 1397-1424.
doi: 10.1137/16M1088326. |
[29] |
E. M. Izhikevich,
Neural Excitability, Spiking and Bursting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1171-1266.
doi: 10.1142/S0218127400000840. |
[30] |
E. M. Izhikevich,
Synchronization of elliptic bursters, SIAM Rev., 43 (2001), 315-344.
doi: 10.1137/S0036144500382064. |
[31] |
M. Krupa and P. Szmolyan,
Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.
doi: 10.1006/jdeq.2000.3929. |
[32] |
M. Kunze, Non-smooth dynamical systems, Lecture Notes in Mathematics, 1744, Springer, 2000.
doi: 10.1007/BFb0103843. |
[33] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd edition, Springer, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[34] |
J. Llibre, E. Ponce and C. Valls,
Uniqueness and non-uniqueness of limit cycles for piecewise linear differential systems with three zones and no symmetry, J. Nonlinear Sci., 25 (2015), 861-887.
doi: 10.1007/s00332-015-9244-y. |
[35] |
J. Llibre and A. E. Teruel, Introduction to the Qualitative Theory of Differential Systems: Planar, Symmetric and Continuous Piecewise Linear Systems, Springer Basel, Birkhäuser Advanced Texts, Basel, 2014.
doi: 10.1007/978-3-0348-0657-2. |
[36] |
O. Makarenkov,
Bifurcation of Limit Cycles from a Fold-Fold Singularity in Planar Switched Systems, SIAM J. Appl. Dyn. Syst, 16 (2017), 1340-1371.
doi: 10.1137/16M1070943. |
[37] |
H. P. McKean,
Nagumo's equation, Adv. Math, 4 (1970), 209-223.
doi: 10.1016/0001-8708(70)90023-X. |
[38] |
J. Nagumo, S. Arimoto and S. Yoshizawa,
An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070.
doi: 10.1109/JRPROC.1962.288235. |
[39] |
L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, 2001. |
[40] |
E. Ponce, J. Ros and E. Vela, Limit cycle and boundary equilibrium bifurcations in continuous planar piecewise linear systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1530008, 18pp.
doi: 10.1142/S0218127415300086. |
[41] |
M. Ringkvist and Y. Zhou,
On Existence and Nonexistence of Limit Cycles for FitzHugh-Nagumo Class Models, LNCIS, 321 (2005), 337-351.
doi: 10.1007/10984413_21. |
[42] |
C. Rocșoreanu, A. Georgescu and N. Giurgiteanu, The FitzHugh-Nagumo Model. Bifurcation and Dynamics, , Springer Science+Business Media Dordrecht, 2000. |
[43] |
J. Stensby,
Saddle node bifurcation at a nonhyperbolic limit cycle in a phase locked loop, J. of Franklin Inst., 330 (1993), 775-786.
doi: 10.1016/0016-0032(93)90076-7. |
[44] |
A. Tonnelier,
The McKean caricature of the FitzHugh-Nagumo model Ⅰ. The space-clamped system, SIAM J. Appl. Math., 63 (2002), 459-484.
doi: 10.1137/S0036139901393500. |
[45] |
A. T. Winfree, The Geometry of Biological Time, Interdisciplinary Applied Mathematics, 12, Springer-Verlag, New York, 2001. |




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