Complex mixed-mode oscillations in a Bonhoeffer–van der Pol oscillator under weak periodic perturbation

2012 ◽  
Vol 241 (18) ◽  
pp. 1518-1526 ◽  
Author(s):  
Kuniyasu Shimizu ◽  
Yuto Saito ◽  
Munehisa Sekikawa ◽  
Naohiko Inaba
Keyword(s):  
Mixed Mode ◽  
Periodic Perturbation ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Mixed Mode Oscillations

Canard-induced mixed mode oscillations as a mechanism for the Bonhoeffer-van der Pol circuit under parametric perturbation

Circuit World ◽  
2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Yue Yu ◽  
Cong Zhang ◽  
Zhenyu Chen ◽  
Zhengdi Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbit ◽  
Large Amplitude ◽  
Mixed Mode ◽  
Periodic Perturbation ◽  
Van Der Pol ◽  
Content Type ◽  
Stable Focus ◽  
Mixed Mode Oscillations ◽  
Transition Mechanism

Purpose This paper aims to investigate the singular Hopf bifurcation and mixed mode oscillations (MMOs) in the perturbed Bonhoeffer-van der Pol (BVP) circuit. There is a singular periodic orbit constructed by the switching between the stable focus and large amplitude relaxation cycles. Using a generalized fast/slow analysis, the authors show the generation mechanism of two distinct kinds of MMOs. Design/methodology/approach The parametric modulation can be used to generate complicated dynamics. The BVP circuit is constructed as an example for second-order differential equation with periodic perturbation. Then the authors draw the bifurcation parameter diagram in terms of a containing two attractive regions, i.e. the stable relaxation cycle and the stable focus. The transition mechanism and characteristic features are investigated intensively by one-fast/two-slow analysis combined with bifurcation theory. Findings Periodic perturbation can suppress nonlinear circuit dynamic to a singular periodic orbit. The combination of these small oscillations with the large amplitude oscillations that occur due to canard cycles yields such MMOs. The results connect the theory of the singular Hopf bifurcation enabling easier calculations of where the oscillations occur. Originality/value By treating the perturbation as the second slow variable, the authors obtain that the MMOs are due to the canards in a supercritical case or in a subcritical case. This study can reveal the transition mechanism for multi-time scale characteristics in perturbed circuit. The information gained from such results can be extended to periodically perturbed circuits.

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.

Mixed-mode oscillations from a constrained extended Bonhoeffer–van der Pol oscillator with a diode

2021 ◽  
Vol 31 (7) ◽  
pp. 073133
Author(s):  
Naohiko Inaba ◽  
Takuji Kousaka ◽  
Tadashi Tsubone ◽  
Hideaki Okazaki ◽  
Hidetaka Ito
Keyword(s):  
Mixed Mode ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Mixed Mode Oscillations

scholarly journals Controlling Canard Cycles

2021 ◽  
Author(s):  
Hildeberto Jardón-Kojakhmetov ◽  
Christian Kuehn
Keyword(s):  
Mixed Mode ◽  
Blow Up ◽  
Van Der Pol Oscillator ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Van Der Pol ◽  
Critical Manifold ◽  
Mixed Mode Oscillations ◽  
Special Solutions ◽  
And Control

AbstractCanard cycles are periodic orbits that appear as special solutions of fast-slow systems (or singularly perturbed ordinary differential equations). It is well known that canard cycles are difficult to detect, hard to reproduce numerically, and that they are sensible to exponentially small changes in parameters. In this paper, we combine techniques from geometric singular perturbation theory, the blow-up method, and control theory, to design controllers that stabilize canard cycles of planar fast-slow systems with a folded critical manifold. As an application, we propose a controller that produces stable mixed-mode oscillations in the van der Pol oscillator.

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