scholarly journals Complete mixed-mode oscillation synchronization in weakly coupled nonautonomous Bonhoeffer–van der Pol oscillators

2018 ◽  
Vol 2018 (6) ◽  
Author(s):  
Naohiko Inaba ◽  
Hidetaka Ito ◽  
Kuniyasu Shimizu ◽  
Hiroomi Hikawa
Keyword(s):  
Mixed Mode ◽  
Van Der Pol ◽  
Weakly Coupled ◽  
Mode Oscillation ◽  
Van Der Pol Oscillators ◽  
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.

Coupled Oscillatory Systems with 𝔻4 Symmetry and Application to van der Pol Oscillators

2016 ◽  
Vol 26 (08) ◽  
pp. 1650141 ◽  
Author(s):  
Adrian C. Murza ◽  
Pei Yu
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Coupled Oscillators ◽  
Invariant Manifolds ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
Oscillatory Systems ◽  
Weakly Coupled ◽  
Form Theory ◽  
Van Der Pol Oscillators

In this paper, we study the dynamics of autonomous ODE systems with [Formula: see text] symmetry. First, we consider eight weakly-coupled oscillators and establish the condition for the existence of stable heteroclinic cycles in most generic [Formula: see text]-equivariant systems. Then, we analyze the action of [Formula: see text] on [Formula: see text] and study the pattern of periodic solutions arising from Hopf bifurcation. We identify the type of periodic solutions associated with the pairs [Formula: see text] of spatiotemporal or spatial symmetries, and prove their existence by using the [Formula: see text] Theorem due to Hopf bifurcation and the [Formula: see text] symmetry. In particular, we give a rigorous proof for the existence of a fourth branch of periodic solutions in [Formula: see text]-equivariant systems. Further, we apply our theory to study a concrete case: two coupled van der Pol oscillators with [Formula: see text] symmetry. We use normal form theory to analyze the periodic solutions arising from Hopf bifurcation. Among the families of the periodic solutions, we pay particular attention to the phase-locked oscillations, each of them being embedded in one of the invariant manifolds, and identify the in-phase, completely synchronized motions. We derive their explicit expressions and analyze their stability in terms of the parameters.

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