American Institute of Mathematical Sciences

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

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

* 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.

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
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References:
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
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
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
Representation of a three-zonal periodic orbit of system (7)

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