Limit cycles and relaxation oscillations

1983 ◽  
pp. 66-71
Keyword(s):  
Limit Cycles ◽  
Relaxation Oscillations

On the Existence of Limit Cycles and Relaxation Oscillations in a 3D van der Pol-like Memristor Oscillator

2017 ◽  
Vol 27 (07) ◽  
pp. 1750102
Author(s):  
Marcelo Messias ◽  
Anderson L. Maciel
Keyword(s):  
Phase Space ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Relaxation Oscillator ◽  
Van Der Pol Oscillator ◽  
Relaxation Oscillations ◽  
Van Der Pol ◽  
Piecewise Linear System ◽  
Parameter Values

We study a van der Pol-like memristor oscillator, obtained by substituting a Chua’s diode with an active controlled memristor in a van der Pol oscillator with Chua’s diode. The mathematical model for the studied circuit is given by a three-dimensional piecewise linear system of ordinary differential equations, depending on five parameters. We show that this system has a line of equilibria given by the [Formula: see text]-axis and the phase space [Formula: see text] is foliated by invariant planes transverse to this line, which implies that the dynamics is essentially two-dimensional. We also show that in each of these invariant planes may occur limit cycles and relaxation oscillations (that is, nonsinusoidal repetitive (periodic) solutions), depending on the parameter values. Hence, the oscillator studied here, constructed with a memristor, is also a relaxation oscillator, as the original van der Pol oscillator, although with a main difference: in the case of the memristor oscillator, an infinity of oscillations are produced, one in each invariant plane, depending on the initial condition considered. We also give conditions for the nonexistence of oscillations, depending on the position of the invariant planes in the phase space.

scholarly journals On the Limit Cycles of a Class of Planar Singular Perturbed Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Yuhai Wu ◽  
Jingjing Zhou
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Relaxation Oscillation ◽  
Turning Points ◽  
Two Dimensional ◽  
Relaxation Oscillations ◽  
Perturbed Systems

Relaxation oscillations of two-dimensional planar singular perturbed systems with a layer equation exhibiting canard cycles are studied. The canard cycles under consideration contain two turning points and two jump points. We suppose that there exist three parameters permitting generic breaking at both the turning points and the connecting fast orbit. The conditions of one (resp., two, three) relaxation oscillation near the canard cycles are given by studying a map from the space of phase parameters to the space of breaking parameters.

scholarly journals Lac Operon Boolean Models: Dynamical Robustness and Alternative Improvements

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 600 ◽  
Author(s):  
Marco Montalva-Medel ◽  
Thomas Ledger ◽  
Gonzalo A. Ruz ◽  
Eric Goles
Keyword(s):  
Escherichia Coli ◽  
Limit Cycles ◽  
The Other ◽  
Lac Operon ◽  
Boolean Models ◽  
Bistable Behavior

In Veliz-Cuba and Stigler 2011, Boolean models were proposed for the lac operon in Escherichia coli capable of reproducing the operon being OFF, ON and bistable for three (low, medium and high) and two (low and high) parameters, representing the concentration ranges of lactose and glucose, respectively. Of these 6 possible combinations of parameters, 5 produce results that match with the biological experiments of Ozbudak et al., 2004. In the remaining one, the models predict the operon being OFF while biological experiments show a bistable behavior. In this paper, we first explore the robustness of two such models in the sense of how much its attractors change against any deterministic update schedule. We prove mathematically that, in cases where there is no bistability, all the dynamics in both models lack limit cycles while, when bistability appears, one model presents 30% of its dynamics with limit cycles while the other only 23%. Secondly, we propose two alternative improvements consisting of biologically supported modifications; one in which both models match with Ozbudak et al., 2004 in all 6 combinations of parameters and, the other one, where we increase the number of parameters to 9, matching in all these cases with the biological experiments of Ozbudak et al., 2004.

Recurrent canards producing relaxation oscillations

2021 ◽  
Vol 31 (2) ◽  
pp. 023121
Author(s):  
C. Abdulwahed ◽  
F. Verhulst
Keyword(s):  
Relaxation Oscillations
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