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

  • * Corresponding author: V. Carmona

    * Corresponding author: V. Carmona 

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)

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  • 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.

    Mathematics Subject Classification: Primary: 34C05, 34C23, 34C25; Secondary: 37G15, 37G25.

    Citation:

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  • 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

    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

    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

    Figure 4.  Representation of a three-zonal periodic orbit of system (7)

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