Fast–slow variable analysis of the transition to mixed‐mode oscillations and chaos in the peroxidase reaction

1988 ◽  
Vol 89 (10) ◽  
pp. 6506-6514 ◽  
Author(s):  
Raima Larter ◽  
Curtis G. Steinmetz ◽  
Baltazar D. Aguda
Keyword(s):  
Mixed Mode ◽  
Slow Variable ◽  
Mixed Mode Oscillations ◽  
Peroxidase Reaction ◽  
Variable Analysis

scholarly journals Complexity of a Peroxidase-Oxidase Reaction Model

2021 ◽  
Author(s):  
Jason Gallas ◽  
Marcus Hauser ◽  
Lars Folke Olsen
Keyword(s):  
Chemical Reaction ◽  
Mixed Mode ◽  
Period Doubling ◽  
Reaction Model ◽  
Chaotic Behaviour ◽  
Mixed Mode Oscillations ◽  
Oscillating Reaction ◽  
Oxidase Reaction

The peroxidase-oxidase oscillating reaction was the first (bio)chemical reaction to show chaotic behaviour. The reaction is rich in bifurcation scenarios, from period-doubling to peak-adding mixed mode oscillations. Here, we study...

Bursting patterns and mixed-mode oscillations in reduced Purkinje model

2018 ◽  
Vol 32 (05) ◽  
pp. 1850043 ◽  
Author(s):  
Feibiao Zhan ◽  
Shenquan Liu ◽  
Jing Wang ◽  
Bo Lu
Keyword(s):  
Purkinje Cell ◽  
Mixed Mode ◽  
Cell Model ◽  
Dynamical Behavior ◽  
Characteristic Index ◽  
Mixed Mode Oscillations ◽  
Codimension One ◽  
Bifurcation Phenomenon ◽  
Lyapunov Coefficient ◽  
Potential Link

Bursting discharge is a ubiquitous behavior in neurons, and abundant bursting patterns imply many physiological information. There exists a closely potential link between bifurcation phenomenon and the number of spikes per burst as well as mixed-mode oscillations (MMOs). In this paper, we have mainly explored the dynamical behavior of the reduced Purkinje cell and the existence of MMOs. First, we adopted the codimension-one bifurcation to illustrate the generation mechanism of bursting in the reduced Purkinje cell model via slow–fast dynamics analysis and demonstrate the process of spike-adding. Furthermore, we have computed the first Lyapunov coefficient of Hopf bifurcation to determine whether it is subcritical or supercritical and depicted the diagrams of inter-spike intervals (ISIs) to examine the chaos. Moreover, the bifurcation diagram near the cusp point is obtained by making the codimension-two bifurcation analysis for the fast subsystem. Finally, we have a discussion on mixed-mode oscillations and it is further investigated using the characteristic index that is Devil’s staircase.

Intermittency, Secondary Bifurcation and Mixed Mode Oscillations in a Swirl-Stabilized Annular Combustor: Experiments and Modeling

2021 ◽  
Author(s):  
Samarjeet Singh ◽  
Amitesh Roy ◽  
K V Reeja ◽  
Asalatha A. S. Nair ◽  
Swetaprovo Chaudhuri ◽  
...  
Keyword(s):  
Mixed Mode ◽  
Mixed Mode Oscillations ◽  
Annular Combustor ◽  
Secondary Bifurcation

One variable map prediction of Belousov–Zhabotinskii mixed mode oscillations

1984 ◽  
Vol 80 (11) ◽  
pp. 5610-5615 ◽  
Author(s):  
John Rinzel ◽  
Ira B. Schwartz
Keyword(s):  
Mixed Mode ◽  
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.

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