scholarly journals Mixed-mode oscillation-incrementing bifurcations and a devil’s staircase from a nonautonomous, constrained Bonhoeffer–van der Pol oscillator

2018 ◽  
Vol 2018 (10) ◽  
Author(s):  
Hiroaki Takahashi ◽  
Takuji Kousaka ◽  
Hiroyuki Asahara ◽  
Nataliya Stankevich ◽  
Naohiko Inaba
Keyword(s):  
Mixed Mode ◽  
Van Der Pol Oscillator ◽  
Devil's Staircase ◽  
Van Der Pol ◽  
Devil’S Staircase ◽  
Mode Oscillation ◽  
Mixed Mode Oscillation

Bifurcation Structures of Nested Mixed-Mode Oscillations

2021 ◽  
Vol 31 (08) ◽  
pp. 2150121
Author(s):  
Munehisa Sekikawa ◽  
Naohiko Inaba
Keyword(s):  
Mixed Mode ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Mixed Mode Oscillations ◽  
Return Maps ◽  
Mode Oscillation ◽  
Mixed Mode Oscillation ◽  
Bifurcation Structures ◽  
First Return Maps ◽  
Strong Order

In recently published work [Inaba & Kousaka, 2020a; Inaba & Tsubone, 2020b], we discovered significant mixed-mode oscillation (MMO) bifurcation structures in which MMOs are nested. Simple mixed-mode oscillation-incrementing bifurcations (MMOIBs) are known to generate [Formula: see text] oscillations for successive [Formula: see text] between regions of [Formula: see text]- and [Formula: see text]-oscillations, where [Formula: see text] and [Formula: see text] are adjacent simple MMOs, e.g. [Formula: see text] and [Formula: see text], where [Formula: see text] is an integer. MMOIBs are universal phenomena of evidently strong order and have been studied extensively in chemistry, physics, and engineering. Nested MMOIBs are phenomena that are more complex, but have an even stronger order, generating chaotic MMO windows that include sequences [Formula: see text] for successive [Formula: see text], where [Formula: see text] and [Formula: see text] are adjacent MMOIB-generated MMOs, i.e. [Formula: see text] and [Formula: see text] for integer [Formula: see text]. Herein, we investigate the bifurcation structures of nested MMOIB-generated MMOs exhibited by a classical forced Bonhoeffer–van der Pol oscillator. We use numerical methods to prepare two- and one-parameter bifurcation diagrams of the system with [Formula: see text], and 3 for successive [Formula: see text] for the case [Formula: see text]. Our analysis suggests that nested MMOs could be widely observed and are clearly ordered phenomena. We then define the first return maps for nested MMOs, which elucidate the appearance of successively nested MMOIBs.

Analysis of mixed-mode oscillation-incrementing bifurcations generated in a nonautonomous constrained Bonhoeffer–van der Pol oscillator

2017 ◽  
Vol 353-354 ◽  
pp. 48-57 ◽  
Author(s):  
Takuji Kousaka ◽  
Yutsuki Ogura ◽  
Kuniyasu Shimizu ◽  
Hiroyuki Asahara ◽  
Naohiko Inaba
Keyword(s):  
Mixed Mode ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Mode Oscillation ◽  
Mixed Mode Oscillation

Experimental and Numerical Observation of Successive Mixed-Mode Oscillation-Incrementing Bifurcations in an Extended Bonhoeffer–van der Pol Oscillator

2018 ◽  
Vol 28 (14) ◽  
pp. 1830047 ◽  
Author(s):  
Kuniyasu Shimizu ◽  
Naohiko Inaba
Keyword(s):  
Large Class ◽  
Constant Temperature ◽  
Mixed Mode ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Mode Oscillation ◽  
Mixed Mode Oscillation

Mixed-mode oscillation-incrementing bifurcations (MMOIBs) are a universal phenomenon appearing in a large class of mixed-mode oscillation (MMO)-generating dynamics that appear to be governed by asymmetric Farey arithmetic. This report presents experimental and numerical observations of the simplest MMOIBs generated by an extended Bonhoeffer–van der Pol circuit. The results indicate that maintaining a constant temperature during the circuit experiments is required to yield stable MMOs and MMOIBs. We also clarify how the firing number variations related to MMOIB phenomena follow asymmetric Farey arithmetic.

STABLE OSCILLATIONS AND DEVIL'S STAIRCASE IN THE VAN DER POL OSCILLATOR

2000 ◽  
Vol 10 (01) ◽  
pp. 155-164 ◽  
Author(s):  
T. GILBERT ◽  
R. W. GAMMON
Keyword(s):  
Rotation Number ◽  
Variable Parameter ◽  
Relaxation Oscillator ◽  
Van Der Pol Oscillator ◽  
Driving Frequency ◽  
Devil's Staircase ◽  
Van Der Pol ◽  
Devil’S Staircase ◽  
Staircase Structure ◽  
Parameter Values

A forced van der Pol relaxation oscillator is studied experimentally in the regime of stable oscillations. The variable parameter is chosen to be the driving frequency. For a range of parameter values, we show that the rotation number varies continuously from 0 to 1. This work provides experimental evidence that period-adding bifurcations to chaos previously reported by Kennedy and Chua are intimately connected to the existence of a regime of stable oscillations where the rotation number shows a Devil's-staircase structure.

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