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September  2019, 39(9): 5275-5299. doi: 10.3934/dcds.2019215

Saddle-node of limit cycles in planar piecewise linear systems and applications

Victoriano Carmona 1,, , Soledad Fernández-García 2, and  Antonio E. Teruel 3,

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

* Corresponding author: V. Carmona

Received  September 2018 Revised  March 2019 Published  May 2019

Fund Project: The first author is supported by Ministerio de Economía y Competitividad through the project MTM2015-65608-P and by Junta de Andaucía by project P12-FQM-1658. The second author is supported by University of Seville VPPI-US and partially supported by Ministerio de Economía y Competitividad through the project MTM2015-65608-P and by Junta de Andaucía by project P12-FQM-1658. The third author is supported by Ministerio de Economía y Competitividad through the projects MTM2014-54275-P and MTM2017-83568-P (AEI/ERDF, EU)

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.

Keywords: Piecewise linear systems, bifurcations, saddle-node of limit cycles, neuroscience.
Mathematics Subject Classification: Primary: 34C05, 34C23, 34C25; Secondary: 37G15, 37G25.
Citation: Victoriano Carmona, Soledad Fernández-García, Antonio E. Teruel. Saddle-node of limit cycles in planar piecewise linear systems and applications. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5275-5299. doi: 10.3934/dcds.2019215
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.Google Scholar

[2] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillators, Pergamon Press, Oxford, 1966. Google Scholar
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

C. Chicone, Bifurcations of nonlinear oscillations and frequency entrainment near resonance, SIAM J. Math. Anal., 23 (1992), 1577-1608. doi: 10.1137/0523087. Google Scholar

[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. Google Scholar

[13]

L. O. Chua, Memristor: the missing circuit element, IEEE Trans. Circuit Theory, 18 (1971), 507-519. doi: 10.1109/TCT.1971.1083337. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[29]

E. M. Izhikevich, Neural Excitability, Spiking and Bursting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840. Google Scholar

[30]

E. M. Izhikevich, Synchronization of elliptic bursters, SIAM Rev., 43 (2001), 315-344. doi: 10.1137/S0036144500382064. Google Scholar

[31]

M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368. doi: 10.1006/jdeq.2000.3929. Google Scholar

[32]

M. Kunze, Non-smooth dynamical systems, Lecture Notes in Mathematics, 1744, Springer, 2000. doi: 10.1007/BFb0103843. Google Scholar

[33]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd edition, Springer, 2004. doi: 10.1007/978-1-4757-3978-7. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

H. P. McKean, Nagumo's equation, Adv. Math, 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X. Google Scholar

[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. Google Scholar

[39]

L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, 2001.Google Scholar

[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. Google Scholar

[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. Google Scholar

[42]

C. Rocșoreanu, A. Georgescu and N. Giurgiteanu, The FitzHugh-Nagumo Model. Bifurcation and Dynamics, , Springer Science+Business Media Dordrecht, 2000.Google Scholar

[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. Google Scholar

[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. Google Scholar

[45]

A. T. Winfree, The Geometry of Biological Time, Interdisciplinary Applied Mathematics, 12, Springer-Verlag, New York, 2001.Google Scholar

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.Google Scholar

[2] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillators, Pergamon Press, Oxford, 1966. Google Scholar
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

C. Chicone, Bifurcations of nonlinear oscillations and frequency entrainment near resonance, SIAM J. Math. Anal., 23 (1992), 1577-1608. doi: 10.1137/0523087. Google Scholar

[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. Google Scholar

[13]

L. O. Chua, Memristor: the missing circuit element, IEEE Trans. Circuit Theory, 18 (1971), 507-519. doi: 10.1109/TCT.1971.1083337. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[29]

E. M. Izhikevich, Neural Excitability, Spiking and Bursting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840. Google Scholar

[30]

E. M. Izhikevich, Synchronization of elliptic bursters, SIAM Rev., 43 (2001), 315-344. doi: 10.1137/S0036144500382064. Google Scholar

[31]

M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368. doi: 10.1006/jdeq.2000.3929. Google Scholar

[32]

M. Kunze, Non-smooth dynamical systems, Lecture Notes in Mathematics, 1744, Springer, 2000. doi: 10.1007/BFb0103843. Google Scholar

[33]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd edition, Springer, 2004. doi: 10.1007/978-1-4757-3978-7. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

H. P. McKean, Nagumo's equation, Adv. Math, 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X. Google Scholar

[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. Google Scholar

[39]

L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, 2001.Google Scholar

[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. Google Scholar

[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. Google Scholar

[42]

C. Rocșoreanu, A. Georgescu and N. Giurgiteanu, The FitzHugh-Nagumo Model. Bifurcation and Dynamics, , Springer Science+Business Media Dordrecht, 2000.Google Scholar

[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. Google Scholar

[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. Google Scholar

[45]

A. T. Winfree, The Geometry of Biological Time, Interdisciplinary Applied Mathematics, 12, Springer-Verlag, New York, 2001.Google Scholar

Figure 1.  Schematic representation of the bifurcation diagram in $ m, a_C $ parameter plane in case $ t_L<0,t_R>0, $ and $ t_L t_R(t_L^2-t_R^2)<0 $ for $ m $ and $ a_C $ sufficiently small. Solid line represents the saddle-node curve $ a_C = a_C^*(m) $. In the region $ a_C>a_C^*(m), $ two limit cycles with opposite stability and close to $ \Gamma_0 $ exist and in the region $ a_C<a_C^*(m) $ no limit cycles close to $ \Gamma_0 $ exist
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Figure 2.  Saddle-node bifurcation in the McKean model (11) with $ C = 0.25, \beta = 0.5, w_0 = 0, a = 1, \delta = 0.25, t_c = 0.1 $ and $ t_r = 0.8 $. According to Corollary 1, the bifurcation takes place at $ I^* = 1.263\ldots $ In panel (a), the parameter $ I = 1.2 $ is smaller than the bifurcation value, then no limit cycles exit near the equilibrium point which is a local attractor. In panel (b) the parameter is just on the bifurcation, $ I = I^* $, and a non-hyperbolic limit cycle appears. This limit cycle is stable from the outside and unstable from the inside. In panel (c) the parameter $ I = 1.265 $ is greater than the bifurcation value and then two concentric limit cycles perturb from the non-hyperbolic one. The outer limit cycle is stable whereas the inner one is unstable. The limit cycles perturbing from the saddle-node limit cycle move away one each other as the parameter increases far from the perturbation value $ I^* $. In panel (d), for $ I = 1.3 $, the inner limit cycle becomes a two zonal limit cycle whereas the outer one becomes a four zones limit cycle
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Figure 3.  Fine tuning of the external impulse, $ I $, in order to facilitate annihilation and single-pulse triggering in the McKean model (13) with $ C = 0.25, \beta = 0.5, w_0 = 0, a = 1, \delta = 0.25, t_c = 0.1 $ and $ t_r = 0.8 $, see Appendix B. (a) An oscillatory behavior is ceased by injecting a pulse, see panel (b). The activity is restarted again by injecting a new pulse
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Figure 4.  Representation of a three-zonal periodic orbit of system (7)
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