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

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.

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.

scholarly journals Mixed-mode dynamics of certain bistable oscillators: behavioural mapping, approximations for motion and links with van der Pol oscillators

2015 ◽  
Vol 471 (2184) ◽  
pp. 20150638 ◽  
Author(s):  
Ivana Kovacic ◽  
Matthew Cartmell ◽  
Miodrag Zukovic
Keyword(s):  
Mixed Mode ◽  
Autonomous Systems ◽  
Low Frequency ◽  
Harmonic Excitation ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Restoring Force ◽  
Van Der Pol Oscillators ◽  
Flow Oscillations ◽  
Fast Flow

This study is concerned with a new generalized mathematical model for single degree-of-freedom bistable oscillators with harmonic excitation of low-frequency, linear viscous damping and a restoring force that contains a negative linear term and a positive nonlinear term which is a power-form function of the generalized coordinate. Comprehensive numerical mapping of the range of bifurcatory behaviour shows that such non-autonomous systems can experience mixed-mode oscillations, including bursting oscillations (fast flow oscillations around the outer curves of a slow flow), and relaxation oscillations like a classical (autonomous) van der Pol oscillator. After studying the global system dynamics the focus of the investigations is on cubic oscillators of this type. Approximate techniques are presented to quantify their response, i.e. to determine approximations for both the slow and fast flows. In addition, a clear analogy between the behaviour of two archetypical oscillators—the non-autonomous bistable oscillator operating at low frequency and the strongly damped autonomous van der Pol oscillator—is established for the first time.

Sign in / Sign up

Export Citation Format

Share Document