Chaos via duck solution breakdown in a piecewise linear van der Pol oscillator driven by an extremely small periodic perturbation

2004 ◽  
Author(s):  
M SEKIKAWA
Keyword(s):  
Piecewise Linear ◽  
Periodic Perturbation ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Small Periodic ◽  
Small Periodic Perturbation

scholarly journals Remarkable similarities of two pairs of stable and saddle canards in a van der Pol oscillator under extremely weak periodic perturbation

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
Yuta Nagata ◽  
Naohiko Inaba ◽  
Munehisa Sekikawa ◽  
Tetsuro Endo ◽  
Ken’ichi Fujimoto ◽  
...  
Keyword(s):  
Periodic Perturbation ◽  
Van Der Pol Oscillator ◽  
Van Der Pol

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.

Sudden change from chaos to oscillation death in the Bonhoeffer–van der Pol oscillator under weak periodic perturbation

2011 ◽  
Vol 84 (5) ◽  
Author(s):  
Munehisa Sekikawa ◽  
Kuniyasu Shimizu ◽  
Naohiko Inaba ◽  
Hiroki Kita ◽  
Tetsuro Endo ◽  
...  
Keyword(s):  
Sudden Change ◽  
Periodic Perturbation ◽  
Van Der Pol Oscillator ◽  
Van Der Pol
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